chapter 8 resource masters - farragutcareeracademy.org · ©glencoe/mcgraw-hill iv glencoe geometry...
TRANSCRIPT
Chapter 8Resource Masters
Geometry
Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.
Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3
ANSWERS FOR WORKBOOKS The answers for Chapter 8 of these workbookscan be found in the back of this Chapter Resource Masters booklet.
Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.
Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027
ISBN: 0-07-860185-1 GeometryChapter 8 Resource Masters
1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03
© Glencoe/McGraw-Hill iii Glencoe Geometry
Contents
Vocabulary Builder . . . . . . . . . . . . . . . . vii
Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix
Lesson 8-1Study Guide and Intervention . . . . . . . . 417–418Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 419Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Reading to Learn Mathematics . . . . . . . . . . 421Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 422
Lesson 8-2Study Guide and Intervention . . . . . . . . 423–424Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 425Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 426Reading to Learn Mathematics . . . . . . . . . . 427Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 428
Lesson 8-3Study Guide and Intervention . . . . . . . . 429–430Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 431Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 432Reading to Learn Mathematics . . . . . . . . . . 433Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 434
Lesson 8-4Study Guide and Intervention . . . . . . . . 435–436Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 437Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 438Reading to Learn Mathematics . . . . . . . . . . 439Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 440
Lesson 8-5Study Guide and Intervention . . . . . . . . 441–442Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 443Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 444Reading to Learn Mathematics . . . . . . . . . . 445Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 446
Lesson 8-6Study Guide and Intervention . . . . . . . . 447–448Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 449Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 450Reading to Learn Mathematics . . . . . . . . . . 451Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 452
Lesson 8-7Study Guide and Intervention . . . . . . . . 453–454Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 455Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 456Reading to Learn Mathematics . . . . . . . . . . 457Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 458
Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 459–460Chapter 8 Test, Form 2A . . . . . . . . . . . 461–462Chapter 8 Test, Form 2B . . . . . . . . . . . 463–464Chapter 8 Test, Form 2C . . . . . . . . . . . 465–466Chapter 8 Test, Form 2D . . . . . . . . . . . 467–468Chapter 8 Test, Form 3 . . . . . . . . . . . . 469–470Chapter 8 Open-Ended Assessment . . . . . . 471Chapter 8 Vocabulary Test/Review . . . . . . . 472Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 473Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 474Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 475Chapter 8 Cumulative Review . . . . . . . . . . . 476Chapter 8 Standardized Test Practice . 477–478
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 8 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing in theGeometry TeacherWorks CD-ROM.
Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.
WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.
Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.
WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.
Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.
WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.
Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.
WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.
Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.
WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.
Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.
WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.
© Glencoe/McGraw-Hill v Glencoe Geometry
Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.
WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.
Assessment OptionsThe assessment masters in the Chapter 8Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.
Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions
and is intended for use with basic levelstudents.
• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.
• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.
• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.
All of the above tests include a free-response Bonus question.
• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.
• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.
Intermediate Assessment• Four free-response quizzes are included
to offer assessment at appropriateintervals in the chapter.
• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.
Continuing Assessment• The Cumulative Review provides
students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.
• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.
Answers• Page A1 is an answer sheet for the
Standardized Test Practice questionsthat appear in the Student Edition onpages 458–459. This improves students’familiarity with the answer formats theymay encounter in test taking.
• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.
• Full-size answer keys are provided forthe assessment masters in this booklet.
Reading to Learn MathematicsVocabulary Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
88
© Glencoe/McGraw-Hill vii Glencoe Geometry
Voca
bula
ry B
uild
erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.
Vocabulary Term Found on Page Definition/Description/Example
diagonals
isosceles trapezoid
kite
median
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
parallelogram
rectangle
rhombus
square
trapezoid
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
88
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 8. As you
study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 8.1Interior Angle Sum Theorem
Theorem 8.2Exterior Angle Sum Theorem
Theorem 8.3
Theorem 8.4
Theorem 8.5
Theorem 8.7
Theorem 8.8
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 8.12
Theorem 8.13
Theorem 8.15
Theorem 8.17
Theorem 8.18
Theorem 8.19
Theorem 8.20
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
88
Study Guide and InterventionAngles of Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 417 Glencoe Geometry
Less
on
8-1
Sum of Measures of Interior Angles The segments that connect thenonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals fromone vertex of an n-gon separates the polygon into n � 2 triangles. The sum of the measuresof the interior angles of the polygon can be found by adding the measures of the interiorangles of those n � 2 triangles.
Interior Angle If a convex polygon has n sides, and S is the sum of the measures of its interior angles, Sum Theorem then S � 180(n � 2).
A convex polygon has 13 sides. Find the sum of the measuresof the interior angles.S � 180(n � 2)
� 180(13 � 2)� 180(11)� 1980
The measure of aninterior angle of a regular polygon is120. Find the number of sides.The number of sides is n, so the sum of themeasures of the interior angles is 120n.
S � 180(n � 2)120n � 180(n � 2)120n � 180n � 360
�60n � �360n � 6
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the sum of the measures of the interior angles of each convex polygon.
1. 10-gon 2. 16-gon 3. 30-gon
4. 8-gon 5. 12-gon 6. 3x-gon
The measure of an interior angle of a regular polygon is given. Find the numberof sides in each polygon.
7. 150 8. 160 9. 175
10. 165 11. 168.75 12. 135
13. Find x.
7x �
(4x � 10)�
(4x � 5)�
(6x � 10)�
(5x � 5)�
A B
C
D
E
© Glencoe/McGraw-Hill 418 Glencoe Geometry
Sum of Measures of Exterior Angles There is a simple relationship among theexterior angles of a convex polygon.
Exterior Angle If a polygon is convex, then the sum of the measures of the exterior angles, Sum Theorem one at each vertex, is 360.
Find the sum of the measures of the exterior angles, one at eachvertex, of a convex 27-gon.For any convex polygon, the sum of the measures of its exterior angles, one at each vertex,is 360.
Find the measure of each exterior angle of regular hexagon ABCDEF.The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then
6n � 360n � 60
Find the sum of the measures of the exterior angles of each convex polygon.
1. 10-gon 2. 16-gon 3. 36-gon
Find the measure of an exterior angle for each convex regular polygon.
4. 12-gon 5. 36-gon 6. 2x-gon
Find the measure of an exterior angle given the number of sides of a regularpolygon.
7. 40 8. 18 9. 12
10. 24 11. 180 12. 8
A B
C
DE
F
Study Guide and Intervention (continued)
Angles of Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeAngles of Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 419 Glencoe Geometry
Less
on
8-1
Find the sum of the measures of the interior angles of each convex polygon.
1. nonagon 2. heptagon 3. decagon
The measure of an interior angle of a regular polygon is given. Find the numberof sides in each polygon.
4. 108 5. 120 6. 150
Find the measure of each interior angle using the given information.
7. 8.
9. quadrilateral STUW with �S � �T, 10. hexagon DEFGHI with �U � �W, m�S � 2x � 16, �D � �E � �G � �H, �F � �I,m�U � x � 14 m�D � 7x, m�F � 4x
Find the measures of an interior angle and an exterior angle for each regularpolygon.
11. quadrilateral 12. pentagon 13. dodecagon
Find the measures of an interior angle and an exterior angle given the number ofsides of each regular polygon. Round to the nearest tenth if necessary.
14. 8 15. 9 16. 13
H G
FI
D ES T
UW
2x �
(2x � 20)� (3x � 10)�
(2x � 10)�
L M
NP
x �
x �
(2x � 15)�
(2x � 15)�
A B
CD
© Glencoe/McGraw-Hill 420 Glencoe Geometry
Find the sum of the measures of the interior angles of each convex polygon.
1. 11-gon 2. 14-gon 3. 17-gon
The measure of an interior angle of a regular polygon is given. Find the numberof sides in each polygon.
4. 144 5. 156 6. 160
Find the measure of each interior angle using the given information.
7. 8. quadrilateral RSTU with m�R � 6x � 4, m�S � 2x � 8
Find the measures of an interior angle and an exterior angle for each regularpolygon. Round to the nearest tenth if necessary.
9. 16-gon 10. 24-gon 11. 30-gon
Find the measures of an interior angle and an exterior angle given the number ofsides of each regular polygon. Round to the nearest tenth if necessary.
12. 14 13. 22 14. 40
15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. Thebasis of the classification is the shapes of the faces of the crystal. Turquoise belongs tothe triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram.Find the sum of the measures of the interior angles of one such face.
R S
TU
x �
(2x � 15)� (3x � 20)�
(x � 15)�
J K
MN
Practice Angles of Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Reading to Learn MathematicsAngles of Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
© Glencoe/McGraw-Hill 421 Glencoe Geometry
Less
on
8-1
Pre-Activity How does a scallop shell illustrate the angles of polygons?
Read the introduction to Lesson 8-1 at the top of page 404 in your textbook.
• How many diagonals of the scallop shell shown in your textbook can bedrawn from vertex A?
• How many diagonals can be drawn from one vertex of an n-gon? Explainyour reasoning.
Reading the Lesson1. Write an expression that describes each of the following quantities for a regular n-gon. If
the expression applies to regular polygons only, write regular. If it applies to all convexpolygons, write all.
a. the sum of the measures of the interior angles
b. the measure of each interior angle
c. the sum of the measures of the exterior angles (one at each vertex)
d. the measure of each exterior angle
2. Give the measure of an interior angle and the measure of an exterior angle of each polygon.a. equilateral triangle c. squareb. regular hexagon d. regular octagon
3. Underline the correct word or phrase to form a true statement about regular polygons.a. As the number of sides increases, the sum of the measures of the interior angles
(increases/decreases/stays the same).b. As the number of sides increases, the measure of each interior angle
(increases/decreases/stays the same).c. As the number of sides increases, the sum of the measures of the exterior angles
(increases/decreases/stays the same).d. As the number of sides increases, the measure of each exterior angle
(increases/decreases/stays the same).e. If a regular polygon has more than four sides, each interior angle will be a(n)
(acute/right/obtuse) angle, and each exterior angle will be a(n) (acute/right/obtuse) angle.
Helping You Remember4. A good way to remember a new mathematical idea or formula is to relate it to something
you already know. How can you use your knowledge of the Angle Sum Theorem (for atriangle) to help you remember the Interior Angle Sum Theorem?
© Glencoe/McGraw-Hill 422 Glencoe Geometry
TangramsThe tangram puzzle is composed of seven pieces that form a square, as shown at theright. This puzzle has been a popularamusement for Chinese students for hundreds and perhaps thousands of years.
Make a careful tracing of the figure above. Cut out the pieces and rearrange them to form each figure below. Record each answer by drawing lines within each figure.
1. 2.
3. 4.
5. 6.
7. Create a different figure using the seven tangram pieces. Trace the outline.Then challenge another student to solve the puzzle.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-18-1
Study Guide and InterventionParallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 423 Glencoe Geometry
Less
on
8-2
Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are fourimportant properties of parallelograms.
If PQRS is a parallelogram, then
The opposite sides of a P�Q� � S�R� and P�S� �Q�R�parallelogram are congruent.
The opposite angles of a �P � �R and �S � �Qparallelogram are congruent.
The consecutive angles of a �P and �S are supplementary; �S and �R are supplementary; parallelogram are supplementary. �R and �Q are supplementary; �Q and �P are supplementary.
If a parallelogram has one right If m�P � 90, then m�Q � 90, m�R � 90, and m�S � 90.angle, then it has four right angles.
If ABCD is a parallelogram, find a and b.A�B� and C�D� are opposite sides, so A�B� � C�D�.2a � 34a � 17
�A and �C are opposite angles, so �A � �C.8b � 112b � 14
Find x and y in each parallelogram.
1. 2.
3. 4.
5. 6.
72x
30x 150
2y60�
55� 5x�
2y�
6x� 12x�
3y�3y
12
6x
8y
88
6x�
4y�
3x�
BA
D C34
2a8b�
112�
P Q
RS
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 424 Glencoe Geometry
Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.
If ABCD is a parallelogram, then:
The diagonals of a parallelogram bisect each other. AP � PC and DP � PB
Each diagonal separates a parallelogram �ACD � �CAB and �ADB � �CBDinto two congruent triangles.
Find x and y in parallelogram ABCD.The diagonals bisect each other, so AE � CE and DE � BE.6x � 24 4y � 18x � 4 y � 4.5
Find x and y in each parallelogram.
1. 2. 3.
4. 5. 6.
Complete each statement about �ABCD.Justify your answer.
7. �BAC �
8. D�E� �
9. �ADC �
10. A�D� ||
A B
CDE
x
17
4y3x� 2y
12
3x
30�10
y
2x�
60� 4y�
4x
2y28
3x4y
812
A BE
CD
6x
1824
4y
A B
CD
P
Study Guide and Intervention (continued)
Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
ExampleExample
ExercisesExercises
Skills PracticeParallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 425 Glencoe Geometry
Less
on
8-2
Complete each statement about �DEFG. Justify your answer.
1. D�G� ||
2. D�E� �
3. G�H� �
4. �DEF �
5. �EFG is supplementary to .
6. �DGE �
ALGEBRA Use �WXYZ to find each measure or value.
7. m�XYZ � 8. m�WZY �
9. m�WXY � 10. a �
COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram HJKL given each set of vertices.
11. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 12. H(�1, 4), J(3, 3), K(3, �2), L(�1, �1)
13. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogramare supplementary.
D C
BA
YZ
XWA
50�30
2a
70�
?
?
?
?
?
? FG
ED H
© Glencoe/McGraw-Hill 426 Glencoe Geometry
Complete each statement about �LMNP. Justify your answer.
1. L�Q� �
2. �LMN �
3. �LMP �
4. �NPL is supplementary to .
5. L�M� �
ALGEBRA Use �RSTU to find each measure or value.
6. m�RST � 7. m�STU �
8. m�TUR � 9. b �
COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram PRYZ given each set of vertices.
10. P(2, 5), R(3, 3), Y(�2, �3), Z(�3, �1) 11. P(2, 3), R(1, �2), Y(�5, �7), Z(�4, �2)
12. PROOF Write a paragraph proof of the following.Given: �PRST and �PQVUProve: �V � �S
13. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the pavingstone for a sidewalk. If m�1 is 130, find m�2, m�3, and m�4.
1 2
34
P R
S
U V
Q
T
TU
SRB30� 23
4b � 1
25�
?
?
?
?
?P N
MQ
L
Practice Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Reading to Learn MathematicsParallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
© Glencoe/McGraw-Hill 427 Glencoe Geometry
Less
on
8-2
Pre-Activity How are parallelograms used to represent data?
Read the introduction to Lesson 8-2 at the top of page 411 in your textbook.
• What is the name of the shape of the top surface of each wedge of cheese?
• Are the three polygons shown in the drawing similar polygons? Explain yourreasoning.
Reading the Lesson
1. Underline words or phrases that can complete the following sentences to make statementsthat are always true. (There may be more than one correct choice for some of the sentences.)
a. Opposite sides of a parallelogram are (congruent/perpendicular/parallel).
b. Consecutive angles of a parallelogram are (complementary/supplementary/congruent).
c. A diagonal of a parallelogram divides the parallelogram into two(acute/right/obtuse/congruent) triangles.
d. Opposite angles of a parallelogram are (complementary/supplementary/congruent).
e. The diagonals of a parallelogram (bisect each other/are perpendicular/are congruent).
f. If a parallelogram has one right angle, then all of its other angles are(acute/right/obtuse) angles.
2. Let ABCD be a parallelogram with AB � BC and with no right angles.
a. Sketch a parallelogram that matches the descriptionabove and draw diagonal B�D�.
In parts b–f, complete each sentence.
b. A�B� || and A�D� || .
c. A�B� � and B�C� � .
d. �A � and �ABC � .
e. �ADB � because these two angles are
angles formed by the two parallel lines and and the
transversal .
f. �ABD � .
Helping You Remember
3. A good way to remember new theorems in geometry is to relate them to theorems youlearned earlier. Name a theorem about parallel lines that can be used to remember thetheorem that says, “If a parallelogram has one right angle, it has four right angles.”
DA
CB
© Glencoe/McGraw-Hill 428 Glencoe Geometry
TessellationsA tessellation is a tiling pattern made of polygons. The pattern can be extended so that the polygonal tiles cover the plane completely withno gaps. A checkerboard and a honeycomb pattern are examples oftessellations. Sometimes the same polygon can make more than onetessellation pattern. Both patterns below can be formed from anisosceles triangle.
Draw a tessellation using each polygon.
1. 2. 3.
Draw two different tessellations using each polygon.
4. 5.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-28-2
Study Guide and InterventionTests for Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 429 Glencoe Geometry
Less
on
8-3
Conditions for a Parallelogram There are many ways to establish that a quadrilateral is a parallelogram.
If: If:
both pairs of opposite sides are parallel, A�B� || D�C� and A�D� || B�C�,
both pairs of opposite sides are congruent, A�B� � D�C� and A�D� � B�C�,
both pairs of opposite angles are congruent, �ABC � �ADC and �DAB � �BCD,
the diagonals bisect each other, A�E� � C�E� and D�E� � B�E�,
one pair of opposite sides is congruent and parallel, A�B� || C�D� and A�B� � C�D�, or A�D� || B�C� and A�D� �B�C�,
then: the figure is a parallelogram. then: ABCD is a parallelogram.
Find x and y so that FGHJ is a parallelogram.FGHJ is a parallelogram if the lengths of the opposite sides are equal.
6x � 3 � 15 4x � 2y � 26x � 12 4(2) � 2y � 2x � 2 8 � 2y � 2
�2y � �6y � 3
Find x and y so that each quadrilateral is a parallelogram.
1. 2.
3. 4.
5. 6. 6y�
3x�
(x � y)�2x�
30�
24�
9x�6y
45�185x�5y�
25�
11x�
5y� 55�2x � 2
2y 8
12
J H
GF 6x � 3
154x � 2y 2
A B
CD
E
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 430 Glencoe Geometry
Parallelograms on the Coordinate Plane On the coordinate plane, the DistanceFormula and the Slope Formula can be used to test if a quadrilateral is a parallelogram.
Determine whether ABCD is a parallelogram.The vertices are A(�2, 3), B(3, 2), C(2, �1), and D(�3, 0).
Method 1: Use the Slope Formula, m � �yx
2
2
�
�
yx
1
1�.
slope of A�D� � ��2
3�
�
(0�3)
� � �31� � 3 slope of B�C� � �
23�
�
(�21)
� � �31� � 3
slope of A�B� � �3
2�
�
(�32)
� � ��15� slope of C�D� � �
2�
�
1(�
�
03)
� � ��15�
Opposite sides have the same slope, so A�B� || C�D� and A�D� || B�C�. Both pairs of opposite sidesare parallel, so ABCD is a parallelogram.
Method 2: Use the Distance Formula, d � �(x2 ��x1)2 �� ( y2 �� y1)2�.
AB � �(�2 �� 3)2 �� (3 ��2)2� � �25 ��1� or �26�
CD � �(2 � (��3))2�� (�1� � 0)2� � �25 ��1� or �26�
AD � �(�2 �� (�3))�2 � (3� � 0)2� � �1 � 9� or �10�
BC � �(3 � 2�)2 � (�2 � (��1))2� � �1 � 9� or �10�Both pairs of opposite sides have the same length, so ABCD is a parallelogram.
Determine whether a figure with the given vertices is a parallelogram. Use themethod indicated.
1. A(0, 0), B(1, 3), C(5, 3), D(4, 0); 2. D(�1, 1), E(2, 4), F(6, 4), G(3, 1);Slope Formula Slope Formula
3. R(�1, 0), S(3, 0), T(2, �3), U(�3, �2); 4. A(�3, 2), B(�1, 4), C(2, 1), D(0, �1);Distance Formula Distance and Slope Formulas
5. S(�2, 4), T(�1, �1), U(3, �4), V(2, 1); 6. F(3, 3), G(1, 2), H(�3, 1), I(�1, 4);Distance and Slope Formulas Midpoint Formula
7. A parallelogram has vertices R(�2, �1), S(2, 1), and T(0, �3). Find all possiblecoordinates for the fourth vertex.
x
y
O
C
A
D
B
Study Guide and Intervention (continued)
Tests for Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
ExampleExample
ExercisesExercises
Skills PracticeTests for Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 431 Glencoe Geometry
Less
on
8-3
Determine whether each quadrilateral is a parallelogram. Justify your answer.
1. 2.
3. 4.
COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use the method indicated.
5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula
6. S(�2, 1), R(1, 3), T(2, 0), Z(�1, �2); Distance and Slope Formula
7. W(2, 5), R(3, 3), Y(�2, �3), N(�3, 1); Midpoint Formula
ALGEBRA Find x and y so that each quadrilateral is a parallelogram.
8. 9.
10. 11.
3x � 14
x � 20
3y � 2y � 20
(4x � 35)�
(3x � 10)�(2y � 5)�
(y � 15)�
2y � 11
y � 3 4x � 3
3x
x � 16
y � 192y
2x � 8
© Glencoe/McGraw-Hill 432 Glencoe Geometry
Determine whether each quadrilateral is a parallelogram. Justify your answer.
1. 2.
3. 4.
COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use the method indicated.
5. P(�5, 1), S(�2, 2), F(�1, �3), T(2, �2); Slope Formula
6. R(�2, 5), O(1, 3), M(�3, �4), Y(�6, �2); Distance and Slope Formula
ALGEBRA Find x and y so that each quadrilateral is a parallelogram.
7. 8.
9. 10.
11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes areparallelograms?
�4y � 2
x � 12
�2x � 6
y � 23�4x � 6
�6x
7y � 312y � 7
3y � 5�3x � 4
�4x � 22y � 8(5x � 29)�
(7x � 11)�(3y � 15)�
(5y � 9)�
118� 62�
118�62�
Practice Tests for Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Reading to Learn MathematicsTests for Parallelograms
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
© Glencoe/McGraw-Hill 433 Glencoe Geometry
Less
on
8-3
Pre-Activity How are parallelograms used in architecture?
Read the introduction to Lesson 8-3 at the top of page 417 in your textbook.
Make two observations about the angles in the roof of the covered bridge.
Reading the Lesson1. Which of the following conditions guarantee that a quadrilateral is a parallelogram?
A. Two sides are parallel.B. Both pairs of opposite sides are congruent.C. The diagonals are perpendicular.D. A pair of opposite sides is both parallel and congruent.E. There are two right angles.F. The sum of the measures of the interior angles is 360.G. All four sides are congruent.H. Both pairs of opposite angles are congruent.I. Two angles are acute and the other two angles are obtuse.J. The diagonals bisect each other.K. The diagonals are congruent.L. All four angles are right angles.
2. Determine whether there is enough given information to know that each figure is aparallelogram. If so, state the definition or theorem that justifies your conclusion.
a. b.
c. d.
Helping You Remember3. A good way to remember a large number of mathematical ideas is to think of them in
groups. How can you state the conditions as one group about the sides of quadrilateralsthat guarantee that the quadrilateral is a parallelogram?
© Glencoe/McGraw-Hill 434 Glencoe Geometry
Tests for ParallelogramsBy definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sidesare parallel. What conditions other than both pairs of opposite sides parallel will guaranteethat a quadrilateral is a parallelogram? In this activity, several possibilities will beinvestigated by drawing quadrilaterals to satisfy certain conditions. Remember that any test that seems to work is not guaranteed to work unless it can be formally proven.
Complete.
1. Draw a quadrilateral with one pair of opposite sides congruent.Must it be a parallelogram?
2. Draw a quadrilateral with both pairs of opposite sides congruent.Must it be a parallelogram?
3. Draw a quadrilateral with one pair of opposite sides parallel and the other pair of opposite sides congruent. Must it be a parallelogram?
4. Draw a quadrilateral with one pair of opposite sides both parallel and congruent. Must it be a parallelogram?
5. Draw a quadrilateral with one pair of opposite angles congruent.Must it be a parallelogram?
6. Draw a quadrilateral with both pairs of opposite angles congruent.Must it be a parallelogram?
7. Draw a quadrilateral with one pair of opposite sides parallel and one pair of opposite angles congruent. Must it be a parallelogram?
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-38-3
Study Guide and InterventionRectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 435 Glencoe Geometry
Less
on
8-4
Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles.
A rectangle has all the properties of a parallelogram.
• Opposite sides are parallel.• Opposite angles are congruent.• Opposite sides are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.
Also:
• All four angles are right angles. �UTS, �TSR, �SRU, and �RUT are right angles.• The diagonals are congruent. T�R� � U�S�
T S
RU
Q
In rectangle RSTUabove, US � 6x � 3 and RT � 7x � 2.Find x.The diagonals of a rectangle bisect eachother, so US � RT.
6x � 3 � 7x � 23 � x � 25 � x
In rectangle RSTUabove, m�STR � 8x � 3 and m�UTR �16x � 9. Find m�STR.�UTS is a right angle, so m�STR � m�UTR � 90.
8x � 3 � 16x � 9 � 9024x � 6 � 90
24x � 96x � 4
m�STR � 8x � 3 � 8(4) � 3 or 35
Example 1Example 1 Example 2Example 2
ExercisesExercises
ABCD is a rectangle.
1. If AE � 36 and CE � 2x � 4, find x.
2. If BE � 6y � 2 and CE � 4y � 6, find y.
3. If BC � 24 and AD � 5y � 1, find y.
4. If m�BEA � 62, find m�BAC.
5. If m�AED � 12x and m�BEC � 10x � 20, find m�AED.
6. If BD � 8y � 4 and AC � 7y � 3, find BD.
7. If m�DBC � 10x and m�ACB � 4x2� 6, find m� ACB.
8. If AB � 6y and BC � 8y, find BD in terms of y.
9. In rectangle MNOP, m�1 � 40. Find the measure of each numbered angle.
M N
OP
K
1 23
45
6
B C
DA
E
© Glencoe/McGraw-Hill 436 Glencoe Geometry
Prove that Parallelograms Are Rectangles The diagonals of a rectangle arecongruent, and the converse is also true.
If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
In the coordinate plane you can use the Distance Formula, the Slope Formula, andproperties of diagonals to show that a figure is a rectangle.
Determine whether A(�3, 0), B(�2, 3), C(4, 1),and D(3, �2) are the vertices of a rectangle.
Method 1: Use the Slope Formula.
slope of A�B� � ��2
3�
�
(0�3)
� � �31� or 3 slope of A�D� � �
3�
�
2(�
�
03)
� � ��62� or ��
13�
slope of C�D� � ��32��
41
� � ���
31� or 3 slope of B�C� � �
41�
�
(�32)
� � ��62� or ��
13�
Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides areperpendicular, so ABCD is a rectangle.
Method 2: Use the Midpoint and Distance Formulas.
The midpoint of A�C� is ���32� 4�, �
0 �2
1�� � ��
12�, �
12�� and the midpoint of B�D� is
���22� 3�, �
3 �2
2�� � ��
12�, �
12��. The diagonals have the same midpoint so they bisect each other.
Thus, ABCD is a parallelogram.
AC � �(�3 �� 4)2 �� (0 ��1)2� � �49 ��1� or �50�BD � �(�2 �� 3)2 �� (3 ��(�2))2� � �25 ��25� or �50�The diagonals are congruent. ABCD is a parallelogram with diagonals that bisect eachother, so it is a rectangle.
Determine whether ABCD is a rectangle given each set of vertices. Justify youranswer.
1. A(�3, 1), B(�3, 3), C(3, 3), D(3, 1) 2. A(�3, 0), B(�2, 3), C(4, 5), D(3, 2)
3. A(�3, 0), B(�2, 2), C(3, 0), D(2, �2) 4. A(�1, 0), B(0, 2), C(4, 0), D(3, �2)
5. A(�1, �5), B(�3, 0), C(2, 2), D(4, �3) 6. A(�1, �1), B(0, 2), C(4, 3), D(3, 0)
7. A parallelogram has vertices R(�3, �1), S(�1, 2), and T(5, �2). Find the coordinates ofU so that RSTU is a rectangle.
x
y
O
D
B
A
E C
Study Guide and Intervention (continued)
Rectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
ExampleExample
ExercisesExercises
Skills PracticeRectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 437 Glencoe Geometry
Less
on
8-4
ALGEBRA ABCD is a rectangle.
1. If AC � 2x � 13 and DB � 4x � 1, find x.
2. If AC � x � 3 and DB � 3x � 19, find AC.
3. If AE � 3x � 3 and EC � 5x � 15, find AC.
4. If DE � 6x � 7 and AE � 4x � 9, find DB.
5. If m�DAC � 2x � 4 and m�BAC � 3x � 1, find x.
6. If m�BDC � 7x � 1 and m�ADB � 9x � 7, find m�BDC.
7. If m�ABD � x2 � 7 and m�CDB � 4x � 5, find x.
8. If m�BAC � x2 � 3 and m�CAD � x � 15, find m�BAC.
PRST is a rectangle. Find each measure if m�1 � 50.
9. m�2 10. m�3
11. m�4 12. m�5
13. m�6 14. m�7
15. m�8 16. m�9
COORDINATE GEOMETRY Determine whether TUXY is a rectangle given each setof vertices. Justify your answer.
17. T(�3, �2), U(�4, 2), X(2, 4), Y(3, 0)
18. T(�6, 3), U(0, 6), X(2, 2), Y(�4, �1)
19. T(4, 1), U(3, �1), X(�3, 2), Y(�2, 4)
T
1 2 3 4
567 89
S
RP
D C
BAE
© Glencoe/McGraw-Hill 438 Glencoe Geometry
ALGEBRA RSTU is a rectangle.
1. If UZ � x � 21 and ZS � 3x � 15, find US.
2. If RZ � 3x � 8 and ZS � 6x � 28, find UZ.
3. If RT � 5x � 8 and RZ � 4x � 1, find ZT.
4. If m�SUT � 3x � 6 and m�RUS � 5x � 4, find m�SUT.
5. If m�SRT � x2 � 9 and m�UTR � 2x � 44, find x.
6. If m�RSU � x2 � 1 and m�TUS � 3x � 9, find m�RSU.
GHJK is a rectangle. Find each measure if m�1 � 37.
7. m�2 8. m�3
9. m�4 10. m�5
11. m�6 12. m�7
COORDINATE GEOMETRY Determine whether BGHL is a rectangle given each setof vertices. Justify your answer.
13. B(�4, 3), G(�2, 4), H(1, �2), L(�1, �3)
14. B(�4, 5), G(6, 0), H(3, �6), L(�7, �1)
15. B(0, 5), G(4, 7), H(5, 4), L(1, 2)
16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for aJapanese Zen garden. Is it sufficient to know that opposite sides of the garden plot arecongruent and parallel to determine that the garden plot is rectangular? Explain.
K
2 1 5
436
7
J
HG
U T
SRZ
Practice Rectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
Reading to Learn MathematicsRectangles
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
© Glencoe/McGraw-Hill 439 Glencoe Geometry
Less
on
8-4
Pre-Activity How are rectangles used in tennis?
Read the introduction to Lesson 8-4 at the top of page 424 in your textbook.
Are the singles court and doubles court similar rectangles? Explain youranswer.
Reading the Lesson1. Determine whether each sentence is always, sometimes, or never true.
a. If a quadrilateral has four congruent angles, it is a rectangle.
b. If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a rectangle.
c. The diagonals of a rectangle bisect each other.
d. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a rectangle.
e. Consecutive angles of a rectangle are complementary.
f. Consecutive angles of a rectangle are congruent.
g. If the diagonals of a quadrilateral are congruent, the quadrilateral is a rectangle.
h. A diagonal of a rectangle bisects two of its angles.
i. A diagonal of a rectangle divides the rectangle into two congruent right triangles.
j. If the diagonals of a quadrilateral bisect each other and are congruent, thequadrilateral is a rectangle.
k. If a parallelogram has one right angle, it is a rectangle.
l. If a parallelogram has four congruent sides, it is a rectangle.
2. ABCD is a rectangle with AD � AB.Name each of the following in this figure.
a. all segments that are congruent to B�E�
b. all angles congruent to �1
c. all angles congruent to �7
d. two pairs of congruent triangles
Helping You Remember3. It is easier to remember a large number of geometric relationships and theorems if you
are able to combine some of them. How can you combine the two theorems aboutdiagonals that you studied in this lesson?
A1
2 3 4 5
678D
CBE
© Glencoe/McGraw-Hill 440 Glencoe Geometry
Counting Squares and RectanglesEach puzzle below contains many squares and/or rectangles.Count them carefully. You may want to label each region so youcan list all possibilities.
How many rectangles are in the figure at the right?
Label each small region with a letter. Then list each rectangle by writing the letters of regions it contains.
A, B, C, D, AB, CD, AC, BD, ABCD
There are 9 rectangles.
How many squares are in each figure?
1. 2. 3.
How many rectangles are in each figure?
1. 2. 3.
A B
C D
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-48-4
ExampleExample
Study Guide and InterventionRhombi and Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 441 Glencoe Geometry
Less
on
8-5
Properties of Rhombi A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also aparallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties.
The diagonals are perpendicular. M�H� ⊥ R�O�
Each diagonal bisects a pair of opposite angles. M�H� bisects �RMO and �RHO.R�O� bisects �MRH and �MOH.
If the diagonals of a parallelogram are If RHOM is a parallelogram and R�O� ⊥ M�H�,perpendicular, then the figure is a rhombus. then RHOM is a rhombus.
In rhombus ABCD, m�BAC � 32. Find the measure of each numbered angle.ABCD is a rhombus, so the diagonals are perpendicular and �ABEis a right triangle. Thus m�4 � 90 and m�1 � 90 � 32 or 58. Thediagonals in a rhombus bisect the vertex angles, so m�1 � m�2.Thus, m�2 � 58.
A rhombus is a parallelogram, so the opposite sides are parallel. �BAC and �3 arealternate interior angles for parallel lines, so m�3 � 32.
ABCD is a rhombus.
1. If m�ABD � 60, find m�BDC. 2. If AE � 8, find AC.
3. If AB � 26 and BD � 20, find AE. 4. Find m�CEB.
5. If m�CBD � 58, find m�ACB. 6. If AE � 3x � 1 and AC � 16, find x.
7. If m�CDB � 6y and m�ACB � 2y � 10, find y.
8. If AD � 2x � 4 and CD � 4x � 4, find x.
9. a. What is the midpoint of F�H�?
b. What is the midpoint of G�J�?
c. What kind of figure is FGHJ? Explain.
d. What is the slope of F�H�?
e. What is the slope of G�J�?
f. Based on parts c, d, and e, what kind of figure is FGHJ ? Explain.
x
y
O
J
GF
H
C
A D
EB
CA
D
E
B
32�
1 2
34
M O
HRB
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 442 Glencoe Geometry
Properties of Squares A square has all the properties of a rhombus and all theproperties of a rectangle.
Find the measure of each numbered angle of square ABCD.
Using properties of rhombi and rectangles, the diagonals are perpendicular and congruent. �ABE is a right triangle, so m�1 � m�2 � 90.
Each vertex angle is a right angle and the diagonals bisect the vertex angles,so m�3 � m�4 � m�5 � 45.
Determine whether the given vertices represent a parallelogram, rectangle,rhombus, or square. Explain your reasoning.
1. A(0, 2), B(2, 4), C(4, 2), D(2, 0)
2. D(�2, 1), E(�1, 3), F(3, 1), G(2, �1)
3. A(�2, �1), B(0, 2), C(2, �1), D(0, �4)
4. A(�3, 0), B(�1, 3), C(5, �1), D(3, �4)
5. S(�1, 4), T(3, 2), U(1, �2), V(�3, 0)
6. F(�1, 0), G(1, 3), H(4, 1), I(2, �2)
7. Square RSTU has vertices R(�3, �1), S(�1, 2), and T(2, 0). Find the coordinates ofvertex U.
C
A D
1 2345
EB
Study Guide and Intervention (continued)
Rhombi and Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
ExercisesExercises
ExampleExample
Skills PracticeRhombi and Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 443 Glencoe Geometry
Less
on
8-5
Use rhombus DKLM with AM � 4x, AK � 5x � 3, and DL � 10.
1. Find x. 2. Find AL.
3. Find m�KAL. 4. Find DM.
Use rhombus RSTV with RS � 5y � 2, ST � 3y � 6, and NV � 6.
5. Find y. 6. Find TV.
7. Find m�NTV. 8. Find m�SVT.
9. Find m�RST. 10. Find m�SRV.
COORDINATE GEOMETRY Given each set of vertices, determine whether �QRSTis a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.
11. Q(3, 5), R(3, 1), S(�1, 1), T(�1, 5)
12. Q(�5, 12), R(5, 12), S(�1, 4), T(�11, 4)
13. Q(�6, �1), R(4, �6), S(2, 5), T(�8, 10)
14. Q(2, �4), R(�6, �8), S(�10, 2), T(�2, 6)
R V
NS T
M L
AD K
© Glencoe/McGraw-Hill 444 Glencoe Geometry
Use rhombus PRYZ with RK � 4y � 1, ZK � 7y � 14,PK � 3x � 1, and YK � 2x � 6.
1. Find PY. 2. Find RZ.
3. Find RY. 4. Find m�YKZ.
Use rhombus MNPQ with PQ � 3�2�, PA � 4x � 1, and AM � 9x � 6.
5. Find AQ. 6. Find m�APQ.
7. Find m�MNP. 8. Find PM.
COORDINATE GEOMETRY Given each set of vertices, determine whether �BEFGis a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.
9. B(�9, 1), E(2, 3), F(12, �2), G(1, �4)
10. B(1, 3), E(7, �3), F(1, �9), G(�5, �3)
11. B(�4, �5), E(1, �5), F(�7, �1), G(�2, �1)
12. TESSELATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.
AN
M Q
P
K
P
YR
Z
Practice Rhombi and Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
Reading to Learn MathematicsRhombi and Squares
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
© Glencoe/McGraw-Hill 445 Glencoe Geometry
Less
on
8-5
Pre-Activity How can you ride a bicycle with square wheels?
Read the introduction to Lesson 8-5 at the top of page 431 in your textbook.
If you draw a diagonal on the surface of one of the square wheels shown inthe picture in your textbook, how can you describe the two triangles that areformed?
Reading the Lesson
1. Sketch each of the following.
a. a quadrilateral with perpendicular b. a quadrilateral with congruent diagonals that is not a rhombus diagonals that is not a rectangle
c. a quadrilateral whose diagonals are perpendicular and bisect each other,but are not congruent
2. List all of the following special quadrilaterals that have each listed property:parallelogram, rectangle, rhombus, square.
a. The diagonals are congruent.
b. Opposite sides are congruent.
c. The diagonals are perpendicular.
d. Consecutive angles are supplementary.
e. The quadrilateral is equilateral.
f. The quadrilateral is equiangular.
g. The diagonals are perpendicular and congruent.
h. A pair of opposite sides is both parallel and congruent.
3. What is the common name for a regular quadrilateral? Explain your answer.
Helping You Remember
4. A good way to remember something is to explain it to someone else. Suppose that yourclassmate Luis is having trouble remembering which of the properties he has learned inthis chapter apply to squares. How can you help him?
© Glencoe/McGraw-Hill 446 Glencoe Geometry
Creating Pythagorean PuzzlesBy drawing two squares and cutting them in a certain way, youcan make a puzzle that demonstrates the Pythagorean Theorem.A sample puzzle is shown. You can create your own puzzle byfollowing the instructions below.
1. Carefully construct a square and label the length of a side as a. Then construct a smaller square to the right of it and label the length of a side as b,as shown in the figure above. The bases should be adjacent and collinear.
2. Mark a point X that is b units from the left edge of the larger square. Then draw the segments from the upper left corner of the larger square to point X, and from point X to the upper right corner of the smaller square.
3. Cut out and rearrange your five pieces to form a larger square. Draw a diagram to show your answer.
4. Verify that the length of each side is equal to
�a2 � b�2�.
a
a
b X
b
b
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-58-5
Study Guide and InterventionTrapezoids
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 447 Glencoe Geometry
Less
on
8-6
Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called basesand the nonparallel sides are called legs. If the legs are congruent,the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent. STUR is an isosceles trapezoid.
S�R� � T�U�; �R � �U, �S ��T
The vertices of ABCD are A(�3, �1), B(�1, 3),C(2, 3), and D(4, �1). Verify that ABCD is a trapezoid.
slope of A�B� � ��
31�
�
(�(�
13))
� � �42� � 2
slope of A�D� � ��
41�
�
(�(�
31))
� � �07� � 0
slope of B�C� � �2
3�
�
(�31)
� � �03� � 0
slope of C�D� � ��41��
23
� � ��24� � �2
Exactly two sides are parallel, A�D� and B�C�, so ABCD is a trapezoid. AB � CD, so ABCD isan isosceles trapezoid.
In Exercises 1–3, determine whether ABCD is a trapezoid. If so, determinewhether it is an isosceles trapezoid. Explain.
1. A(�1, 1), B(2, 1), C(3, �2), and D(2, �2)
2. A(3, �3), B(�3, �3), C(�2, 3), and D(2, 3)
3. A(1, �4), B(�3, �3), C(�2, 3), and D(2, 2)
4. The vertices of an isosceles trapezoid are R(�2, 2), S(2, 2), T(4, �1), and U(�4, �1).Verify that the diagonals are congruent.
x
y
O
B
DA
C
T
R U
base
base
Sleg
AB � �(�3 �� (�1))�2 � (��1 � 3�)2�� �4 � 1�6� � �20� � 2�5�
CD � �(2 � 4�)2 � (�3 � (��1))2�� �4 � 1�6� � �20� � 2�5�
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 448 Glencoe Geometry
Medians of Trapezoids The median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases, and its length is one-half the sum of the lengths of the bases.
In trapezoid HJKL, MN � �12�(HJ � LK ).
M�N� is the median of trapezoid RSTU. Find x.
MN � �12�(RS � UT)
30 � �12�(3x � 5 � 9x � 5)
30 � �12�(12x)
30 � 6x5 � x
M�N� is the median of trapezoid HJKL. Find each indicated value.
1. Find MN if HJ � 32 and LK � 60.
2. Find LK if HJ � 18 and MN � 28.
3. Find MN if HJ � LK � 42.
4. Find m�LMN if m�LHJ � 116.
5. Find m�JKL if HJKL is isosceles and m�HLK � 62.
6. Find HJ if MN � 5x � 6, HJ � 3x � 6, and LK � 8x.
7. Find the length of the median of a trapezoid with vertices A(�2, 2), B(3, 3), C(7, 0), and D(�3, �2).
L K
NM
H J
U T
NM
R 3x � 5
30
9x � 5
S
L K
NM
H J
Study Guide and Intervention (continued)
Trapezoids
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
ExampleExample
ExercisesExercises
Skills PracticeTrapezoids
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 449 Glencoe Geometry
Less
on
8-6
COORDINATE GEOMETRY ABCD is a quadrilateral with vertices A(�4, �3),B(3, �3), C(6, 4), D(�7, 4).
1. Verify that ABCD is a trapezoid.
2. Determine whether ABCD is an isosceles trapezoid. Explain.
COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0),G(8, �5), H(�4, 4).
3. Verify that EFGH is a trapezoid.
4. Determine whether EFGH is an isosceles trapezoid. Explain.
COORDINATE GEOMETRY LMNP is a quadrilateral with vertices L(�1, 3), M(�4, 1),N(�6, 3), P(0, 7).
5. Verify that LMNP is a trapezoid.
6. Determine whether LMNP is an isosceles trapezoid. Explain.
ALGEBRA Find the missing measure(s) for the given trapezoid.
7. For trapezoid HJKL, S and T are 8. For trapezoid WXYZ, P and Q aremidpoints of the legs. Find HJ. midpoints of the legs. Find WX.
9. For trapezoid DEFG, T and U are 10. For isosceles trapezoid QRST, find themidpoints of the legs. Find TU, m�E, length of the median, m�Q, and m�S.and m�G.
ST
RQ125�
60
25
D E
FG
T U85�
35�42
14
W X
YZ
P Q12
19
H J
KL
S T72
86
© Glencoe/McGraw-Hill 450 Glencoe Geometry
COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(�3, �3), S(5, 1),T(10, �2), U(�4, �9).
1. Verify that RSTU is a trapezoid.
2. Determine whether RSTU is an isosceles trapezoid. Explain.
COORDINATE GEOMETRY BGHJ is a quadrilateral with vertices B(�9, 1), G(2, 3),H(12, �2), J(�10, �6).
3. Verify that BGHJ is a trapezoid.
4. Determine whether BGHJ is an isosceles trapezoid. Explain.
ALGEBRA Find the missing measure(s) for the given trapezoid.
5. For trapezoid CDEF, V and Y are 6. For trapezoid WRLP, B and C aremidpoints of the legs. Find CD. midpoints of the legs. Find LP.
7. For trapezoid FGHI, K and M are 8. For isosceles trapezoid TVZY, find the midpoints of the legs. Find FI, m�F, length of the median, m�T, and m�Z.and m�I.
9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in theshape of an isosceles trapezoid with the longer base at the bottom of the stairs and theshorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14, findthe width of the stairs halfway to the top.
10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in aclassroom. The carpenter knows the lengths of both bases of the desktop. What othermeasurements, if any, does the carpenter need?
V
Y Z
T 60�34
5
H
K M
G
IF
140� 125�
21
36
W R
LP
B C42
66F E
DC
V Y28
18
Practice Trapezoids
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
Reading to Learn MathematicsTrapezoids
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
© Glencoe/McGraw-Hill 451 Glencoe Geometry
Less
on
8-6
Pre-Activity How are trapezoids used in architecture?
Read the introduction to Lesson 8-6 at the top of page 439 in your textbook.
How might trapezoids be used in the interior design of a home?
Reading the Lesson
1. In the figure at the right, EFGH is a trapezoid, I is the midpoint of F�E�, and J is the midpoint of G�H�. Identify each of the following segments or angles in the figure.
a. the bases of trapezoid EFGH
b. the two pairs of base angles of trapezoid EFGH
c. the legs of trapezoid EFGH
d. the median of trapezoid EFGH
2. Determine whether each statement is true or false. If the statement is false, explain why.
a. A trapezoid is a special kind of parallelogram.
b. The diagonals of a trapezoid are congruent.
c. The median of a trapezoid is parallel to the legs.
d. The length of the median of a trapezoid is the average of the length of the bases.
e. A trapezoid has three medians.
f. The bases of an isosceles trapezoid are congruent.
g. An isosceles trapezoid has two pairs of congruent angles.
h. The median of an isosceles trapezoid divides the trapezoid into two smaller isoscelestrapezoids.
Helping You Remember
3. A good way to remember a new geometric theorem is to relate it to one you alreadyknow. Name and state in words a theorem about triangles that is similar to the theoremin this lesson about the median of a trapezoid.
HE
JIGF
© Glencoe/McGraw-Hill 452 Glencoe Geometry
Quadrilaterals in Construction
Quadrilaterals are often used in construction work.
1. The diagram at the right represents a roof frame and shows many quadrilaterals. Find the following shapes in the diagram and shade in their edges.
a. isosceles triangle
b. scalene triangle
c. rectangle
d. rhombus
e. trapezoid (not isosceles)
f. isosceles trapezoid
2. The figure at the right represents a window. The wooden part between the panes of glass is 3 inches wide. The frame around the outer edge is 9 inches wide. The outside measurements of the frame are 60 inches by 81 inches. The height of the top and bottom panes is the same. The top three panes are the same size.
a. How wide is the bottom pane of glass?
b. How wide is each top pane of glass?
c. How high is each pane of glass?
3. Each edge of this box has been reinforced with a piece of tape. The box is 10 inches high, 20 inches wide, and 12 inches deep.What is the length of the tape that has been used?
10 in.
12 in.
20 in.
60 in.
81 in.
9 in.
3 in.
3 in.
3 in.
9 in.
9 in.
jack rafters
hip rafter
common rafters
Roof Frame
ridge boardvalley rafter
plate
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-68-6
Study Guide and InterventionCoordinate Proof with Quadrilaterals
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 453 Glencoe Geometry
Less
on
8-7
Position Figures Coordinate proofs use properties of lines and segments to provegeometric properties. The first step in writing a coordinate proof is to place the figure on thecoordinate plane in a convenient way. Use the following guidelines for placing a figure onthe coordinate plane.
1. Use the origin as a vertex, so one set of coordinates is (0, 0), or use the origin as the center of the figure.
2. Place at least one side of the quadrilateral on an axis so you will have some zero coordinates.
3. Try to keep the quadrilateral in the first quadrant so you will have positive coordinates.
4. Use coordinates that make the computations as easy as possible. For example, use even numbers if you are going to be finding midpoints.
Position and label a rectangle with sides aand b units long on the coordinate plane.
• Place one vertex at the origin for R, so one vertex is R(0, 0).• Place side R�U� along the x-axis and side R�S� along the y-axis,
with the rectangle in the first quadrant.• The sides are a and b units, so label two vertices S(0, a) and
U(b, 0).• Vertex T is b units right and a units up, so the fourth vertex is T(b, a).
Name the missing coordinates for each quadrilateral.
1. 2. 3.
Position and label each quadrilateral on the coordinate plane.
4. square STUV with 5. parallelogram PQRS 6. rectangle ABCD withside s units with congruent diagonals length twice the width
x
y
D(2a, 0)
C(2a, a)B(0, a)
A(0, 0)x
y
S(a, 0)
R(a, b)Q(0, b)
P(0, 0)x
y
V(s, 0)
U(s, s)T(0, s)
S(0, 0)
x
y
D(a, 0)
C(?, c)B(b, ?)
A(0, 0)x
y
Q(a, 0)
P(a � b, c)N(?, ?)
M(0, 0)x
y
B(c, 0)
C(?, ?)D(0, c)
A(0, 0)
x
y
U(b, 0)
T(b, a)S(0, a)
R(0, 0)
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 454 Glencoe Geometry
Prove Theorems After a figure has been placed on the coordinate plane and labeled, acoordinate proof can be used to prove a theorem or verify a property. The Distance Formula,the Slope Formula, and the Midpoint Theorem are often used in a coordinate proof.
Write a coordinate proof to show that the diagonals of a square are perpendicular.The first step is to position and label a square on the coordinate plane. Place it in the first quadrant, with one side on each axis.Label the vertices and draw the diagonals.
Given: square RSTUProve: S�U� ⊥ R�T�
Proof: The slope of S�U� is �0a��
a0� � �1, and the slope of R�T� is �aa
��
00� � 1.
The product of the two slopes is �1, so S�U� ⊥ R�T�.
Write a coordinate proof to show that the length of the median of a trapezoid ishalf the sum of the lengths of the bases.
D(2a, 0)
B(2b, 2c)
M N
C(2d, 2c)
A(0, 0)
x
y
U(a, 0)
T(a, a)S(0, a)
R(0, 0)
Study Guide and Intervention (continued)
Coordinate Proof With Quadrilaterals
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
ExampleExample
ExerciseExercise
Skills PracticeCoordinate Proof with Quadrilaterals
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 455 Glencoe Geometry
Less
on
8-7
Position and label each quadrilateral on the coordinate plane.
1. rectangle with length 2a units and 2. isosceles trapezoid with height a units,height a units bases c � b units and b � c units
Name the missing coordinates for each quadrilateral.
3. rectangle 4. rectangle
5. parallelogram 6. isosceles trapezoid
Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.
7. The segments joining the midpoints of the sides of a rhombus form a rectangle.
B(0, 2b)
D(0, �2b)P
NM
Q
C(2a, 0)
A(�2a, 0)
x
y
T(b, 0)
W(?, c)Y(?, ?)
S(a, 0)Ox
y
S(a, ?)
R(?, ?)P(b, c)
T(0, 0)O
x
y
E(5b, 0)
F(5b, 2b)G(?, ?)
D(0, 0)Ox
y
U(c, 0)
C(?, ?)D(0, a)
A(0, 0)O
B(b � c, 0)
B(b, a) C(c, a)
A(0, 0)B(2a, 0)
D(0, a) C(2a, a)
A(0, 0)
© Glencoe/McGraw-Hill 456 Glencoe Geometry
Position and label each quadrilateral on the coordinate plane.
1. parallelogram with side length b units 2. isosceles trapezoid with height b units,and height a units bases 2c � a units and 2c � a units
Name the missing coordinates for each quadrilateral.
3. parallelogram 4. isosceles trapezoid
Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.
5. The opposite sides of a parallelogram are congruent.
6. THEATER A stage is in the shape of a trapezoid. Write a coordinate proof to prove that T�R� and S�F� are parallel.
x
y
T(10, 0)
S(30, 25)F(0, 25)
R(20, 0)O
B(a, 0)
D(b, c) C(a � b, c)
A(0, 0)
x
y
W(a, 0)
Z(b, c)Y(?, ?)
X(?, ?) Ox
y
J(2b, 0)
K(?, a)L(c, ?)
H(0, 0)O
B(a � 2c, 0)
D(a, b) C(2c, b)
A(0, 0)B(b, 0)
D(c, a) C(b � c, a)
A(0, 0)
Practice Coordinate Proof with Quadrilaterals
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
Reading to Learn MathematicsCoordinate Proof with Quadrilaterals
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
© Glencoe/McGraw-Hill 457 Glencoe Geometry
Less
on
8-7
Pre-Activity How can you use a coordinate plane to prove theorems aboutquadrilaterals?
Read the introduction to Lesson 8-7 at the top of page 447 in your textbook.
What special kinds of quadrilaterals can be placed on a coordinate system sothat two sides of the quadrilateral lie along the axes?
Reading the Lesson1. Find the missing coordinates in each figure. Then write the coordinates of the four
vertices of the quadrilateral.
a. isosceles trapezoid b. parallelogram
2. Refer to quadrilateral EFGH.
a. Find the slope of each side.
b. Find the length of each side.
c. Find the slope of each diagonal.
d. Find the length of each diagonal.
e. What can you conclude about the sides of EFGH?
f. What can you conclude about the diagonals of EFGH?
g. Classify EFGH as a parallelogram, a rhombus, or a square. Choose the most specificterm. Explain how your results from parts a-f support your conclusion.
Helping You Remember3. What is an easy way to remember how best to draw a diagram that will help you devise
a coordinate proof?
x
y
O
E
FG
H
x
y
Q(b, ?)
P(?, c � a)
N(?, a)
M(?, ?)Ox
y
U(a, ?)
T(b, ?)S(?, c)
R(?, ?) O
© Glencoe/McGraw-Hill 458 Glencoe Geometry
Coordinate ProofsAn important part of planning a coordinate proof is correctly placing and labeling thegeometric figure on the coordinate plane.
Draw a diagram you would use in a coordinate proof of thefollowing theorem.
The median of a trapezoid is parallel to the bases of the trapezoid.
In the diagram, note that one vertex, A, is placed at the origin. Also,the coordinates of B, C, and D use 2a, 2b, and 2c in order to avoid the use of fractions when finding the coordinates of the midpoints,M and N.
When doing coordinate proofs, the following strategies may be helpful.
1. If you are asked to prove that segments are parallel or perpendicular, use slopes.
2. If you are asked to prove that segments are congruent or have related measures,use the distance formula.
3. If you are asked to prove that a segment is bisected, use the midpoint formula.
For each of the following theorems, a diagram has been provided to be used in aformal proof. Name the missing coordinates in the diagram. Then, using the Givenand the Prove statements, prove the theorem.
1. The median of a trapezoid is parallel to the bases of the trapezoid. (Use the diagramgiven in the example above.)
Coordinates of M and N:
Given: Trapezoid ABCD has median M�N�.Prove: B�C� || M�N� and A�D� || M�N�
Proof:
2. The medians to the legs of an isosceles triangle are congruent.
Coordinates of T and K:
Given: �ABC is isosceles with medians T�B� and K�A�.Prove: T�B� � K�A�
Proof:x
y
O
T K
A(0, 0) B(4a, 0)
C(2a, 2b)
x
y
O
M N
A(0, 0) D(2a, 0)
C(2d, 2c)B(2b, 2c)
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
8-78-7
ExampleExample
Chapter 8 Test, Form 188
© Glencoe/McGraw-Hill 459 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Find the sum of the measures of the interior angles of a convex 30-gon.A. 5400 B. 5040 C. 360 D. 168
2. Find the sum of the measures of the exterior angles of a convex 21-gon.A. 21 B. 180 C. 360 D. 3420
3. If the measure of each interior angle of a regular polygon is 108, find themeasure of each exterior angle.A. 18 B. 72 C. 90 D. 108
4. For parallelogram ABCD, find x.A. 4 B. 10.25C. 16 D. 21.5
5. Which of the following is a property of a parallelogram?A. The diagonals are congruent. B. The diagonals bisect the angles.C. The diagonals are perpendicular. D. The diagonals bisect each other.
6. Find x and y so that ABCD will be a parallelogram.A. x � 6, y � 42B. x � 6, y � 22C. x � 20, y � 42D. x � 20, y � 22
7. Find x so that this quadrilateral is a parallelogram.A. 44 B. 46C. 90 D. 134
8. ABCD is a parallelogram with A(0, 0), B(2, 4), and C(10, 4). Find the possiblecoordinates of D.A. (8, 0) B. (10, 0) C. (0, 4) D. (10, 8)
9. Which of the following is a property of all rectangles?A. four congruent sides B. diagonals bisect the anglesC. diagonals are perpendicular D. four right angles
10. ABCD is a rectangle with diagonals A�C� and B�D�. If AC � 2x � 10 and BD � 56, find x.A. 23 B. 33 C. 78 D. 122
11. ABCD is a rectangle with B(�5, 0), C(7, 0) and D(7, 3). Find the coordinates of A.A. (�5, 7) B. (3, 5)C. (�5, 3) D. (7, �3)
x� 134�
134� 46�
4x�(y � 10)�32�
24�
5x � 12
3x � 20
A D
CB
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 460 Glencoe Geometry
Chapter 8 Test, Form 1 (continued)88
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. Given rhombus ABCD, find m�1.A. 45 B. 60C. 90 D. 120
13. Find m�PRS in square PQRS.A. 30 B. 45C. 60 D. 90
14. Choose a pair of base angles of trapezoid ABCD.A. �A, �C B. �B, �DC. �A, �D D. �D, �C
15. In trapezoid DEFG, find m�D.A. 44 B. 72C. 108 D. 136
16. The bases of a trapezoid are 12 and 26. Find the length of the median.A. 12 B. 19 C. 26 D. 38
17. If the length of one base of a trapezoid is 44, the median is 36, and the otherbase is 2x � 10, find x.A. 9 B. 17 C. 21 D. 40
18. Given trapezoid ABCD with median E�F�, which of the following is true?
A. EF � �12�AD B. AE � FD
C. AB � EF D. EF � �BC �
2AD
�
19. ABCD is a rectangle with A(0, 0), B(b, 0), and D(0, a). Find the coordinates of C.A. C(a, b) B. C(b, a) C. C(�b, a) D. C(a � b, a)
20. To prove that the diagonals of a square bisect each other, you would position andlabel a square in the coordinate plane and then find which of the following?A. measures of the angles B. midpoints of the diagonalsC. lengths of the diagonals D. slopes of the diagonals
Bonus Find x and m�WYZ in rhombus XYZW.
X Y
ZW
(3x � 20)�
(x2)�
DA
B CFE
D G
E F136�
72�
D C
A B
P S
Q R
1
A D
CB
B:
NAME DATE PERIOD
Chapter 8 Test, Form 2A88
© Glencoe/McGraw-Hill 461 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Find the sum of the measures of the interior angles of a convex 45-gon.A. 8100 B. 7740 C. 360 D. 172
2. Find x.A. 30 B. 66C. 102 D. 138
3. Find the sum of the measures of the exterior angles of a convex 39-gon.A. 39 B. 90 C. 180 D. 360
4. Which of the following is a property of a parallelogram?A. Each pair of opposite sides is congruent.B. Only one pair of opposite angles is congruent.C. Each pair of opposite angles is supplementary.D. There are four right angles.
5. Find m�1 in parallelogram ABCD.A. 60 B. 54C. 36 D. 18
6. ABCD is a parallelogram with diagonals intersecting at E. If AE � 3x � 12and EC � 27, find x.A. 5 B. 17 C. 27 D. 47
7. Find x and y so that this quadrilateral is a parallelogram.A. x � 13, y � 24 B. x � 13, y � 6C. x � 7, y � 24 D. x � 7, y � 6
8. Find x so that this quadrilateral is a parallelogram.A. 12 B. 24C. 36 D. 132
9. Given A(8, 2), B(6, �4), C(�5, �4), find the coordinates of D so that ABCD isa parallelogram.A. D(�5, 2) B. D(�3, 2) C. D(�2, 2) D. D(�4, 8)
10. ABCD is a rectangle. If AC � 5x � 2 and BD � x � 22, find x.A. 5 B. 6 C. 11 D. 26
11. Which of the following is true for all rectangles?A. The diagonals are perpendicular.B. The diagonals bisect the angles.C. The consecutive sides are congruent.D. The consecutive sides are perpendicular.
(2x � 60)�
(4x � 12)�
(2x � 30)�
(5x � 9)�
(y � 12)�1–2y �
120�
36�
1
A D
CB
(2x � 10)�(x � 40)�
(x � 20)�x�
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 462 Glencoe Geometry
Chapter 8 Test, Form 2A (continued)88
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. ABCD is a rectangle with B(�4, 6), C(�4, 2), and D(10, 2). Find thecoordinates of A.A. A(6, 4) B. A(10, 4) C. A(2, 6) D. A(10, 6)
13. Find m�1 in rhombus GHJK.A. 22 B. 44C. 68 D. 90
14. The diagonals of square ABCD intersect at E. If AE � 2x � 6 and BD � 6x � 10, find AC.A. 11 B. 28 C. 56 D. 90
15. ABCD is an isosceles trapezoid with A(10, �1), B(8, 3), and C(�1, 3). Find thecoordinates of D.A. (�3, �1) B. (�10, �11) C. (�1, 8) D. (�3, 3)
16. For isosceles trapezoid MNOP, find m�MNP.A. 44 B. 64C. 80 D. 116
17. The length of one base of a trapezoid is 19 inches and the length of themedian is 16 inches. Find the length of the other base.A. 35 in. B. 19 in. C. 17.5 in. D. 13 in.
18. E�F� is the median of isosceles trapezoid ABCD.Find x.A. 2x � 46 B. 32C. 46 D. 68
19. What type of quadrilateral has vertices at (0, 0), (a, b), (c, b), and (c � a, 0)?A. parallelogram B. rectangleC. rhombus D. trapezoid
20. To prove that the diagonals of a rhombus are perpendicular to each other, youwould position and label a rhombus on a coordinate plane and then findwhich of the following?A. measures of the angles B. slopes of the diagonalsC. lengths of the diagonals D. midpoints of the diagonals
Bonus Find the possible value(s) of x in rectangle JKLM.
J K
LM
x2 � 8
3x � 36
A D
CB
E F3x � 72
x � 20
P
ON
M 36�44�
22�
1G H
JK
B:
NAME DATE PERIOD
Chapter 8 Test, Form 2B88
© Glencoe/McGraw-Hill 463 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Find the sum of the measures of the interior angles of a convex 50-gon.A. 9000 B. 8640 C. 360 D. 172.8
2. Find x.A. 16 B. 34C. 50 D. 70
3. Find the sum of the measures of the exterior angles of a convex 65-gon.A. 5.54 B. 90 C. 180 D. 360
4. Which of the following is a property of all parallelograms?A. Each pair of opposite angles is congruent.B. Only one pair of opposite sides is congruent.C. Each pair of opposite angles is supplementary.D. There are four right angles.
5. Find m�1 in parallelogram ABCD.A. 19 B. 38C. 52 D. 56
6. ABCD is a parallelogram with diagonals intersecting at E. If AE � 4x � 8and EC � 36, find x.A. 7 B. 11 C. 15.5 D. 38
7. Find x and y so that this quadrilateral is a parallelogram.A. x � 27, y � 90 B. x � 27, y � 40C. x � 13, y � 90 D. x � 13, y � 40
8. Find x so that this quadrilateral is a parallelogram.
A. 7�13� B. 8
C. 12 D. 66
9. ABCD is a parallelogram with A(5, 4), B(�1, �2), C(8, �2). Find thecoordinates of D.A. D(�5, 4) B. D(8, 2) C. D(14, 4) D. D(4, 1)
10. ABCD is a rectangle. If AB � 7x � 6 and CD � 5x � 30, find x.
A. 5�13� B. 12 C. 13 D. 18
11. Which of the following is true for all rectangles?A. The diagonals are perpendicular.B. The consecutive angles are supplementary.C. The opposite sides are supplementary.D. The opposite angles are complementary.
6x � 6 3x � 30
(4x � 4)�
(3x � 23)�
(y � 60)�
1–3y �
38�
124�1
A D
B C
(2x � 10)�
(x � 30)�
(x � 70)� x �
2x �
2x �
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 464 Glencoe Geometry
Chapter 8 Test, Form 2B (continued)88
12.
13.
14.
15.
16.
17.
18.
19.
20.
12. ABCD is a rectangle with B(�7, 3), C(5, 3), and D(5, �8). Find the coordinatesof A.A. (�8, �7) B. (�7, �8) C. (�5, �3) D. (�8, �5)
13. Find m�1 in rhombus GHJK.A. 90 B. 64C. 52 D. 38
14. The diagonals of square ABCD intersect at E. If AE � 3x � 4 and BD � 10x � 48, find AC.A. 90 B. 52 C. 26 D. 10
15. ABCD is an isosceles trapezoid with A(0, �1), B(�2, 3), and D(6, �1). Findthe coordinates of C.A. C(6, 1) B. C(9, 4) C. C(2, 3) D. C(8, 3)
16. Find m�MNP in isosceles trapezoid MNOP.A. 42 B. 70C. 82 D. 98
17. The length of one base of a trapezoid is 19 meters and the length of themedian is 23 meters. Find the length of the other base.A. 15 m B. 21 m C. 27 m D. 42 m
18. E�F� is the median of isosceles trapezoid ABCD.Find x.A. 22 B. 18.5C. 42.5 D. 82
19. What type of quadrilateral has vertices at (0, 0), (a, b), (a � c, b), and (c, 0)?A. parallelogram B. rectangleC. rhombus D. trapezoid
20. To prove that the diagonals of a rectangle are congruent, you would positionand label a rectangle on a coordinate plane and then find which of thefollowing?A. measures of the angles B. slopes of the diagonalsC. lengths of the diagonals D. midpoints of the diagonals
Bonus The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of the polygon.
A D
CB
E F3x � 52
x � 15
28�42�
P
ON
M
128�
1
G K
JH
B:
NAME DATE PERIOD
Chapter 8 Test, Form 2C88
© Glencoe/McGraw-Hill 465 Glencoe Geometry
Ass
essm
ents
1. Find the sum of the measures of the interior angles of aconvex 60-gon.
2. A convex pentagon has interior angles with measures (5x � 12)°,(2x � 100)°, (4x � 16)°, (6x � 15)°, and (3x � 41)°. Find x.
3. If the measure of each interior angle of a regular polygon is171, find the number of sides of the polygon.
4. In parallelogram ABCD,m�1 � x � 12, and m�2 � 6x � 18.Find m�1.
5. Find the measure of each exterior angle of a regular 45-gon.
6. In parallelogram ABCD, m�A � 58. Find m�B.
7. Find the coordinates of the intersection of the diagonals ofparallelogram XYZW with vertices X(2, 2), Y(3, 6), Z(10, 6),and W(9, 2).
8. Determine whether ABCD is a parallelogram. Justify your answer.
9. Use the Slope Formula to determine whether A(5, 7), B(1, �2),C(�6, �3), and D(2, 5) are the coordinates of the vertices ofparallelogram ABCD.
10. If the slope of A�B� is �14�, the slope of B�C� is ��
23�, and the slope of
C�D� is �14�, find the slope of D�A� so that ABCD will be a
parallelogram.
11. Given rectangle ABCD, find x.
12. ABCD is a parallelogram and A�C� � B�D�. Determine whetherABCD is a rectangle. Justify your answer.
13. ABCD is a rhombus with diagonals intersecting at E. Ifm�ABC � 3m�BAD, find m�EBC.
A B
CD
(2x � 10)�
(3x � 30)�
A D
CB
12 C
D
A
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 466 Glencoe Geometry
Chapter 8 Test, Form 2C (continued)88
14. TUVW is a square with U(10, 2), V(8, 8), and W(2, 6). Find thecoordinates of T.
15. Find m�MNQ in isosceles trapezoid MNOP.
16. ABCD is a quadrilateral with A(8, 3), B(6, 7), C(�1, 5), andD(�6, �1). Determine whether ABCD is a trapezoid. Justifyyour answer.
17. The length of the median of trapezoid EFGH is 13 feet. If thebases have lengths 2x � 4 and 10x � 50, find x.
18. Name the missing coordinates for rhombus ABCD. Thendetermine the relationshipbetween A�C� and B�D�.
For Questions 19–25, write true or false.
19. A rectangle is always a parallelogram.
20. The diagonals of a rhombus are always perpendicular.
21. The diagonals of a square always bisect each other.
22. A trapezoid always has two congruent sides.
23. The median of a trapezoid is always parallel to the bases.
24. A quadrilateral with vertices (a, 0), (b, c), (�b, c), and (�a, 0)is an isosceles trapezoid.
25. If the diagonals of a parallelogram are perpendicular, then theparallelogram is a rectangle.
Bonus In parallelogram ABCD, AB � 2x � 7, BC � x � 3y,CD � x � y, and AD � 2x � y � 1. Find x and y.
x
y
A(?, ?)
B(0, b)
C(�a, 0)
D(0, �b)
O
N
Q
O
PM 59�
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
NAME DATE PERIOD
Chapter 8 Test, Form 2D88
© Glencoe/McGraw-Hill 467 Glencoe Geometry
Ass
essm
ents
1. Find the sum of the measures of the interior angles of aconvex 36-gon.
2. A convex octagon has interior angles with measures (x � 55)°,(3x � 20)°, 4x°, (4x � 10)°, (6x � 55)°, (3x � 52)°, 3x°, and (2x � 30)°. Find x.
3. If the measure of each interior angle of a regular polygon is 176find the number of sides on the polygon.
4. In parallelogram ABCD,m�1 � x � 25, and m�2 � 2x.Find m�2.
5. Find the measure of each exterior angle of a regular 100-gon.
6. If m�A � 63 in parallelogram ABCD, find m�B.
7. Find the coordinates of the intersection of the diagonals ofparallelogram XYZW with vertices X(3, 0), Y(3, 8), Z(�2, 6),and W(�2, �2).
8. Determine whether this quadrilateral is a parallelogram. Justify your answer.
9. Use the Slope Formula to determine whether A(5, 7), B(1, �1),C(�6, �3), and D(�2, 5) are the coordinates of the vertices ofa parallelogram.
10. If the slope of A�B� is �12�, the slope of B�C� is �4, and the slope of
C�D� is �12�, find the slope of D�A� so that ABCD is a parallelogram.
11. Given rectangle ABCD, find x.
12. ABCD is a parallelogram and m�A � 90. Determine whetherABCD is a rectangle. Justify your answer.
B C
DA
(6x � 20)�
(3x � 2)�
1
2A D
CB
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 468 Glencoe Geometry
Chapter 8 Test, Form 2D (continued)88
13. ABCD is a rhombus with diagonals intersecting at E. Ifm�ABC � 4m�BAD, find m�EBC.
14. PQRS is a square with Q(�2, 8), R(5, 7), and S(4, 0). Find thecoordinates of P.
15. Find m�MNQ in isosceles trapezoid MNOP.
16. ABCD is a quadrilateral with A(8, �1), B(10, �7), C(�6, �1),and D(0, 2). Determine whether ABCD is a trapezoid. Justifyyour answer.
17. The length of the median of trapezoid EFGH is 17 centimeters.If the bases have lengths 2x � 4 and 8x � 50, find x.
18. Name the missing coordinates for square ABCD. Then determine thecoordinates of the midpoints of thediagonals.
For Questions 19–25, write true or false.
19. A parallelogram always has four right angles.
20. The diagonals of a rhombus always bisect the angles.
21. A rhombus is always a square.
22. A rectangle is always a square.
23. The diagonals of an isosceles trapezoid are always congruent.
24. The median of a trapezoid always bisects the angles.
25. A quadrilateral with vertices (0, 0), (a, 0), (a � c, b), and (b, c)is a parallelogram.
Bonus The measure of each interior angle of a regular polygon is24 more than 38 times the measure of each exterior angle.Find the number of sides of the polygon.
x
y
A(a, 0)
C(0, a)
D(0, 0)
B(?, ?)
O
M P
ON
Q74�
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
NAME DATE PERIOD
Chapter 8 Test, Form 388
© Glencoe/McGraw-Hill 469 Glencoe Geometry
Ass
essm
ents
1. Find the sum of the measures of the interior angles of aconvex 24-gon.
2. A convex hexagon has interior angles with measures x°,(5x � 103)°, (2x � 60)°, (7x � 31)°, (6x � 6)°, and (9x � 100)°.Find x and the measure of each angle.
3. Find the measure of each exterior angle of a regular 2x-gon.
4. Find m�1 in parallelogram ABCD.
5. ABCD is a parallelogram with diagonals that intersect eachother at E. If AE � x2 and EC � 6x � 8, find all possiblevalues of AC.
6. Determine whether this quadrilateral is a parallelogram. Justify your answer.
7. Given the slope of A�B� is �23� and the slope of B�C� is �2, find the
slopes of C�D� and D�A� so that ABCD will be a parallelogram.
8. In rectangle ABCD, find m�1.
9. The diagonals of rhombus ABCD intersect at E. If m�BAE �
�23�(m�ABE), find m�BCD.
10. The diagonals of square ABCD intersect at E. If AE � 2, findthe perimeter of ABCD.
11. Find AE in isosceles trapezoid ABCD.
12. Points G and H are midpoints of A�F� and D�E� inregular hexagon ABCDEF. If AB � 6 find GH.
B C
F
A DHG
E
A D
CB
E F60�
8
B C
DA(3x � 16)�
1(2x � 10)�
96� 31�1
D C
BA
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 470 Glencoe Geometry
Chapter 8 Test, Form 3 (continued)88
13. If the vertices of trapezoid ABCD are A(10, �1), B(6, 6),C(�8, �1), and D(�2, 6), find the length of the median.
14. A parallelogram has three vertices (0, 0), (b, c), and (d, 0).What are the possible coordinates of the fourth vertex?
15. Determine whether ABCD is a rectangle given A(6, 2), B(2, 10),C(�6, 6), and D(�2, �2). Justify your answer.
16. Use the Distance Formula to determine whether ABCD is aparallelogram given A(1, 6), B(7, 6), C(2, �3), and D(�4, �3).
17. Name the missing coordinates for ABCD. Then determine therelationship between AC and thesegment formed by connecting the midpoints of B�C� and A�B�.
For Questions 18 and 19, complete the two-column proof bysupplying the missing information for each correspondinglocation.Given: ABCD is a parallelogram.
B�Q� � D�S�, P�A� � R�C�Prove: PQRS is a parallelogram.
Statements Reasons
1. ABCD is a �. 1. Given2. A�D� � C�B� 2.3. P�A� � R�C� 3. Given4. P�D� � R�B� 4. Seg. Sub. Prop.5. A�B� � C�D� 5. Opp. sides of a � are �.6. B�Q� � D�S� 6. Given7. A�Q� � C�S� 7. Seg. Sub. Prop.8. �B � �D, �A � �C 8. Opp. � of a � are �.9. �QBR � �SDP, �PAQ � �RCS 9. SAS
10. Q�P� � R�S�, Q�R� � P�S� 10. CPCTC11. PQRS is a parallelogram. 11.
20. In isosceles trapezoid ABCD, AE � 2x � 5,EC � 3x � 12, and BD � 4x � 20. Find x.
Bonus If three of the interior angles of a convex hexagon eachmeasure 140, a fourth angle measures 84, and themeasure of the fifth angle is 3 times the measure of thesixth angle, find the measure of the sixth angle.
B C
EDA
(Question 19)
(Question 18)
A P
R
D
C
SQ
B
x
y
A(2a, 0)
B(0, 2b)
C(?, ?)
D(0, �2b)
O
13.
14.
15.
16.
17.
18.
19.
20.
B:
NAME DATE PERIOD
Chapter 8 Open-Ended Assessment88
© Glencoe/McGraw-Hill 471 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. a. Draw a regular convex polygon and a convex polygon that is notregular, each with the same number of sides.
b. Label the measures of each exterior angle on your figures.
c. Find the sum of the exterior angles for each figure. What conjecture canbe made?
2. Draw a rectangle. Connect the midpoints of the consecutive sides. Whattype of quadrilateral is formed? How do you know?
3. Draw an example to show why one pair of opposite sides congruent and theother pair of opposite sides parallel is not sufficient to form aparallelogram.
4. a. Name a property that is true for a square and not always true for arectangle.
b. Name a property that is true for a square and not always true for arhombus.
c. Name a property that is true for a rectangle and not always true for aparallelogram.
5. John drew this figure to use in a coordinate proof.Explain whether John’s diagram could actually be used to prove that the diagonals of a rhombus are perpendicular.
x
y
D(a, 0)
B(b, c)
A(0, 0)
C(a � b, c)
O
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 472 Glencoe Geometry
Chapter 8 Vocabulary Test/Review88
Choose from the terms above to complete each sentence.
1. A quadrilateral with only one pair of opposite sides paralleland the other pair of opposite sides congruent is a(n) .
2. A quadrilateral with two pairs of opposite sides parallel is a(n).
3. A quadrilateral with only one pair of opposite sides parallel isa(n) .
4. A quadrilateral that is both a rectangle and a rhombus is a(n).
5. A quadrilateral with four congruent sides is a(n) .
6. A quadrilateral with four right angles is a(n) .
7. A quadrilateral with two pairs of congruent consecutive sidesis a(n) .
8. Segments that join opposite vertices in a quadrilateral arecalled .
9. The segment joining the midpoints of the nonparallel sides ofa trapezoid is called the .
10. The parallel sides of a trapezoid are called the .
Define each term.
11. base angles of an isosceles trapezoid
12. legs of a trapezoid
?
?
?
?
?
?
?
?
?
?1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
basesdiagonalsisosceles trapezoid
kitemedianparallelogram
rectanglerhombus
squaretrapezoid
NAME DATE PERIOD
SCORE
Chapter 8 Quiz (Lessons 8–1 and 8–2)
88
© Glencoe/McGraw-Hill 473 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
1. Find the sum of the measures of the interior angles of aconvex 70-gon.
2. If the measure of each interior angle of a regular polygon is172, find the number of sides of the polygon.
3. If the measure of each exterior angle of a regular polygon is18, find the number of sides of the polygon.
4. Given parallelogram ABCD with C(5, 4), find the coordinatesof A if the diagonals intersect at (2, 7).
5. STANDARDIZED TEST PRACTICE Find m�1 in parallelogram ABCD.A. 64 B. 58C. 46 D. 36
110�
46� 1
D C
BA
Chapter 8 Quiz (Lessons 8–3 and 8–4)
88
1.
2.
3.
4.
5.
1. Determine whether this quadrilateral is a parallelogram. Justify your answer.
For Questions 2–4, write true or false.
2. A quadrilateral with two pairs of parallel sides is always arectangle.
3. The diagonals of a parallelogram are always perpendicular.
4. The slope of A�B� and C�D� is �35� and the slope of B�C� and A�D� is
��53�. ABCD is a parallelogram.
5. Find m�1, m�2, and m�3 in the rectangle at the right.
34�
1
2
3
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 474 Glencoe Geometry
Chapter 8 Quiz (Lessons 8–5 and 8–6)
88
1.
2.
3.
4.
5.
1. Given M(4, 0), N(0, 6), P(�4, 0) and Q(0, �6), determine whetherMNPQ is a trapezoid, a parallelogram, a square, a rhombus, or aquadrilateral. Choose the most specific term. Explain.
For Questions 2 and 3, use trapezoid MNPQ.
2. Find m�N.
3. Find m�Q.
4. True or false. A quadrilateral that is a rectangle and arhombus is a square.
5. Find x if the bases of a trapezoid have lengths 2x � 4 and 8x � 10 and the length of the median is 3x � 21.
M Q
PN 130�
72�
NAME DATE PERIOD
SCORE
Chapter 8 Quiz (Lesson 8–7)
88
1.
2.
3.
4.
5.
1. ABCD is a rhombus with A(a, 0), B(0, b), and D(0, �b) Findthe possible coordinates of C.
2. ABCD is a square with A(a, 0), B(0, a), and C(�a, 0). Find the possible coordinates of D.
3. Given rectangle ABCD, find the coordinates of A.
4. Name the coordinates of the endpoints of the median of trapezoid ABCD.
5. Name the missing coordinates for parallelogram ABCD. Thendetermine the lengths of each of the four sides.
x
y
D(a, 0)
B(b, c)
A(0, 0)
C(?, ?)
O
x
y
A(2a, 0)
C(�2d, 2c)
Y X
D(�2e, 0)
B(2b, 2c)
O
x
y
A(?, ?)
C(a, �b)D(�d, �b)
B(a, c)
O
NAME DATE PERIOD
SCORE
Chapter 8 Mid-Chapter Test (Lessons 8–1 through 8–4)
88
© Glencoe/McGraw-Hill 475 Glencoe Geometry
Ass
essm
ents
1. Find the measure of each exterior angle of a regular 56-gon. Round to thenearest tenth.A. 3.2 B. 6.4 C. 173.6 D. 9720
2. Given BE � 2x � 6 and ED � 5x � 12 in parallelogram ABCD, find BD.A. 6 B. 12C. 18 D. 36
3. If the slope of P�Q� is �23� and the slope of R�S� is �
23�, find the slope of Q�R� and S�P�
so that PQRS is a rectangle.
A. ��32� B. �
32� C. ��
23� D. �
23�
4. Find m�1 in rectangle RSTW.A. 163 B. 146C. 34 D. 17
5. Find the sum of the measures of the interior angles of a convex 48-gon.A. 172.5 B. 360 C. 8280 D. 8640
R S
TW
17�
1
D
ECB
A
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
Part II
6. Find x.
7. ABCD is a parallelogram with m�A � 138. Find m�B.
8. Determine whether ABCD is a parallelogram if AB � 6,BC � 12, CD � 6, and DA � 12.
9. In rectangle XYZW, find m�1, m�2,m�3, and m�4.
10. In quadrilateral ABCD, B�C� � A�D� and B�D� and E�F� bisect each other at G.By what theorem is ABCD aparallelogram? D
E
F
G
CB
A
Y Z
WX12� 1 2
34
(3x � 2)�(2x � 12)�
(x � 10)�(x � 30)�
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 476 Glencoe Geometry
Chapter 8 Cumulative Review(Chapters 1–8)
88
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
1. Point B lies on RT��� between points R and T. Name twoopposite rays. (Lesson 1-4)
2. Write an equation in slope-intercept form for the line that contains (�12, 9) and is perpendicular to the line y � �
23�x � 5.
(Lesson 3-4)
3. If �ABC � �WXY, AB � 72, BC � 65, CA � 13, XY � 7x � 12,and WX � 19y � 34, find x and y. (Lesson 4-3)
4. Freda bought two bells for just over $90 before tax. State theassumption you would make to write an indirect proof to showthat at least one of the bells costs more than $45. (Lesson 5-3)
5. A plot of land has dimensions 82 feet, 132 feet, 75 feet, and120 feet. A draftsman applies a scale of 2 inches � 8 feet tomake a scale drawing of the land. Find the dimensions of thedrawing. (Lesson 6-2)
6. Triangle QRT has vertices Q(1, 2), R(5, �1), and T(�2, �4),and A�B� is a midsegment parallel to R�T�. Find the coordinatesof A and B. (Lesson 6-4)
7. Name the angles of elevation and depression. (Lesson 7-5)
8. Use the Law of Sines to find c to the nearest whole number. (Lesson 7-6)
For Questions 10 and 11, use rectangle ABDC and parallelogram CDGF.
9. If m�CDF � 21, m�FDG � 34,CF � 3x � 8, and DG � 7, find m�GFC,m�FCD, and x. (Lesson 8-2)
10. Find AE if AD � 12y � 7 and BC � 21 � 2y. (Lesson 8-4)
11. Quadrilateral LMNP has vertices L(3, 0), M(7, �3), N(�7, �9),and P(�4, �3). Verify that LMNP is a trapezoid. Explain.(Lesson 8-6)
A B
DC
F G
E
A
B C
bc
69�
96�
46.89
W X
ZY
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–8)
© Glencoe/McGraw-Hill 477 Glencoe Geometry
1. Find the coordinates of X if V(0.5, 5) is the midpoint of U�X� withU(15, �10). (Lesson 1-3)
A. (�14, 20) B. (7.75, �2.5)C. (0, 0) D. (15.5, �5)
2. Which of the following are possible measures for vertical angles Gand H? (Lesson 2-8)
E. m�G � 125 and m�H � 55F. m�G � 125 and m�H � 125G. m�G � 55 and m�H � 45H. m�G � 55 and m�H � 152.5
3. Determine which lines are parallel.(Lesson 3-5)
A. NS��� || PT��� B. NP��� || ST���
C. QR��� || ST��� D. NP��� || QR���
4. Find the coordinates of B, the midpoint of A�C�, if A(2a, b) and C(0, �b). (Lesson 4-7)
E. (2a, �b) F. (a, b) G. (a, 0) H. (0, b)
5. If R�V� is an angle bisector, find m�UVT. (Lesson 5-1)
A. 10 B. 34C. 68 D. 136
6. Which is not a valid postulate or theorem for proving twotriangles similar? (Lesson 6-3)
E. SSA Theorem F. SAS TheoremG. SSS Theorem H. AA Postulate
7. Find m�K to the nearest tenth.(Lesson 7-7)
A. 22.3 B. 32.6C. 54.8 D. 125.2
8. Which statement ensures that quadrilateral QRST is a parallelogram? (Lesson 8-3)
E. �Q � �S F. Q�R� � T�S� and Q�R� || T�S�G. Q�T� || R�S� H. m�Q � m�S � 180
Q R
ST
41
2719H
J K
(2y � 14)�
(4y � 6)�T VS
R
U
67�
113�
65�
N P
ST
RQ
NAME DATE PERIOD
SCORE 88
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7.
8. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
© Glencoe/McGraw-Hill 478 Glencoe Geometry
Standardized Test Practice (continued)
9. AB��� || JK���. Find the slope of AB��� if J(14, 8) andK(�7, 5). (Lesson 3-3)
10. If �UVW is an isosceles triangle, U�V� � W�U�,UV � 16b � 40, VW � 6b, and WU � 10b � 2,find b. (Lesson 4-1)
11. If �RTU � �GEF and S�U� and F�H� arealtitudes of �RTU and �GEF respectively, findFH. (Lesson 6-5)
12. Find the geometric mean between 25 and 4.(Lesson 7-1)
13. Find the sum of the measures of the interiorangles for a convex heptagon. (Lesson 8-1)
35 50 16
ST
E
FGH
R
U
NAME DATE PERIOD
88
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
9. 11.
11. 12.
13.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
14. A polygon has six congruent sides. Lines containing two of itssides contain points in its interior. Name the polygon by itsnumber of sides, and then classify it as convex or concave andregular or irregular. (Lesson 1-6)
15. If R�T� � Q�M� and RT � 88.9 centimeters, find QM. (Lesson 2-7)
16. Which segment is the shortest segment from D to JM���? (Lesson 5-4)
D
J K L H M
14.
15.
16.
1 / 7 7
1 1 . 2
9 0 0
1 0
Standardized Test PracticeStudent Record Sheet (Use with pages 458–459 of the Student Edition.)
88
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5
3 6 DCBADCBA
DCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Question 11, also enter your answer by writing each number or symbol in abox. Then fill in the corresponding oval for that number or symbol.
8 11
9
10
11 (grid in)
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Record your answers for Questions 12–13 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
An
gle
s o
f P
oly
go
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill41
7G
lenc
oe G
eom
etry
Lesson 8-1
Sum
of
Mea
sure
s o
f In
teri
or
An
gle
sT
he
segm
ents
th
at c
onn
ect
the
non
con
secu
tive
sid
es o
f a
poly
gon
are
cal
led
dia
gon
als.
Dra
win
g al
l of
th
e di
agon
als
from
one
vert
ex o
f an
n-g
onse
para
tes
the
poly
gon
in
to n
�2
tria
ngl
es.T
he
sum
of
the
mea
sure
sof
th
e in
teri
or a
ngl
es o
f th
e po
lygo
n c
an b
e fo
un
d by
add
ing
the
mea
sure
s of
th
e in
teri
oran
gles
of
thos
e n
�2
tria
ngl
es.
Inte
rio
r A
ng
leIf
a co
nvex
pol
ygon
has
nsi
des,
and
Sis
the
sum
of
the
mea
sure
s of
its
inte
rior
angl
es,
Su
m T
heo
rem
then
S�
180(
n�
2).
A c
onve
x p
olyg
on h
as
13 s
ides
.Fin
d t
he
sum
of
the
mea
sure
sof
th
e in
teri
or a
ngl
es.
S�
180(
n�
2)�
180(
13 �
2)�
180(
11)
�19
80
Th
e m
easu
re o
f an
inte
rior
an
gle
of a
reg
ula
r p
olyg
on i
s12
0.F
ind
th
e n
um
ber
of
sid
es.
Th
e n
um
ber
of s
ides
is
n,s
o th
e su
m o
f th
em
easu
res
of t
he
inte
rior
an
gles
is
120n
.S
�18
0(n
�2)
120n
�18
0(n
�2)
120n
�18
0n�
360
�60
n�
�36
0n
�6
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d t
he
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es o
f ea
ch c
onve
x p
olyg
on.
1.10
-gon
2.16
-gon
3.30
-gon
1440
2520
5040
4.8-
gon
5.12
-gon
6.3x
-gon
1080
1800
180(
3x�
2)
Th
e m
easu
re o
f an
in
teri
or a
ngl
e of
a r
egu
lar
pol
ygon
is
give
n.F
ind
th
e n
um
ber
of s
ides
in
eac
h p
olyg
on.
7.15
08.
160
9.17
5
1218
72
10.1
6511
.168
.75
12.1
35
2432
8
13.F
ind
x.
20
7x�
( 4x
� 1
0)�
( 4x
� 5
) � ( 6x
� 1
0)�
( 5x
� 5
) �
AB
C
D
E
©G
lenc
oe/M
cGra
w-H
ill41
8G
lenc
oe G
eom
etry
Sum
of
Mea
sure
s o
f Ex
teri
or
An
gle
sT
her
e is
a s
impl
e re
lati
onsh
ip a
mon
g th
eex
teri
or a
ngl
es o
f a
con
vex
poly
gon
.
Ext
erio
r A
ng
leIf
a po
lygo
n is
con
vex,
the
n th
e su
m o
f th
e m
easu
res
of t
he e
xter
ior
angl
es,
Su
m T
heo
rem
one
at e
ach
vert
ex,
is 3
60.
Fin
d t
he
sum
of
the
mea
sure
s of
th
e ex
teri
or a
ngl
es,o
ne
at e
ach
vert
ex,o
f a
con
vex
27-g
on.
For
an
yco
nve
x po
lygo
n,t
he
sum
of
the
mea
sure
s of
its
ext
erio
r an
gles
,on
e at
eac
h v
erte
x,is
360
.
Fin
d t
he
mea
sure
of
each
ext
erio
r an
gle
of
regu
lar
hex
agon
AB
CD
EF
.T
he
sum
of
the
mea
sure
s of
th
e ex
teri
or a
ngl
es i
s 36
0 an
d a
hex
agon
h
as 6
an
gles
.If
nis
th
e m
easu
re o
f ea
ch e
xter
ior
angl
e,th
en
6n�
360
n�
60
Fin
d t
he
sum
of
the
mea
sure
s of
th
e ex
teri
or a
ngl
es o
f ea
ch c
onve
x p
olyg
on.
1.10
-gon
2.16
-gon
3.36
-gon
360
360
360
Fin
d t
he
mea
sure
of
an e
xter
ior
angl
e fo
r ea
ch c
onve
x re
gula
r p
olyg
on.
4.12
-gon
5.36
-gon
6.2x
-gon
3010
�18 x0 �
Fin
d t
he
mea
sure
of
an e
xter
ior
angl
e gi
ven
th
e n
um
ber
of
sid
es o
f a
regu
lar
pol
ygon
.
7.40
8.18
9.12
920
30
10.2
411
.180
12.8
152
45
AB
C
DE
F
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
An
gle
s o
f P
oly
go
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
An
gle
s o
f P
oly
go
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill41
9G
lenc
oe G
eom
etry
Lesson 8-1
Fin
d t
he
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es o
f ea
ch c
onve
x p
olyg
on.
1.n
onag
on2.
hep
tago
n3.
deca
gon
1260
900
1440
Th
e m
easu
re o
f an
in
teri
or a
ngl
e of
a r
egu
lar
pol
ygon
is
give
n.F
ind
th
e n
um
ber
of s
ides
in
eac
h p
olyg
on.
4.10
85.
120
6.15
0
56
12
Fin
d t
he
mea
sure
of
each
in
teri
or a
ngl
e u
sin
g th
e gi
ven
in
form
atio
n.
7.8.
m�
A�
115,
m�
B�
65,
m�
L�
100,
m�
M�
110,
m�
C�
115,
m�
D�
65m
�N
�70
,m�
P�
80
9.qu
adri
late
ral
ST
UW
wit
h �
S�
�T
,10
.hex
agon
DE
FG
HI
wit
h
�U
��
W,m
�S
�2x
�16
,�
D�
�E
� �
G�
�H
,�F
��
I,m
�U
�x
�14
m�
D�
7x,m
�F
�4x
m�
S�
116,
m�
T�
116,
m�
D�
140,
m�
E�
140,
m�
U�
64,m
�W
�64
m�
F�
80,m
�G
�14
0,m
�H
�14
0,m
�I�
80
Fin
d t
he
mea
sure
s of
an
in
teri
or a
ngl
e an
d a
n e
xter
ior
angl
e fo
r ea
ch r
egu
lar
pol
ygon
.
11.q
uad
rila
tera
l12
.pen
tago
n13
.dod
ecag
on
90,9
010
8,72
150,
30
Fin
d t
he
mea
sure
s of
an
in
teri
or a
ngl
e an
d a
n e
xter
ior
angl
e gi
ven
th
e n
um
ber
of
sid
es o
f ea
ch r
egu
lar
pol
ygon
.Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
14.8
15.9
16.1
3
135,
4514
0,40
152.
3,27
.7
HG
FI
DE
ST
UW
2x�( 2x
� 2
0)�
( 3x
� 1
0)�
( 2x
� 1
0)�
LM
NP
x�
x�
( 2x
� 1
5)�
( 2x
� 1
5)�
AB
CD
©G
lenc
oe/M
cGra
w-H
ill42
0G
lenc
oe G
eom
etry
Fin
d t
he
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es o
f ea
ch c
onve
x p
olyg
on.
1.11
-gon
2.14
-gon
3.17
-gon
1620
2160
2700
Th
e m
easu
re o
f an
in
teri
or a
ngl
e of
a r
egu
lar
pol
ygon
is
give
n.F
ind
th
e n
um
ber
of s
ides
in
eac
h p
olyg
on.
4.14
45.
156
6.16
0
1015
18
Fin
d t
he
mea
sure
of
each
in
teri
or a
ngl
e u
sin
g th
e gi
ven
in
form
atio
n.
7.8.
quad
rila
tera
l R
ST
Uw
ith
m
�R
�6x
�4,
m�
S�
2x�
8
m�
J�
115,
m�
K�
130,
m�
M�
50,m
�N
�65
m�
R�
128,
m�
S�
52,
m�
T�
128,
m�
U�
52
Fin
d t
he
mea
sure
s of
an
in
teri
or a
ngl
e an
d a
n e
xter
ior
angl
e fo
r ea
ch r
egu
lar
pol
ygon
.Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
9.16
-gon
10.2
4-go
n11
.30-
gon
157.
5,22
.516
5,15
168,
12
Fin
d t
he
mea
sure
s of
an
in
teri
or a
ngl
e an
d a
n e
xter
ior
angl
e gi
ven
th
e n
um
ber
of
sid
es o
f ea
ch r
egu
lar
pol
ygon
.Rou
nd
to
the
nea
rest
ten
th i
f n
eces
sary
.
12.1
413
.22
14.4
0
154.
3,25
.716
3.6,
16.4
171,
9
15.C
RYST
ALL
OG
RA
PHY
Cry
stal
s ar
e cl
assi
fied
acc
ordi
ng
to s
even
cry
stal
sys
tem
s.T
he
basi
s of
th
e cl
assi
fica
tion
is
the
shap
es o
f th
e fa
ces
of t
he
crys
tal.
Tu
rqu
oise
bel
ongs
to
the
tric
lin
ic s
yste
m.E
ach
of
the
six
face
s of
tu
rqu
oise
is
in t
he
shap
e of
a p
aral
lelo
gram
.F
ind
the
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es o
f on
e su
ch f
ace.
360
RS
TU
x�
( 2x
� 1
5)�
( 3x
� 2
0)�
( x �
15)
�
JK
MN
Pra
ctic
e (
Ave
rag
e)
An
gle
s o
f P
oly
go
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csA
ng
les
of
Po
lyg
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
©G
lenc
oe/M
cGra
w-H
ill42
1G
lenc
oe G
eom
etry
Lesson 8-1
Pre-
Act
ivit
yH
ow d
oes
a sc
allo
p s
hel
l il
lust
rate
th
e an
gles
of
pol
ygon
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-1
at
the
top
of p
age
404
in y
our
text
book
.
•H
ow m
any
diag
onal
s of
th
e sc
allo
p sh
ell
show
n i
n y
our
text
book
can
be
draw
n f
rom
ver
tex
A?
9•
How
man
y di
agon
als
can
be
draw
n f
rom
on
e ve
rtex
of
an n
-gon
? E
xpla
inyo
ur r
easo
ning
.n
�3;
Sam
ple
an
swer
:An
n-g
on
has
nve
rtic
es,
but
dia
go
nal
s ca
nn
ot
be
dra
wn
fro
m a
ver
tex
to it
self
or
toei
ther
of
the
two
ad
jace
nt
vert
ices
.Th
eref
ore
,th
e n
um
ber
of
dia
go
nal
s fr
om
on
e ve
rtex
is 3
less
th
an t
he
nu
mb
er o
fve
rtic
es.
Rea
din
g t
he
Less
on
1.W
rite
an
exp
ress
ion
th
at d
escr
ibes
eac
h o
f th
e fo
llow
ing
quan
titi
es f
or a
reg
ula
r n
-gon
.If
the
expr
essi
on a
ppli
es t
o re
gula
r po
lygo
ns
only
,wri
te r
egu
lar.
If i
t ap
plie
s to
all
con
vex
poly
gon
s,w
rite
all
.
a.th
e su
m o
f th
e m
easu
res
of t
he
inte
rior
an
gles
180(
n�
2);
all
b.
the
mea
sure
of
each
in
teri
or a
ngl
e�18
0(n n
�2)
�;
reg
ula
r
c.th
e su
m o
f th
e m
easu
res
of t
he
exte
rior
an
gles
(on
e at
eac
h v
erte
x)36
0;al
l
d.
the
mea
sure
of
each
ext
erio
r an
gle
�3 n60 �;
reg
ula
r
2.G
ive
the
mea
sure
of
an in
teri
or a
ngle
and
the
mea
sure
of
an e
xter
ior
angl
e of
eac
h po
lygo
n.a.
equ
ilat
eral
tri
angl
e60
;12
0c.
squ
are
90;
90b
.re
gula
r h
exag
on12
0;60
d.r
egu
lar
octa
gon
135;
45
3.U
nde
rlin
e th
e co
rrec
t w
ord
or p
hra
se t
o fo
rm a
tru
e st
atem
ent
abou
t re
gula
r po
lygo
ns.
a.A
s th
e n
um
ber
of s
ides
in
crea
ses,
the
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es(i
ncr
ease
s/de
crea
ses/
stay
s th
e sa
me)
.b
.A
s th
e n
um
ber
of s
ides
in
crea
ses,
the
mea
sure
of
each
in
teri
or a
ngl
e(i
ncr
ease
s/de
crea
ses/
stay
s th
e sa
me)
.c.
As
the
nu
mbe
r of
sid
es i
ncr
ease
s,th
e su
m o
f th
e m
easu
res
of t
he
exte
rior
an
gles
(in
crea
ses/
decr
ease
s/st
ays
the
sam
e).
d.
As
the
nu
mbe
r of
sid
es i
ncr
ease
s,th
e m
easu
re o
f ea
ch e
xter
ior
angl
e(i
ncr
ease
s/de
crea
ses/
stay
s th
e sa
me)
.e.
If a
reg
ula
r po
lygo
n h
as m
ore
than
fou
r si
des,
each
in
teri
or a
ngl
e w
ill
be a
(n)
(acu
te/r
ight
/obt
use)
ang
le,a
nd e
ach
exte
rior
ang
le w
ill b
e a(
n) (
acut
e/ri
ght/
obtu
se)
angl
e.
Hel
pin
g Y
ou
Rem
emb
er4.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al i
dea
or f
orm
ula
is
to r
elat
e it
to
som
eth
ing
you
alr
eady
kn
ow.H
ow c
an y
ou u
se y
our
know
ledg
e of
th
e A
ngl
e S
um
Th
eore
m (
for
atr
ian
gle)
to
hel
p yo
u r
emem
ber
the
Inte
rior
An
gle
Su
m T
heo
rem
?S
amp
le a
nsw
er:
Th
e su
m o
f th
e m
easu
res
of
the
(in
teri
or)
an
gle
s o
f a
tria
ng
le is
180
.E
ach
tim
e an
oth
er s
ide
is a
dd
ed,t
he
po
lyg
on
can
be
sub
div
ided
into
on
em
ore
tri
ang
le,s
o 1
80 is
ad
ded
to
th
e in
teri
or
ang
le s
um
.
©G
lenc
oe/M
cGra
w-H
ill42
2G
lenc
oe G
eom
etry
Tan
gra
ms
Th
e ta
ngr
am p
uzz
le i
s co
mpo
sed
of s
even
pi
eces
th
at f
orm
a s
quar
e,as
sh
own
at
the
righ
t.T
his
pu
zzle
has
bee
n a
pop
ula
ram
use
men
t fo
r C
hin
ese
stu
den
ts f
or
hu
ndr
eds
and
perh
aps
thou
san
ds o
f ye
ars.
Mak
e a
care
ful
trac
ing
of t
he
figu
re a
bov
e.C
ut
out
the
pie
ces
and
rea
rran
ge
them
to
form
eac
h f
igu
re b
elow
.Rec
ord
eac
h a
nsw
er b
y d
raw
ing
lin
es w
ith
in
each
fig
ure
.
1.2.
3.4.
5.6.
7.C
reat
e a
diff
eren
t fi
gure
usi
ng
the
seve
n t
angr
am p
iece
s.T
race
th
e ou
tlin
e.T
hen
ch
alle
nge
an
oth
er s
tude
nt
to s
olve
th
e pu
zzle
.S
ee s
tud
ents
’wo
rk.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-1
8-1
Answers (Lesson 8-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Par
alle
log
ram
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill42
3G
lenc
oe G
eom
etry
Lesson 8-2
Sid
es a
nd
An
gle
s o
f Pa
ralle
log
ram
sA
qu
adri
late
ral
wit
h
both
pai
rs o
f op
posi
te s
ides
par
alle
l is
a p
aral
lelo
gram
.Her
e ar
e fo
ur
impo
rtan
t pr
oper
ties
of
para
llel
ogra
ms.
If P
QR
Sis
a p
aral
lelo
gra
m,t
hen
The
opp
osite
sid
es o
f a
P�Q�
� S�
R�an
d P�
S��
Q�R�
para
llelo
gram
are
con
grue
nt.
The
opp
osite
ang
les
of a
�
P�
�R
and
�S
��
Qpa
ralle
logr
am a
re c
ongr
uent
.
The
con
secu
tive
angl
es o
f a
�P
and
�S
are
supp
lem
enta
ry;
�S
and
�R
are
supp
lem
enta
ry;
para
llelo
gram
are
sup
plem
enta
ry.
�R
and
�Q
are
supp
lem
enta
ry;
�Q
and
�P
are
supp
lem
enta
ry.
If a
para
llelo
gram
has
one
rig
ht
If m
�P
�90
, th
en m
�Q
�90
, m
�R
�90
, an
d m
�S
�90
.an
gle,
the
n it
has
four
rig
ht a
ngle
s.
If A
BC
Dis
a p
aral
lelo
gram
,fin
d a
and
b.
A �B�
and
C�D�
are
oppo
site
sid
es,s
o A�
B��
C�D�
.2a
�34
a�
17
�A
and
�C
are
oppo
site
an
gles
,so
�A
��
C.
8b�
112
b�
14
Fin
d x
and
yin
eac
h p
aral
lelo
gram
.
1.2.
x�
30;
y�
22.5
x�
15;
y�
11
3.4.
x�
2;y
�4
x�
10;
y�
40
5.6.
x�
13;
y�
32.5
x�
5;y
�18
0
72x
30x
150
2y60
�55�
5x�
2y�
6x�
12x�
3y�
3y 12
6x
8y 88
6x�
4y�
3x�
BA
DC
34
2 a8b
�
112�
PQ
RS
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill42
4G
lenc
oe G
eom
etry
Dia
go
nal
s o
f Pa
ralle
log
ram
sT
wo
impo
rtan
t pr
oper
ties
of
para
llel
ogra
ms
deal
wit
h t
hei
r di
agon
als.
If A
BC
Dis
a p
aral
lelo
gra
m,t
hen
:
The
dia
gona
ls o
f a
para
llelo
gram
bis
ect
each
oth
er.
AP
�P
Can
d D
P�
PB
Eac
h di
agon
al s
epar
ates
a p
aral
lelo
gram
�
AC
D�
�C
AB
and
�A
DB
��
CB
Din
to t
wo
cong
ruen
t tr
iang
les.
Fin
d x
and
yin
par
alle
logr
am A
BC
D.
Th
e di
agon
als
bise
ct e
ach
oth
er,s
o A
E�
CE
and
DE
�B
E.
6x�
244y
�18
x�
4y
�4.
5
Fin
d x
and
yin
eac
h p
aral
lelo
gram
.
1.2.
3.
x�
4;y
�2
x�
7;y
�14
x�
15;
y�
7.5
4.5.
6.
x�
3�1 3� ;
y�
10�
3�x
�15
;y
�6�
2�x
�15
;y
��
241
�
Com
ple
te e
ach
sta
tem
ent
abou
t �
AB
CD
.J
ust
ify
you
r an
swer
.
7.�
BA
C�
�A
CD
If li
nes
are
|| ,th
en a
lt.i
nt.
�ar
e �
.
8.D�
E��
B�E�
Th
e d
iag
s.o
f a
par
alle
log
ram
bis
ect
each
oth
er.
9.�
AD
C�
�C
BA
Th
e d
iag
on
al o
f a
par
alle
log
ram
div
ides
th
e p
aral
lelo
gra
m in
to 2
��
s.
10.A�
D�||
C�B�
Op
po
site
sid
es o
f a
par
alle
log
ram
are
|| .
AB
CD
E
x
17
4y
3x�
2y12
3x
30�
10
y
2x�60
�4 y
�
4x
2y28
3x4y
812
AB
E
CD
6x
1824
4y
AB
CD
P
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Par
alle
log
ram
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Par
alle
log
ram
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill42
5G
lenc
oe G
eom
etry
Lesson 8-2
Com
ple
te e
ach
sta
tem
ent
abou
t �
DE
FG
.Ju
stif
y yo
ur
answ
er.
1.D�
G�||
E�F�;
op
p.s
ides
of
�ar
e || .
2.D�
E��
G�F�;
op
p.s
ides
of
�ar
e �
.
3.G�
H��
E�H�
;d
iag
.of
�b
isec
t ea
ch o
ther
.
4.�
DE
F�
�F
GD
;o
pp
.�o
f �
are
�.
5.�
EF
Gis
su
pple
men
tary
to
.�
DE
Fo
r �
FG
D;
con
s.�
in �
are
sup
pl.
6.�
DG
E�
�F
EG
;d
iag
.of
�se
par
ates
�in
to 2
��
s.
ALG
EBR
AU
se �
WX
YZ
to f
ind
eac
h m
easu
re o
r va
lue.
7.m
�X
YZ
�8.
m�
WZ
Y�
9.m
�W
XY
�10
.a�
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e co
ord
inat
es o
f th
e in
ters
ecti
on o
f th
ed
iago
nal
s of
par
alle
logr
am H
JK
Lgi
ven
eac
h s
et o
f ve
rtic
es.
11.H
(1,1
),J
(2,3
),K
(6,3
),L
(5,1
)12
.H(�
1,4)
,J(3
,3),
K(3
,�2)
,L(�
1,�
1)
(3.5
,2)
(1,1
)
13.P
RO
OF
Wri
te a
par
agra
ph p
roof
of
the
theo
rem
Con
secu
tive
an
gles
in
a p
aral
lelo
gram
are
supp
lem
enta
ry.
Giv
en:
�A
BC
DP
rove
:�
Aan
d �
Bar
e su
pp
lem
enta
ry.
�B
and
�C
are
sup
ple
men
tary
.�
Can
d �
Dar
e su
pp
lem
enta
ry.
�D
and
�A
are
sup
ple
men
tary
.P
roo
f:W
e ar
e g
iven
�A
BC
D,s
o w
e kn
ow
th
at A�
B�|| C�
D�an
d B�
C�|| D�
A�by
the
def
initi
on
of
a p
aral
lelo
gra
m.W
e al
so k
now
th
at if
tw
o p
aral
lel l
ines
are
cut
by a
tra
nsv
ersa
l,th
en c
on
secu
tive
inte
rio
r an
gle
s ar
e su
pp
lem
enta
ry.
So
,�A
and
�B
,�B
and
�C
,�C
and
�D
,an
d �
Dan
d �
Aar
e p
airs
of
sup
ple
men
tary
an
gle
s.
DC
BA
1560
6012
0
YZ
XW
A 50�
30
2 a 70�
?
?
?
???F
G
ED
H
©G
lenc
oe/M
cGra
w-H
ill42
6G
lenc
oe G
eom
etry
Com
ple
te e
ach
sta
tem
ent
abou
t �
LM
NP
.Ju
stif
y yo
ur
answ
er.
1.L�
Q��
N�Q�
;d
iag
.of
�b
isec
t ea
ch o
ther
.
2.�
LM
N�
�N
PL
;o
pp
.�o
f �
are
�.
3.�
LM
P�
�N
PM
;d
iag
.of
�se
par
ates
�in
to 2
��
s.
4.�
NP
Lis
su
pple
men
tary
to
.�
MN
Po
r �
PL
M;
con
s.�
in �
are
sup
pl.
5.L�
M��
N�P�
;o
pp
.sid
es o
f �
are
�.
ALG
EBR
AU
se �
RS
TU
to f
ind
eac
h m
easu
re o
r va
lue.
6.m
�R
ST
�7.
m�
ST
U�
8.m
�T
UR
�9.
b�
CO
OR
DIN
ATE
GEO
MET
RYF
ind
th
e co
ord
inat
es o
f th
e in
ters
ecti
on o
f th
ed
iago
nal
s of
par
alle
logr
am P
RY
Zgi
ven
eac
h s
et o
f ve
rtic
es.
10.P
(2,5
),R
(3,3
),Y
(�2,
�3)
,Z(�
3,�
1)11
.P(2
,3),
R(1
,�2)
,Y(�
5,�
7),Z
(�4,
�2)
(0,1
)(�
1.5,
�2)
12.P
RO
OF
Wri
te a
par
agra
ph p
roof
of
the
foll
owin
g.G
iven
:�P
RS
Tan
d �
PQ
VU
Pro
ve:�
V�
�S
Pro
of:
We
are
giv
en �
PR
ST
and
�P
QV
U.S
ince
op
po
site
ang
les
of
a p
aral
lelo
gra
m a
re c
on
gru
ent,
�P
��
Van
d
�P
��
S.S
ince
co
ng
ruen
ce o
f an
gle
s is
tra
nsi
tive
,�
V�
�S
by t
he
Tran
siti
ve P
rop
erty
of
Co
ng
ruen
ce.
13.C
ON
STR
UC
TIO
NM
r.R
odri
quez
use
d th
e pa
rall
elog
ram
at
the
righ
t to
de
sign
a h
erri
ngb
one
patt
ern
for
a p
avin
g st
one.
He
wil
l u
se t
he
pavi
ng
ston
e fo
r a
side
wal
k.If
m�
1 is
130
,fin
d m
�2,
m�
3,an
d m
�4.
50,1
30,5
0
12
34P
R
S
UVQ
T
612
5
5512
5T
U
SR
B30
�23
4b �
1
25�
?
?
??
?P
N
MQ
L
Pra
ctic
e (
Ave
rag
e)
Par
alle
log
ram
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csP
aral
lelo
gra
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
©G
lenc
oe/M
cGra
w-H
ill42
7G
lenc
oe G
eom
etry
Lesson 8-2
Pre-
Act
ivit
yH
ow a
re p
aral
lelo
gram
s u
sed
to
rep
rese
nt
dat
a?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-2
at
the
top
of p
age
411
in y
our
text
book
.
•W
hat
is
the
nam
e of
th
e sh
ape
of t
he
top
surf
ace
of e
ach
wed
ge o
f ch
eese
?p
aral
lelo
gra
m•
Are
the
thr
ee p
olyg
ons
show
n in
the
dra
win
g si
mila
r po
lygo
ns?
Exp
lain
you
rre
ason
ing.
No
;sam
ple
an
swer
:Th
eir
sid
es a
re n
ot
pro
po
rtio
nal
.
Rea
din
g t
he
Less
on
1.U
nder
line
wor
ds o
r ph
rase
s th
at c
an c
ompl
ete
the
foll
owin
g se
nten
ces
to m
ake
stat
emen
tsth
at a
re a
lway
s tr
ue.(
The
re m
ay b
e m
ore
than
one
cor
rect
cho
ice
for
som
e of
the
sen
tenc
es.)
a.O
ppos
ite
side
s of
a p
aral
lelo
gram
are
(co
ngr
uen
t/pe
rpen
dicu
lar/
para
llel
).
b.
Con
secu
tive
an
gles
of
a pa
rall
elog
ram
are
(co
mpl
emen
tary
/su
pple
men
tary
/con
gru
ent)
.
c.A
dia
gon
al o
f a
para
llel
ogra
m d
ivid
es t
he
para
llel
ogra
m i
nto
tw
o(a
cute
/rig
ht/
obtu
se/c
ongr
uen
t) t
rian
gles
.
d.
Opp
osit
e an
gles
of
a pa
rall
elog
ram
are
(co
mpl
emen
tary
/su
pple
men
tary
/con
gru
ent)
.
e.T
he
diag
onal
s of
a p
aral
lelo
gram
(bi
sect
eac
h o
ther
/are
per
pen
dicu
lar/
are
con
gru
ent)
.
f.If
a p
aral
lelo
gram
has
on
e ri
ght
angl
e,th
en a
ll o
f it
s ot
her
an
gles
are
(acu
te/r
igh
t/ob
tuse
) an
gles
.
2.L
et A
BC
Dbe
a p
aral
lelo
gram
wit
h A
B�
BC
and
wit
h n
o ri
ght
angl
es.
a.S
ketc
h a
par
alle
logr
am t
hat
mat
ches
th
e de
scri
ptio
nS
amp
le a
nsw
er:
abov
e an
d dr
aw d
iago
nal
B�D�
.
In p
arts
b–f
,com
plet
e ea
ch s
ente
nce
.
b.
A�B�
|| an
d A�
D�||
.
c.A�
B��
and
B �C�
�B
.
d.
�A
�an
d �
AB
C�
.
e.�
AD
B�
beca
use
th
ese
two
angl
es a
re
angl
es f
orm
ed b
y th
e tw
o pa
rall
el l
ines
an
d an
d th
e
tran
sver
sal
.
f.�
AB
D�
.
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
new
th
eore
ms
in g
eom
etry
is
to r
elat
e th
em t
o th
eore
ms
you
lear
ned
ear
lier
.Nam
e a
theo
rem
abo
ut
para
llel
lin
es t
hat
can
be
use
d to
rem
embe
r th
eth
eore
m t
hat
say
s,“I
f a
para
llel
ogra
m h
as o
ne
righ
t an
gle,
it h
as f
our
righ
t an
gles
.”P
erp
end
icu
lar T
ran
sver
sal T
heo
rem
�C
DBBD
���
BC
���
AD
���
inte
rio
ral
tern
ate
�C
BD
�C
DA
�C
A�D�
C�D�
B�C�
C�D�
DA
CB
©G
lenc
oe/M
cGra
w-H
ill42
8G
lenc
oe G
eom
etry
Tess
ella
tio
ns
A t
esse
llat
ion
is
a ti
lin
g pa
tter
n m
ade
of p
olyg
ons.
Th
e pa
tter
n c
an
be e
xten
ded
so t
hat
th
e po
lygo
nal
til
es c
over
th
e pl
ane
com
plet
ely
wit
hn
o ga
ps.A
ch
ecke
rboa
rd a
nd
a h
oney
com
b pa
tter
n a
re e
xam
ples
of
tess
ella
tion
s.S
omet
imes
th
e sa
me
poly
gon
can
mak
e m
ore
than
on
ete
ssel
lati
on p
atte
rn.B
oth
pat
tern
s be
low
can
be
form
ed f
rom
an
isos
cele
s tr
ian
gle.
Dra
w a
tes
sell
atio
n u
sin
g ea
ch p
olyg
on.
1.2.
3.
Dra
w t
wo
dif
fere
nt
tess
ella
tion
s u
sin
g ea
ch p
olyg
on.
4.5.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-2
8-2
Answers (Lesson 8-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Test
s fo
r P
aral
lelo
gra
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill42
9G
lenc
oe G
eom
etry
Lesson 8-3
Co
nd
itio
ns
for
a Pa
ralle
log
ram
Th
ere
are
man
y w
ays
to
esta
blis
h t
hat
a q
uad
rila
tera
l is
a p
aral
lelo
gram
.
If:
If:
both
pai
rs o
f op
posi
te s
ides
are
par
alle
l,A�
B�|| D�
C�an
d A�
D�|| B�
C�,
both
pai
rs o
f op
posi
te s
ides
are
con
grue
nt,
A�B�
� D�
C�an
d A�
D��
B�C�
,
both
pai
rs o
f op
posi
te a
ngle
s ar
e co
ngru
ent,
�A
BC
��
AD
Can
d �
DA
B�
�B
CD
,
the
diag
onal
s bi
sect
eac
h ot
her,
A�E�
� C�
E�an
d D�
E��
B�E�
,
one
pair
of o
ppos
ite s
ides
is c
ongr
uent
and
par
alle
l,A�
B�|| C�
D�an
d A�
B��
C�D�
, or
A�D�
|| B�C�
and
A�D�
�B�
C�,
then
:th
e fig
ure
is a
par
alle
logr
am.
then
:A
BC
Dis
a p
aral
lelo
gram
.
Fin
d x
and
yso
th
at F
GH
Jis
a
par
alle
logr
am.
FG
HJ
is a
par
alle
logr
am i
f th
e le
ngt
hs
of t
he
oppo
site
si
des
are
equ
al.
6x�
3 �
154x
�2y
�2
6x�
124(
2) �
2y�
2x
�2
8 �
2y�
2�
2y�
�6
y�
3
Fin
d x
and
yso
th
at e
ach
qu
adri
late
ral
is a
par
alle
logr
am.
1.x
�7;
y�
42.
x�
5;y
�25
3.x
�31
;y
�5
4.x
�5;
y�
3
5.x
�15
;y
�9
6.x
�30
;y
�15
6y�
3 x�
( x �
y) �
2x�
30�
24�
9x�
6y45
�18
5x�
5y�
25�
11x�
5y�
55�
2x �
2
2 y8
12
JHG
F6x
� 3
154x
� 2
y2
AB
CD
E
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill43
0G
lenc
oe G
eom
etry
Para
llelo
gra
ms
on
th
e C
oo
rdin
ate
Plan
eO
n t
he
coor
din
ate
plan
e,th
e D
ista
nce
For
mu
la a
nd
the
Slo
pe F
orm
ula
can
be
use
d to
tes
t if
a q
uad
rila
tera
l is
a p
aral
lelo
gram
.
Det
erm
ine
wh
eth
er A
BC
Dis
a p
aral
lelo
gram
.T
he
vert
ices
are
A(�
2,3)
,B(3
,2),
C(2
,�1)
,an
d D
(�3,
0).
Met
hod
1:U
se t
he
Slo
pe F
orm
ula
,m�
�y x2 2
� �
y x1 1�
.
slop
e of
A �D�
�� �
23 ��
(0 �3)
��
�3 1��
3sl
ope
of B�
C��
�23�
�(�21)
��
�3 1��
3
slop
e of
A �B�
�� 3
2 �
� (�3 2)
��
��1 5�
slop
e of
C�D�
�� 2�
�1(� �
0 3)�
��
�1 5�
Opp
osit
e si
des
hav
e th
e sa
me
slop
e,so
A�B�
|| C�D�
and
A�D�
|| B�C�
.Bot
h p
airs
of
oppo
site
sid
esar
e pa
rall
el,s
o A
BC
Dis
a p
aral
lelo
gram
.
Met
hod
2:U
se t
he
Dis
tan
ce F
orm
ula
,d�
�(x
2�
�x 1
)2�
�(y
2�
�y 1
)2�
.
AB
��
(�2
��
3)2
��
(3 �
�2)
2�
��
25 �
�1�
or �
26�
CD
��
(2 �
(�
�3)
)2�
�(�
1�
�0)
2�
��
25 �
�1�
or �
26�
AD
��
(�2
��
(�3)
)�
2�
(3�
�0)
2�
��
1 �
9�
or �
10�
BC
��
(3 �
2�
)2�
(�
2 �
(��
1))2
��
�1
�9
�or
�10�
Bot
h p
airs
of
oppo
site
sid
es h
ave
the
sam
e le
ngt
h,s
o A
BC
Dis
a p
aral
lelo
gram
.
Det
erm
ine
wh
eth
er a
fig
ure
wit
h t
he
give
n v
erti
ces
is a
par
alle
logr
am.U
se t
he
met
hod
in
dic
ated
.
1.A
(0,0
),B
(1,3
),C
(5,3
),D
(4,0
);2.
D(�
1,1)
,E(2
,4),
F(6
,4),
G(3
,1);
Slo
pe F
orm
ula
Slo
pe F
orm
ula
yes
yes
3.R
(�1,
0),S
(3,0
),T
(2,�
3),U
(�3,
�2)
;4.
A(�
3,2)
,B(�
1,4)
,C(2
,1),
D(0
,�1)
;D
ista
nce
For
mu
laD
ista
nce
an
d S
lope
For
mu
las
no
yes
5.S
(�2,
4),T
(�1,
�1)
,U(3
,�4)
,V(2
,1);
6.F
(3,3
),G
(1,2
),H
(�3,
1),I
(�1,
4);
Dis
tan
ce a
nd
Slo
pe F
orm
ula
sM
idpo
int
For
mu
la
yes
no
7.A
par
alle
logr
am h
as v
erti
ces
R(�
2,�
1),S
(2,1
),an
d T
(0,�
3).F
ind
all
poss
ible
coor
din
ates
for
th
e fo
urt
h v
erte
x.
(4,�
1),(
0,3)
,or
(�4,
�5)
x
y
O
C
A
D
B
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Test
s fo
r P
aral
lelo
gra
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Test
s fo
r P
aral
lelo
gra
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill43
1G
lenc
oe G
eom
etry
Lesson 8-3
Det
erm
ine
wh
eth
er e
ach
qu
adri
late
ral
is a
par
alle
logr
am.J
ust
ify
you
r an
swer
.
1.2.
Yes;
a p
air
of
op
po
site
sid
esYe
s;b
oth
pai
rs o
f o
pp
osi
teis
par
alle
l an
d c
on
gru
ent.
ang
les
are
con
gru
ent.
3.4.
No
;n
on
e o
f th
e te
sts
for
Yes;
bo
th p
airs
of
op
po
site
sid
esp
aral
lelo
gra
ms
is f
ulf
illed
.ar
e co
ng
ruen
t.
CO
OR
DIN
ATE
GEO
MET
RYD
eter
min
e w
het
her
a f
igu
re w
ith
th
e gi
ven
ver
tice
s is
ap
aral
lelo
gram
.Use
th
e m
eth
od i
nd
icat
ed.
5.P
(0,0
),Q
(3,4
),S
(7,4
),Y
(4,0
);S
lope
For
mu
la
yes
6.S
(�2,
1),R
(1,3
),T
(2,0
),Z
(�1,
�2)
;Dis
tan
ce a
nd
Slo
pe F
orm
ula
yes
7.W
(2,5
),R
(3,3
),Y
(�2,
�3)
,N(�
3,1)
;Mid
poin
t F
orm
ula
no
ALG
EBR
AF
ind
xan
d y
so t
hat
eac
h q
uad
rila
tera
l is
a p
aral
lelo
gram
.
8.9.
x�
24,y
�19
x�
3,y
�14
10.
11.
x�
45,y
�20
x�
17,y
�9
3x �
14
x �
20
3y �
2y
� 2
0( 4
x �
35)
�
( 3x
� 1
0)�
( 2y
� 5
) �
( y �
15)
�
2y �
11
y � 3
4x � 3
3x
x �
16
y �
19
2y
2x �
8
©G
lenc
oe/M
cGra
w-H
ill43
2G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er e
ach
qu
adri
late
ral
is a
par
alle
logr
am.J
ust
ify
you
r an
swer
.
1.2.
Yes;
the
dia
go
nal
s b
isec
tN
o;
no
ne
of
the
test
s fo
rea
ch o
ther
.p
aral
lelo
gra
ms
is f
ulf
illed
.
3.4.
Yes;
bo
th p
airs
of
op
po
site
No
;n
on
e o
f th
e te
sts
for
ang
les
are
con
gru
ent.
par
alle
log
ram
s is
fu
lfill
ed.
CO
OR
DIN
ATE
GEO
MET
RYD
eter
min
e w
het
her
a f
igu
re w
ith
th
e gi
ven
ver
tice
s is
ap
aral
lelo
gram
.Use
th
e m
eth
od i
nd
icat
ed.
5.P
(�5,
1),S
(�2,
2),F
(�1,
�3)
,T(2
,�2)
;Slo
pe F
orm
ula
no
6.R
(�2,
5),O
(1,3
),M
(�3,
�4)
,Y(�
6,�
2);D
ista
nce
an
d S
lope
For
mu
la
yes
ALG
EBR
AF
ind
xan
d y
so t
hat
eac
h q
uad
rila
tera
l is
a p
aral
lelo
gram
.
7.8.
x�
20,y
�12
x�
�6,
y�
13
9.10
.
x�
�3,
y�
2x
��
2,y
��
5
11.T
ILE
DES
IGN
Th
e pa
tter
n s
how
n i
n t
he
figu
re i
s to
con
sist
of
con
gru
ent
para
llel
ogra
ms.
How
can
th
e de
sign
er b
e ce
rtai
n t
hat
th
e sh
apes
are
para
llel
ogra
ms?
Sam
ple
an
swer
:C
on
firm
th
at b
oth
pai
rs o
f o
pp
osi
te �
s ar
e �
.
�4y
� 2
x � 12
�2x
� 6
y �
23
�4x
� 6
�6x
7y �
312
y �
7
3y �
5�
3x
� 4
�4x
� 2
2y �
8( 5
x �
29)
�
( 7x
� 1
1)�
( 3y
� 1
5)�
( 5y
� 9
) �
118�
62�
118�
62�
Pra
ctic
e (
Ave
rag
e)
Test
s fo
r P
aral
lelo
gra
ms
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csTe
sts
for
Par
alle
log
ram
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
©G
lenc
oe/M
cGra
w-H
ill43
3G
lenc
oe G
eom
etry
Lesson 8-3
Pre-
Act
ivit
yH
ow a
re p
aral
lelo
gram
s u
sed
in
arc
hit
ectu
re?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-3
at
the
top
of p
age
417
in y
our
text
book
.
Mak
e tw
o ob
serv
atio
ns
abou
t th
e an
gles
in
th
e ro
of o
f th
e co
vere
d br
idge
.S
amp
le a
nsw
er:
Op
po
site
an
gle
s ap
pea
r co
ng
ruen
t.C
on
secu
tive
an
gle
s ap
pea
r su
pp
lem
enta
ry.
Rea
din
g t
he
Less
on
1.W
hic
h o
f th
e fo
llow
ing
con
diti
ons
guar
ante
e th
at a
qu
adri
late
ral
is a
par
alle
logr
am?
A.
Tw
o si
des
are
para
llel
.B
,D,G
,H,J
,LB
.Bot
h p
airs
of
oppo
site
sid
es a
re c
ongr
uen
t.C
.T
he
diag
onal
s ar
e pe
rpen
dicu
lar.
D.A
pai
r of
opp
osit
e si
des
is b
oth
par
alle
l an
d co
ngr
uen
t.E
.T
her
e ar
e tw
o ri
ght
angl
es.
F.T
he
sum
of
the
mea
sure
s of
th
e in
teri
or a
ngl
es i
s 36
0.G
.All
fou
r si
des
are
con
gru
ent.
H.B
oth
pai
rs o
f op
posi
te a
ngl
es a
re c
ongr
uen
t.I.
Tw
o an
gles
are
acu
te a
nd
the
oth
er t
wo
angl
es a
re o
btu
se.
J.T
he
diag
onal
s bi
sect
eac
h o
ther
.K
.Th
e di
agon
als
are
con
gru
ent.
L.
All
fou
r an
gles
are
rig
ht
angl
es.
2.D
eter
min
e w
het
her
th
ere
is e
nou
gh g
iven
in
form
atio
n t
o kn
ow t
hat
eac
h f
igu
re i
s a
para
llel
ogra
m.I
f so
,sta
te t
he
defi
nit
ion
or
theo
rem
th
at ju
stif
ies
you
r co
ncl
usi
on.
a.b
.
no
Yes;
if t
he
dia
go
nal
s b
isec
t ea
cho
ther
,th
en t
he
qu
adri
late
ral i
s a
par
alle
log
ram
.c.
d.
Yes;
if b
oth
pai
rs o
f o
pp
osi
ten
o
sid
es a
re �
,th
en q
uad
rila
tera
l is
�.
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
larg
e n
um
ber
of m
ath
emat
ical
ide
as i
s to
th
ink
of t
hem
in
grou
ps.H
ow c
an y
ou s
tate
th
e co
ndi
tion
s as
on
e gr
oup
abou
t th
e si
des
of q
uad
rila
tera
lsth
at g
uar
ante
e th
at t
he
quad
rila
tera
l is
a p
aral
lelo
gram
?S
amp
le a
nsw
er:
bo
thp
airs
of
op
po
site
sid
es p
aral
lel,
bo
th p
airs
of
op
po
site
sid
es c
on
gru
ent,
or
on
e p
air
of
op
po
site
sid
es b
oth
par
alle
l an
d c
on
gru
ent
©G
lenc
oe/M
cGra
w-H
ill43
4G
lenc
oe G
eom
etry
Test
s fo
r P
aral
lelo
gra
ms
By
defi
nit
ion
,a q
uad
rila
tera
l is
a p
aral
lelo
gram
if
and
on
ly i
fbo
th p
airs
of
oppo
site
sid
esar
e pa
rall
el.W
hat
con
diti
ons
oth
er t
han
bot
h p
airs
of
oppo
site
sid
es p
aral
lel
wil
l gu
aran
tee
that
a q
uad
rila
tera
l is
a p
aral
lelo
gram
? In
th
is a
ctiv
ity,
seve
ral
poss
ibil
itie
s w
ill
bein
vest
igat
ed b
y dr
awin
g qu
adri
late
rals
to
sati
sfy
cert
ain
con
diti
ons.
Rem
embe
r th
at a
ny
test
th
at s
eem
s to
wor
k is
not
gu
aran
teed
to
wor
k u
nle
ss i
t ca
n b
e fo
rmal
ly p
rove
n.
Com
ple
te.
1.D
raw
a q
uad
rila
tera
l w
ith
on
e pa
ir o
f op
posi
te s
ides
con
gru
ent.
Mu
st i
t be
a p
aral
lelo
gram
?n
o
2.D
raw
a q
uad
rila
tera
l w
ith
bot
h p
airs
of
oppo
site
sid
es c
ongr
uen
t.M
ust
it
be a
par
alle
logr
am?
yes
3.D
raw
a q
uad
rila
tera
l w
ith
on
e pa
ir o
f op
posi
te s
ides
par
alle
l an
d th
e ot
her
pai
r of
opp
osit
e si
des
con
gru
ent.
Mu
st i
t be
a
para
llel
ogra
m?
no
4.D
raw
a q
uad
rila
tera
l w
ith
on
e pa
ir o
f op
posi
te s
ides
bot
h p
aral
lel
and
con
gru
ent.
Mu
st i
t be
a p
aral
lelo
gram
?ye
s
5.D
raw
a q
uad
rila
tera
l w
ith
on
e pa
ir o
f op
posi
te a
ngl
es c
ongr
uen
t.M
ust
it
be a
par
alle
logr
am?
no
6.D
raw
a q
uad
rila
tera
l w
ith
bot
h p
airs
of
oppo
site
an
gles
con
gru
ent.
Mu
st i
t be
a p
aral
lelo
gram
?ye
s
7.D
raw
a q
uad
rila
tera
l w
ith
on
e pa
ir o
f op
posi
te s
ides
par
alle
l an
d on
e pa
ir o
f op
posi
te a
ngl
es c
ongr
uen
t.M
ust
it
be a
par
alle
logr
am?
yes
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-3
8-3
Answers (Lesson 8-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Rec
tan
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill43
5G
lenc
oe G
eom
etry
Lesson 8-4
Pro
per
ties
of
Rec
tan
gle
sA
rec
tan
gle
is a
qu
adri
late
ral
wit
h f
our
righ
t an
gles
.Her
e ar
e th
e pr
oper
ties
of
rect
angl
es.
A r
ecta
ngl
e h
as a
ll t
he
prop
erti
es o
f a
para
llel
ogra
m.
•O
ppos
ite
side
s ar
e pa
rall
el.
•O
ppos
ite
angl
es a
re c
ongr
uen
t.•
Opp
osit
e si
des
are
con
gru
ent.
•C
onse
cuti
ve a
ngl
es a
re s
upp
lem
enta
ry.
•T
he
diag
onal
s bi
sect
eac
h o
ther
.
Als
o:
•A
ll f
our
angl
es a
re r
igh
t an
gles
.�
UT
S,�
TS
R,�
SR
U,a
nd
�R
UT
are
righ
t an
gles
.•
Th
e di
agon
als
are
con
gru
ent.
T �R�
� U�
S�
TS R
U
Q
In r
ecta
ngl
e R
ST
Uab
ove,
US
�6x
�3
and
RT
�7x
�2.
Fin
d x
.T
he
diag
onal
s of
a r
ecta
ngl
e bi
sect
eac
hot
her
,so
US
�R
T.
6x�
3�
7x�
23
�x
�2
5�
x
In r
ecta
ngl
e R
ST
Uab
ove,
m�
ST
R�
8x�
3 an
d m
�U
TR
�16
x�
9.F
ind
m�
ST
R.
�U
TS
is a
rig
ht
angl
e,so
m
�S
TR
�m
�U
TR
�90
.
8x�
3 �
16x
�9
�90
24x
�6
�90
24x
�96
x�
4
m�
ST
R�
8x�
3 �
8(4)
�3
or 3
5
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
AB
CD
is a
rec
tan
gle.
1.If
AE
�36
an
d C
E�
2x�
4,fi
nd
x.20
2.If
BE
�6y
�2
and
CE
�4y
�6,
fin
d y.
2
3.If
BC
�24
an
d A
D�
5y�
1,fi
nd
y.5
4.If
m�
BE
A�
62,f
ind
m�
BA
C.
59
5.If
m�
AE
D�
12x
and
m�
BE
C�
10x
�20
,fin
d m
�A
ED
.12
0
6.If
BD
�8y
�4
and
AC
�7y
�3,
fin
d B
D.
52
7.If
m�
DB
C�
10x
and
m�
AC
B�
4x2 �
6,fi
nd
m�
AC
B.
30
8.If
AB
�6y
and
BC
�8y
,fin
d B
Din
ter
ms
of y
.10
y
9.In
rec
tan
gle
MN
OP
,m�
1 �
40.F
ind
the
mea
sure
of
each
n
um
bere
d an
gle.
m�
2 �
40;
m�
3 �
50;
m�
4 �
50;
m�
5 �
80;
m�
6 �
100
MN O
P
K 12
345
6
BC D
A
E
©G
lenc
oe/M
cGra
w-H
ill43
6G
lenc
oe G
eom
etry
Pro
ve t
hat
Par
alle
log
ram
s A
re R
ecta
ng
les
Th
e di
agon
als
of a
rec
tan
gle
are
con
gru
ent,
and
the
con
vers
e is
als
o tr
ue.
If th
e di
agon
als
of a
par
alle
logr
am a
re c
ongr
uent
, th
en t
he p
aral
lelo
gram
is a
rec
tang
le.
In t
he
coor
din
ate
plan
e yo
u c
an u
se t
he
Dis
tan
ce F
orm
ula
,th
e S
lope
For
mu
la,a
nd
prop
erti
es o
f di
agon
als
to s
how
th
at a
fig
ure
is
a re
ctan
gle.
Det
erm
ine
wh
eth
er A
(�3,
0),B
(�2,
3),C
(4,1
),an
d D
(3,�
2) a
re t
he
vert
ices
of
a re
ctan
gle.
Met
hod
1:U
se t
he
Slo
pe F
orm
ula
.
slop
e of
A �B�
�� �
23 ��
(0 �3)
��
�3 1�or
3sl
ope
of A�
D��
� 3�
�2(� �
0 3)�
��� 62 �
or �
�1 3�
slop
e of
C�D�
��� 32 ��
41�
��� �
3 1�or
3sl
ope
of B�
C��
� 41 �
� (�3 2)
��
�� 62 �or
��1 3�
Opp
osit
e si
des
are
para
llel
,so
the
figu
re i
s a
para
llel
ogra
m.C
onse
cuti
ve s
ides
are
perp
endi
cula
r,so
AB
CD
is a
rec
tan
gle.
Met
hod
2:U
se t
he
Mid
poin
t an
d D
ista
nce
For
mu
las.
Th
e m
idpo
int
of A�
C�is
���3 2�
4�
,�0
� 21
���
��1 2� ,�1 2� �
and
the
mid
poin
t of
B�D�
is
���2 2�
3�
,�3
� 22
���
��1 2� ,�1 2� �.
Th
e di
agon
als
hav
e th
e sa
me
mid
poin
t so
th
ey b
isec
t ea
ch o
ther
.
Th
us,
AB
CD
is a
par
alle
logr
am.
AC
��
(�3
��
4)2
��
(0 �
�1)
2�
��
49 �
�1�
or �
50�B
D�
�(�
2 �
�3)
2�
�(3
��
(�2)
)2�
��
25 �
�25�
or �
50�T
he
diag
onal
s ar
e co
ngr
uen
t.A
BC
Dis
a p
aral
lelo
gram
wit
h d
iago
nal
s th
at b
isec
t ea
chot
her
,so
it i
s a
rect
angl
e.
Det
erm
ine
wh
eth
er A
BC
Dis
a r
ecta
ngl
e gi
ven
eac
h s
et o
f ve
rtic
es.J
ust
ify
you
ran
swer
.S
amp
le ju
stif
icat
ion
s ar
e g
iven
.
1.A
(�3,
1),B
(�3,
3),C
(3,3
),D
(3,1
)2.
A(�
3,0)
,B(�
2,3)
,C(4
,5),
D(3
,2)
Yes;
con
secu
tive
sid
es a
re ⊥
.N
o;
con
secu
tive
sid
es a
re n
ot
⊥.
3.A
(�3,
0),B
(�2,
2),C
(3,0
),D
(2,�
2)4.
A(�
1,0)
,B(0
,2),
C(4
,0),
D(3
,�2)
No
;co
nse
cuti
ve s
ides
are
no
t ⊥.
Yes;
con
secu
tive
sid
es a
re ⊥
.
5.A
(�1,
�5)
,B(�
3,0)
,C(2
,2),
D(4
,�3)
6.A
(�1,
�1)
,B(0
,2),
C(4
,3),
D(3
,0)
Yes;
the
dia
go
nal
s ar
e co
ng
ruen
t N
o;
the
dia
go
nal
s ar
e n
ot
�.
and
bis
ect
each
oth
er.
7.A
par
alle
logr
am h
as v
erti
ces
R(�
3,�
1),S
(�1,
2),a
nd
T(5
,�2)
.Fin
d th
e co
ordi
nat
es o
fU
so t
hat
RS
TU
is a
rec
tan
gle.
(3,�
5)
x
y O
D
B
A
EC
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Rec
tan
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Rec
tan
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill43
7G
lenc
oe G
eom
etry
Lesson 8-4
ALG
EBR
AA
BC
Dis
a r
ecta
ngl
e.
1.If
AC
�2x
�13
an
d D
B�
4x�
1,fi
nd
x.7
2.If
AC
�x
�3
and
DB
�3x
�19
,fin
d A
C.
14
3.If
AE
�3x
�3
and
EC
�5x
�15
,fin
d A
C.
60
4.If
DE
�6x
�7
and
AE
�4x
�9,
fin
d D
B.
82
5.If
m�
DA
C�
2x�
4 an
d m
�B
AC
�3x
�1,
fin
d x.
17
6.If
m�
BD
C�
7x�
1 an
d m
�A
DB
�9x
�7,
fin
d m
�B
DC
.43
7.If
m�
AB
D�
x2�
7 an
d m
�C
DB
�4x
�5,
fin
d x.
6
8.If
m�
BA
C�
x2�
3 an
d m
�C
AD
�x
�15
,fin
d m
�B
AC
.67
or
84
PR
ST
is a
rec
tan
gle.
Fin
d e
ach
mea
sure
if
m�
1 �
50.
9.m
�2
4010
.m�
340
11.m
�4
5012
.m�
540
13.m
�6
4014
.m�
750
15.m
�8
100
16.m
�9
80
CO
OR
DIN
ATE
GEO
MET
RYD
eter
min
e w
het
her
TU
XY
is a
rec
tan
gle
give
n e
ach
set
of v
erti
ces.
Ju
stif
y yo
ur
answ
er.
17.T
(�3,
�2)
,U(�
4,2)
,X(2
,4),
Y(3
,0)
No
;sa
mp
le a
nsw
er:
An
gle
s ar
e n
ot
rig
ht
ang
les.
18.T
(�6,
3),U
(0,6
),X
(2,2
),Y
(�4,
�1)
Yes;
sam
ple
an
swer
:O
pp
osi
te s
ides
are
co
ng
ruen
t an
d d
iag
on
als
are
con
gru
ent.
19.T
(4,1
),U
(3,�
1),X
(�3,
2),Y
(�2,
4)
Yes;
sam
ple
an
swer
:O
pp
osi
te s
ides
are
par
alle
l an
d c
on
secu
tive
sid
esar
e p
erp
end
icu
lar.
T
12
34
56
78
9
SRPD
CBA
E
©G
lenc
oe/M
cGra
w-H
ill43
8G
lenc
oe G
eom
etry
ALG
EBR
AR
ST
Uis
a r
ecta
ngl
e.
1.If
UZ
�x
�21
an
d Z
S�
3x�
15,f
ind
US
.78
2.If
RZ
�3x
�8
and
ZS
�6x
�28
,fin
d U
Z.
44
3.If
RT
�5x
�8
and
RZ
�4x
�1,
fin
d Z
T.
9
4.If
m�
SU
T�
3x�
6 an
d m
�R
US
�5x
�4,
fin
d m
�S
UT
.39
5.If
m�
SR
T�
x2�
9 an
d m
�U
TR
�2x
�44
,fin
d x.
�5
or
7
6.If
m�
RS
U�
x2�
1 an
d m
�T
US
�3x
�9,
fin
d m
�R
SU
.24
or
3
GH
JK
is a
rec
tan
gle.
Fin
d e
ach
mea
sure
if
m�
1 �
37.
7.m
�2
538.
m�
337
9.m
�4
3710
.m�
553
11.m
�6
106
12.m
�7
74
CO
OR
DIN
ATE
GEO
MET
RYD
eter
min
e w
het
her
BG
HL
is a
rec
tan
gle
give
n e
ach
set
of v
erti
ces.
Ju
stif
y yo
ur
answ
er.
13.B
(�4,
3),G
(�2,
4),H
(1,�
2),L
(�1,
�3)
Yes;
sam
ple
an
swer
:O
pp
osi
te s
ides
are
par
alle
l an
d c
on
secu
tive
sid
esar
e p
erp
end
icu
lar.
14.B
(�4,
5),G
(6,0
),H
(3,�
6),L
(�7,
�1)
Yes;
sam
ple
an
swer
:O
pp
osi
te s
ides
are
co
ng
ruen
t an
d d
iag
on
als
are
con
gru
ent.
15.B
(0,5
),G
(4,7
),H
(5,4
),L
(1,2
)
No
;sa
mp
le a
nsw
er:
Dia
go
nal
s ar
e n
ot
con
gru
ent.
16.L
AN
DSC
API
NG
Hu
nti
ngt
on P
ark
offi
cial
s ap
prov
ed a
rec
tan
gula
r pl
ot o
f la
nd
for
aJa
pan
ese
Zen
gar
den
.Is
it s
uff
icie
nt
to k
now
th
at o
ppos
ite
side
s of
th
e ga
rden
plo
t ar
eco
ngr
uen
t an
d pa
rall
el t
o de
term
ine
that
th
e ga
rden
plo
t is
rec
tan
gula
r? E
xpla
in.
No
;if
yo
u o
nly
kn
ow
th
at o
pp
osi
te s
ides
are
co
ng
ruen
t an
d p
aral
lel,
the
mo
st y
ou
can
co
ncl
ud
e is
th
at t
he
plo
t is
a p
aral
lelo
gra
m.
K
21
5
43
67
JHGU
TSR
Z
Pra
ctic
e (
Ave
rag
e)
Rec
tan
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csR
ecta
ng
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
©G
lenc
oe/M
cGra
w-H
ill43
9G
lenc
oe G
eom
etry
Lesson 8-4
Pre-
Act
ivit
yH
ow a
re r
ecta
ngl
es u
sed
in
ten
nis
?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-4
at
the
top
of p
age
424
in y
our
text
book
.
Are
th
e si
ngl
es c
ourt
an
d do
ubl
es c
ourt
sim
ilar
rec
tan
gles
? E
xpla
in y
our
answ
er.
No
;sa
mp
le a
nsw
er:
All
of
thei
r an
gle
s ar
e co
ng
ruen
t,bu
t th
eir
sid
es a
re n
ot
pro
po
rtio
nal
.Th
e d
ou
ble
s co
urt
is w
ider
in r
elat
ion
to
its
len
gth
.
Rea
din
g t
he
Less
on
1.D
eter
min
e w
het
her
eac
h s
ente
nce
is
alw
ays,
som
etim
es,o
r n
ever
tru
e.
a.If
a q
uad
rila
tera
l h
as f
our
con
gru
ent
angl
es,i
t is
a r
ecta
ngl
e.al
way
sb
.If
con
secu
tive
an
gles
of
a qu
adri
late
ral
are
supp
lem
enta
ry,t
hen
th
e qu
adri
late
ral
is a
rec
tan
gle.
som
etim
esc.
Th
e di
agon
als
of a
rec
tan
gle
bise
ct e
ach
oth
er.
alw
ays
d.
If t
he
diag
onal
s of
a q
uad
rila
tera
l bi
sect
eac
h o
ther
,th
e qu
adri
late
ral
is a
re
ctan
gle.
som
etim
ese.
Con
secu
tive
an
gles
of
a re
ctan
gle
are
com
plem
enta
ry.
nev
erf.
Con
secu
tive
an
gles
of
a re
ctan
gle
are
con
gru
ent.
alw
ays
g.If
th
e di
agon
als
of a
qu
adri
late
ral
are
con
gru
ent,
the
quad
rila
tera
l is
a
rect
angl
e.so
met
imes
h.
A d
iago
nal
of
a re
ctan
gle
bise
cts
two
of i
ts a
ngl
es.
som
etim
esi.
A d
iago
nal
of
a re
ctan
gle
divi
des
the
rect
angl
e in
to t
wo
con
gru
ent
righ
t tr
ian
gles
.al
way
sj.
If t
he
diag
onal
s of
a q
uad
rila
tera
l bi
sect
eac
h o
ther
an
d ar
e co
ngr
uen
t,th
equ
adri
late
ral
is a
rec
tan
gle.
alw
ays
k.
If a
par
alle
logr
am h
as o
ne
righ
t an
gle,
it i
s a
rect
angl
e.al
way
sl.
If a
par
alle
logr
am h
as f
our
con
gru
ent
side
s,it
is
a re
ctan
gle.
som
etim
es
2.A
BC
Dis
a r
ecta
ngl
e w
ith
AD
�A
B.
Nam
e ea
ch o
f th
e fo
llow
ing
in t
his
fig
ure
.
a.al
l se
gmen
ts t
hat
are
con
gru
ent
to B�
E�A�
E�,C�
E�,D�
E�b
.al
l an
gles
con
gru
ent
to �
1�
2,�
5,�
6c.
all
angl
es c
ongr
uen
t to
�7
�3,
�4,
�8
d.
two
pair
s of
con
gru
ent
tria
ngl
es�
AE
D�
�B
EC
,�A
EB
��
DE
C,
�B
CD
��
DA
B,�
AB
C�
�C
DA
Hel
pin
g Y
ou
Rem
emb
er3.
It i
s ea
sier
to
rem
embe
r a
larg
e n
um
ber
of g
eom
etri
c re
lati
onsh
ips
and
theo
rem
s if
you
are
able
to
com
bin
e so
me
of t
hem
.How
can
you
com
bin
e th
e tw
o th
eore
ms
abou
tdi
agon
als
that
you
stu
died
in
th
is l
esso
n?
Sam
ple
an
swer
:A
par
alle
log
ram
is a
rect
ang
le if
an
d o
nly
if it
s d
iag
on
als
are
con
gru
ent.
A12
34
5 67
8DC
BE
©G
lenc
oe/M
cGra
w-H
ill44
0G
lenc
oe G
eom
etry
Co
un
tin
g S
qu
ares
an
d R
ecta
ng
les
Eac
h p
uzz
le b
elow
con
tain
s m
any
squ
ares
an
d/or
rec
tan
gles
.C
oun
t th
em c
aref
ull
y.Yo
u m
ay w
ant
to l
abel
eac
h r
egio
n s
o yo
uca
n l
ist
all
poss
ibil
itie
s.
How
man
y re
ctan
gles
are
in
th
e fi
gure
at
th
e ri
ght?
Lab
el e
ach
sm
all
regi
on w
ith
a l
ette
r.T
hen
lis
t ea
ch r
ecta
ngl
e by
wri
tin
g th
e le
tter
s of
reg
ion
s it
con
tain
s.
A,B
,C,D
,AB
,CD
,AC
,BD
,AB
CD
Th
ere
are
9 re
ctan
gles
.
How
man
y sq
uar
es a
re i
n e
ach
fig
ure
?
1.16
2.10
3.8
How
man
y re
ctan
gles
are
in
eac
h f
igu
re?
1.19
2.25
3.21
AB
CD
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-4
8-4
Exam
ple
Exam
ple
Answers (Lesson 8-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Rh
om
bi a
nd
Sq
uar
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill44
1G
lenc
oe G
eom
etry
Lesson 8-5
Pro
per
ties
of
Rh
om
bi
A r
hom
bu
sis
a q
uad
rila
tera
l w
ith
fou
r co
ngr
uen
t si
des.
Opp
osit
e si
des
are
con
gru
ent,
so a
rh
ombu
s is
als
o a
para
llel
ogra
m a
nd
has
all
of
the
prop
erti
es o
f a
para
llel
ogra
m.R
hom
bi
also
hav
e th
e fo
llow
ing
prop
erti
es.
The
dia
gona
ls a
re p
erpe
ndic
ular
.M�
H�⊥
R�O�
Eac
h di
agon
al b
isec
ts a
pai
r of
opp
osite
ang
les.
M�H�
bise
cts
�R
MO
and
�R
HO
.R�
O�bi
sect
s �
MR
Han
d �
MO
H.
If th
e di
agon
als
of a
par
alle
logr
am a
re
If R
HO
Mis
a p
aral
lelo
gram
and
R�O�
⊥M�
H�,
perp
endi
cula
r, th
en t
he f
igur
e is
a r
hom
bus.
then
RH
OM
is a
rho
mbu
s.
In r
hom
bu
s A
BC
D,m
�B
AC
�32
.Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle.
AB
CD
is a
rh
ombu
s,so
th
e di
agon
als
are
perp
endi
cula
r an
d �
AB
Eis
a r
igh
t tr
ian
gle.
Th
us
m�
4 �
90 a
nd
m�
1 �
90 �
32 o
r 58
.Th
edi
agon
als
in a
rh
ombu
s bi
sect
th
e ve
rtex
an
gles
,so
m�
1 �
m�
2.T
hu
s,m
�2
�58
.
A r
hom
bus
is a
par
alle
logr
am,s
o th
e op
posi
te s
ides
are
par
alle
l.�
BA
Can
d �
3 ar
eal
tern
ate
inte
rior
an
gles
for
par
alle
l li
nes
,so
m�
3 �
32.
AB
CD
is a
rh
omb
us.
1.If
m�
AB
D�
60,f
ind
m�
BD
C.
602.
If A
E�
8,fi
nd
AC
.16
3.If
AB
�26
an
d B
D�
20,f
ind
AE
.24
4.F
ind
m�
CE
B.
90
5.If
m�
CB
D�
58,f
ind
m�
AC
B.
326.
If A
E�
3x�
1 an
d A
C�
16,f
ind
x.3
7.If
m�
CD
B�
6yan
d m
�A
CB
�2y
�10
,fin
d y.
10
8.If
AD
�2x
�4
and
CD
�4x
�4,
fin
d x.
4
9.a.
Wh
at i
s th
e m
idpo
int
of F�
H�?
(5,5
)
b.
Wh
at i
s th
e m
idpo
int
of G�
J�?
(5,5
)
c.W
hat
kin
d of
fig
ure
is
FG
HJ
? E
xpla
in.
FG
HJ
is a
par
alle
log
ram
bec
ause
th
e d
iag
on
als
bis
ect
each
oth
er.
d.
Wh
at i
s th
e sl
ope
of F�
H�?
�1 2�
e.W
hat
is
the
slop
e of
G�J�
?�
2
f.B
ased
on
par
ts c
,d,a
nd
e,w
hat
kin
d of
fig
ure
is
FG
HJ
? E
xpla
in.
FG
HJ
is a
par
alle
log
ram
wit
h p
erp
end
icu
lar
dia
go
nal
s,so
it is
a r
ho
mbu
s.
x
y
O
J
GF
HC
AD
EB
CA
DEB
32�
12
34
MO
HR
B
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill44
2G
lenc
oe G
eom
etry
Pro
per
ties
of
Squ
ares
A s
quar
e h
as a
ll t
he
prop
erti
es o
f a
rhom
bus
and
all
the
prop
erti
es o
f a
rect
angl
e.
Fin
d t
he
mea
sure
of
each
nu
mb
ered
an
gle
of
squ
are
AB
CD
.
Usi
ng
prop
erti
es o
f rh
ombi
an
d re
ctan
gles
,th
e di
agon
als
are
perp
endi
cula
r an
d co
ngr
uen
t.�
AB
Eis
a r
igh
t tr
ian
gle,
so m
�1
�m
�2
�90
.
Eac
h v
erte
x an
gle
is a
rig
ht
angl
e an
d th
e di
agon
als
bise
ct t
he
vert
ex a
ngl
es,
so m
�3
�m
�4
�m
�5
�45
.
Det
erm
ine
wh
eth
er t
he
give
n v
erti
ces
rep
rese
nt
a p
ara
llel
ogra
m,r
ecta
ngl
e,rh
ombu
s,or
sq
ua
re.E
xpla
in y
our
reas
onin
g.
1.A
(0,2
),B
(2,4
),C
(4,2
),D
(2,0
)
Sq
uar
e;th
e fo
ur
sid
es a
re �
and
co
nse
cuti
ve s
ides
are
⊥.
2.D
(�2,
1),E
(�1,
3),F
(3,1
),G
(2,�
1)
Rec
tan
gle
;bo
th p
airs
of
op
po
site
sid
es a
re ||
and
co
nse
cuti
ve s
ides
are
⊥.
3.A
(�2,
�1)
,B(0
,2),
C(2
,�1)
,D(0
,�4)
Rh
om
bus;
the
fou
r si
des
are
�an
d c
on
secu
tive
sid
es a
re n
ot
⊥.
4.A
(�3,
0),B
(�1,
3),C
(5,�
1),D
(3,�
4)
Rec
tan
gle
;bo
th p
airs
of
op
po
site
sid
es a
re ||
and
co
nse
cuti
ve s
ides
are
⊥.
5.S
(�1,
4),T
(3,2
),U
(1,�
2),V
(�3,
0)
Sq
uar
e;th
e fo
ur
sid
es a
re �
and
co
nse
cuti
ve s
ides
are
⊥.
6.F
(�1,
0),G
(1,3
),H
(4,1
),I(
2,�
2)
Sq
uar
e;th
e fo
ur
sid
es a
re �
and
co
nse
cuti
ve s
ides
are
⊥.
7.S
quar
e R
ST
Uh
as v
erti
ces
R(�
3,�
1),S
(�1,
2),a
nd
T(2
,0).
Fin
d th
e co
ordi
nat
es o
fve
rtex
U. (
0,�
3)
C
AD
12
34
5
EB
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Rh
om
bi a
nd
Sq
uar
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Rh
om
bi a
nd
Sq
uar
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill44
3G
lenc
oe G
eom
etry
Lesson 8-5
Use
rh
omb
us
DK
LM
wit
h A
M�
4x,A
K�
5x�
3,an
d
DL
�10
.
1.F
ind
x.2.
Fin
d A
L.
35
3.F
ind
m�
KA
L.
4.F
ind
DM
.
9013
Use
rh
omb
us
RS
TV
wit
h R
S�
5y�
2,S
T�
3y�
6,an
d
NV
�6.
5.F
ind
y.6.
Fin
d T
V.
212
7.F
ind
m�
NT
V.
8.F
ind
m�
SV
T.
3060
9.F
ind
m�
RS
T.
10.F
ind
m�
SR
V.
120
60
CO
OR
DIN
ATE
GEO
MET
RYG
iven
eac
h s
et o
f ve
rtic
es,d
eter
min
e w
het
her
�Q
RS
Tis
a r
hom
bus,
a re
cta
ngl
e,or
a s
qu
are
.Lis
t al
l th
at a
pp
ly.E
xpla
in y
our
reas
onin
g.
11.Q
(3,5
),R
(3,1
),S
(�1,
1),T
(�1,
5)
Rh
om
bus,
rect
ang
le,s
qu
are;
all s
ides
are
co
ng
ruen
t an
d t
he
dia
go
nal
sar
e p
erp
end
icu
lar
and
co
ng
ruen
t.
12.Q
(�5,
12),
R(5
,12)
,S(�
1,4)
,T(�
11,4
)
Rh
om
bus;
all s
ides
are
co
ng
ruen
t an
d t
he
dia
go
nal
s ar
e p
erp
end
icu
lar,
but
no
t co
ng
ruen
t.
13.Q
(�6,
�1)
,R(4
,�6)
,S(2
,5),
T(�
8,10
)
Rh
om
bus;
all s
ides
are
co
ng
ruen
t an
d t
he
dia
go
nal
s ar
e p
erp
end
icu
lar,
but
no
t co
ng
ruen
t.
14.Q
(2,�
4),R
(�6,
�8)
,S(�
10,2
),T
(�2,
6)
No
ne;
op
po
site
sid
es a
re c
on
gru
ent,
but
the
dia
go
nal
s ar
e n
eith
erco
ng
ruen
t n
or
per
pen
dic
ula
r.
RV
NS
T
ML
AD
K
©G
lenc
oe/M
cGra
w-H
ill44
4G
lenc
oe G
eom
etry
Use
rh
omb
us
PR
YZ
wit
h R
K�
4y�
1,Z
K�
7y�
14,
PK
�3x
�1,
and
YK
�2x
�6.
1.F
ind
PY
.2.
Fin
d R
Z.
4042
3.F
ind
RY
.4.
Fin
d m
�Y
KZ
.
2990
Use
rh
omb
us
MN
PQ
wit
h P
Q�
3�2�,
PA
�4x
�1,
and
A
M�
9x �
6.
5.F
ind
AQ
.6.
Fin
d m
�A
PQ
.
345
7.F
ind
m�
MN
P.
8.F
ind
PM
.
906
CO
OR
DIN
ATE
GEO
MET
RYG
iven
eac
h s
et o
f ve
rtic
es,d
eter
min
e w
het
her
�B
EF
Gis
a r
hom
bus,
a re
cta
ngl
e,or
a s
qu
are
.Lis
t al
l th
at a
pp
ly.E
xpla
in y
our
reas
onin
g.
9.B
(�9,
1),E
(2,3
),F
(12,
�2)
,G(1
,�4)
Rh
om
bus;
all s
ides
are
co
ng
ruen
t an
d t
he
dia
go
nal
s ar
e p
erp
end
icu
lar,
but
no
t co
ng
ruen
t.
10.B
(1,3
),E
(7,�
3),F
(1,�
9),G
(�5,
�3)
Rh
om
bus,
rect
ang
le,s
qu
are;
all s
ides
are
co
ng
ruen
t an
d t
he
dia
go
nal
sar
e p
erp
end
icu
lar
and
co
ng
ruen
t.
11.B
(�4,
�5)
,E(1
,�5)
,F(�
7,�
1),G
(�2,
�1)
No
ne;
two
of
the
op
po
site
sid
es a
re n
ot
con
gru
ent.
12.T
ESSE
LATI
ON
ST
he
figu
re i
s an
exa
mpl
e of
a t
esse
llat
ion
.Use
a r
ule
r
or p
rotr
acto
r to
mea
sure
th
e sh
apes
an
d th
en n
ame
the
quad
rila
tera
ls
use
d to
for
m t
he
figu
re.
Th
e fi
gu
re c
on
sist
s o
f 6
con
gru
ent
rho
mb
i.
AN M
QP
K
P
YR
Z
Pra
ctic
e (
Ave
rag
e)
Rh
om
bi a
nd
Sq
uar
es
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csR
ho
mb
i an
d S
qu
ares
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
©G
lenc
oe/M
cGra
w-H
ill44
5G
lenc
oe G
eom
etry
Lesson 8-5
Pre-
Act
ivit
yH
ow c
an y
ou r
ide
a b
icyc
le w
ith
sq
uar
e w
hee
ls?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-5
at
the
top
of p
age
431
in y
our
text
book
.
If y
ou d
raw
a d
iago
nal
on
th
e su
rfac
e of
on
e of
th
e sq
uar
e w
hee
ls s
how
n i
nth
e pi
ctur
e in
you
r te
xtbo
ok,h
ow c
an y
ou d
escr
ibe
the
two
tria
ngle
s th
at a
refo
rmed
?S
amp
le a
nsw
er:t
wo
co
ng
ruen
t is
osc
eles
rig
ht
tria
ng
les
Rea
din
g t
he
Less
on
1.S
ketc
h e
ach
of
the
foll
owin
g.S
amp
le a
nsw
ers
are
giv
en.
a.a
quad
rila
tera
l w
ith
per
pen
dicu
lar
b.
a qu
adri
late
ral
wit
h c
ongr
uen
t di
agon
als
that
is
not
a r
hom
bus
diag
onal
s th
at i
s n
ot a
rec
tan
gle
c.a
quad
rila
tera
l w
hos
e di
agon
als
are
perp
endi
cula
r an
d bi
sect
eac
h o
ther
,bu
t ar
e n
ot c
ongr
uen
t
2.L
ist
all
of t
he
foll
owin
g sp
ecia
l qu
adri
late
rals
th
at h
ave
each
lis
ted
prop
erty
:pa
rall
elog
ram
,rec
tan
gle,
rhom
bus,
squ
are.
a.T
he
diag
onal
s ar
e co
ngr
uen
t.re
ctan
gle
,sq
uar
eb
.O
ppos
ite
side
s ar
e co
ngr
uen
t.p
aral
lelo
gra
m,r
ecta
ng
le,r
ho
mbu
s,sq
uar
ec.
Th
e di
agon
als
are
perp
endi
cula
r.rh
om
bus,
squ
are
d.
Con
secu
tive
an
gles
are
su
pple
men
tary
.p
aral
lelo
gra
m,r
ecta
ng
le,r
ho
mbu
s,sq
uar
ee.
Th
e qu
adri
late
ral
is e
quil
ater
al.
rho
mbu
s,sq
uar
ef.
Th
e qu
adri
late
ral
is e
quia
ngu
lar.
rect
ang
le,s
qu
are
g.T
he
diag
onal
s ar
e pe
rpen
dicu
lar
and
con
gru
ent.
squ
are
h.
A p
air
of o
ppos
ite
side
s is
bot
h p
aral
lel
and
con
gru
ent.
par
alle
log
ram
,rec
tan
gle
,rh
om
bus,
squ
are
3.W
hat
is
the
com
mon
nam
e fo
r a
regu
lar
quad
rila
tera
l? E
xpla
in y
our
answ
er.
Sq
uar
e;sa
mp
le a
nsw
er:
All
fou
r si
des
of
a sq
uar
e ar
e co
ng
ruen
t an
d a
ll fo
ur
ang
les
are
con
gru
ent.
Hel
pin
g Y
ou
Rem
emb
er
4.A
goo
d w
ay t
o re
mem
ber
som
eth
ing
is t
o ex
plai
n i
t to
som
eon
e el
se.S
upp
ose
that
you
rcl
assm
ate
Lu
is i
s h
avin
g tr
oubl
e re
mem
beri
ng
wh
ich
of
the
prop
erti
es h
e h
as l
earn
ed i
nth
is c
hap
ter
appl
y to
squ
ares
.How
can
you
hel
p h
im?
Sam
ple
an
swer
:A
sq
uar
e is
a p
aral
lelo
gra
m t
hat
is b
oth
a r
ecta
ng
le a
nd
a r
ho
mbu
s,so
all
pro
per
ties
of
par
alle
log
ram
s,re
ctan
gle
s,an
d r
ho
mb
i ap
ply
to
sq
uar
es.
©G
lenc
oe/M
cGra
w-H
ill44
6G
lenc
oe G
eom
etry
Cre
atin
g P
yth
ago
rean
Pu
zzle
sB
y dr
awin
g tw
o sq
uar
es a
nd
cutt
ing
them
in
a c
erta
in w
ay,y
ouca
n m
ake
a pu
zzle
th
at d
emon
stra
tes
the
Pyt
hag
orea
n T
heo
rem
.A
sam
ple
puzz
le i
s sh
own
.You
can
cre
ate
you
r ow
n p
uzz
le b
yfo
llow
ing
the
inst
ruct
ion
s be
low
.
See
stu
den
ts’w
ork
.A s
amp
le a
nsw
er is
sh
ow
n.
1.C
aref
ull
y co
nst
ruct
a s
quar
e an
d la
bel
the
len
gth
of
a s
ide
as a
.Th
en c
onst
ruct
a s
mal
ler
squ
are
to
the
righ
t of
it
and
labe
l th
e le
ngt
h o
f a
side
as
b,as
sh
own
in
th
e fi
gure
abo
ve.T
he
base
s sh
ould
be
adj
acen
t an
d co
llin
ear.
2.M
ark
a po
int
Xth
at i
s b
un
its
from
th
e le
ft e
dge
of t
he
larg
er s
quar
e.T
hen
dra
w t
he
segm
ents
fr
om t
he
upp
er l
eft
corn
er o
f th
e la
rger
squ
are
to
poin
t X
,an
d fr
om p
oin
t X
to t
he
upp
er r
igh
t co
rner
of
th
e sm
alle
r sq
uar
e.
3.C
ut
out
and
rear
ran
ge y
our
five
pie
ces
to f
orm
a
larg
er s
quar
e.D
raw
a d
iagr
am t
o sh
ow y
our
answ
er.
4.V
erif
y th
at t
he
len
gth
of
each
sid
e is
equ
al t
o
�a2
�b
�2 �.
a
a
bX
b
b
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-5
8-5
Answers (Lesson 8-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Trap
ezo
ids
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill44
7G
lenc
oe G
eom
etry
Lesson 8-6
Pro
per
ties
of
Trap
ezo
ids
A t
rap
ezoi
dis
a q
uad
rila
tera
l w
ith
ex
actl
y on
e pa
ir o
f pa
rall
el s
ides
.Th
e pa
rall
el s
ides
are
cal
led
bas
esan
d th
e n
onpa
rall
el s
ides
are
cal
led
legs
.If
the
legs
are
con
gru
ent,
the
trap
ezoi
d is
an
iso
scel
es t
rap
ezoi
d.I
n a
n i
sosc
eles
tra
pezo
id
both
pai
rs o
f b
ase
angl
esar
e co
ngr
uen
t.S
TU
Ris
an
isos
cele
s tr
apez
oid.
S�R�
�T�U�
; �
R�
�U
, �
S�
�T
Th
e ve
rtic
es o
f A
BC
Dar
e A
(�3,
�1)
,B(�
1,3)
,C
(2,3
),an
d D
(4,�
1).V
erif
y th
at A
BC
Dis
a t
rap
ezoi
d.
slop
e of
A �B�
�� �3 1� �
(� (�1 3) )
��
�4 2��
2
slop
e of
A �D�
��� 41 ��
(�(�31 ))
��
�0 7��
0
slop
e of
B �C�
�� 2
3 �
� (�3 1)
��
�0 3��
0
slop
e of
C �D�
��� 41 ��
23�
��� 24 �
��
2
Exa
ctly
tw
o si
des
are
para
llel
,A�D�
and
B�C�
,so
AB
CD
is a
tra
pezo
id.A
B�
CD
,so
AB
CD
isan
iso
scel
es t
rape
zoid
.
In E
xerc
ises
1–3
,det
erm
ine
wh
eth
er A
BC
Dis
a t
rap
ezoi
d.I
f so
,det
erm
ine
wh
eth
er i
t is
an
iso
scel
es t
rap
ezoi
d.E
xpla
in.
1.A
(�1,
1),B
(2,1
),C
(3,�
2),a
nd
D(2
,�2)
Slo
pe
of
A�B�
�0,
slo
pe
of
D�C�
�0,
slo
pe
of
A�D�
��
1,sl
op
e o
f B�
C��
�3.
Exa
ctly
tw
o s
ides
are
par
alle
l,so
AB
CD
is a
tra
pez
oid
.AD
�3�
2�an
d
BC
��
10�;
AD
�B
C,s
o A
BC
Dis
no
t is
osc
eles
.2.
A(3
,�3)
,B(�
3,�
3),C
(�2,
3),a
nd
D(2
,3)
Slo
pe
of
A�B�
�0,
slo
pe
of
D�C�
�0,
slo
pe
of
B�C�
�6,
slo
pe
of
A�D�
��
6.E
xact
ly 2
sid
es a
re || ,
so A
BC
Dis
a t
rap
ezo
id.B
C�
�37�
and
AD
��
37�;
BC
�A
D,s
o A
BC
Dis
iso
scel
es.
3.A
(1,�
4),B
(�3,
�3)
,C(�
2,3)
,an
d D
(2,2
)
Slo
pe
of
A�B�
��
�1 4� ,sl
op
e o
f D�
C��
��1 4� ,
slo
pe
of
B�C�
�6,
slo
pe
of
A�D�
�6.
Bo
th p
airs
of
op
po
site
sid
es a
re p
aral
lel,
so A
BC
Dis
no
t a
trap
ezo
id.
4.T
he
vert
ices
of
an i
sosc
eles
tra
pezo
id a
re R
(�2,
2),S
(2,2
),T
(4,�
1),a
nd
U(�
4,�
1).
Ver
ify
that
th
e di
agon
als
are
con
gru
ent.
RT
��
((4
��
(�2)
)2�
�(�
�1
�2
�)2
)��
�45�
SU
��
((�
4 �
�(2
))2
��
(�1
��
2)2
�)��
�45�
x
y
O
B
DA
C
T
RU
base
base
Sle
g
AB
��
(�3
��
(�1)
)�
2�
(��
1 �
3�
)2 ��
�4
�1
�6�
��
20��
2�5�
CD
��
(2 �
4�
)2�
(�
3 �
(��
1))2
��
�4
�1
�6�
��
20��
2�5�
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill44
8G
lenc
oe G
eom
etry
Med
ian
s o
f Tr
apez
oid
sT
he
med
ian
of a
tra
pezo
id i
s th
e se
gmen
t th
at jo
ins
the
mid
poin
ts o
f th
e le
gs.I
t is
par
alle
l to
th
e ba
ses,
and
its
len
gth
is
one-
hal
f th
e su
m o
f th
e le
ngt
hs
of t
he
base
s.
In t
rape
zoid
HJ
KL
,MN
��1 2� (
HJ
�L
K).
M�N�
is t
he
med
ian
of
trap
ezoi
d R
ST
U.F
ind
x.
MN
��1 2� (
RS
�U
T)
30�
�1 2� (3x
�5
�9x
�5)
30�
�1 2� (12
x)
30�
6x5
�x
M �N�
is t
he
med
ian
of
trap
ezoi
d H
JK
L.F
ind
eac
h i
nd
icat
ed v
alu
e.
1.F
ind
MN
if H
J�
32 a
nd
LK
�60
.
46
2.F
ind
LK
if H
J�
18 a
nd
MN
�28
.
38
3.F
ind
MN
if H
J�
LK
�42
.
21
4.F
ind
m�
LM
Nif
m�
LH
J�
116.
116
5.F
ind
m�
JK
Lif
HJ
KL
is i
sosc
eles
an
d m
�H
LK
�62
.
62
6.F
ind
HJ
if M
N�
5x�
6,H
J�
3x�
6,an
d L
K�
8x.
24
7.F
ind
the
len
gth
of
the
med
ian
of
a tr
apez
oid
wit
h v
erti
ces
A(�
2,2)
,B(3
,3),
C(7
,0),
and
D(�
3,�
2).
�3 2� �26�
LK
NM
HJ
UT
NM
R3x
� 5
30 9x �
5S
LK
NM
HJ
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Trap
ezo
ids
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Trap
ezo
ids
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill44
9G
lenc
oe G
eom
etry
Lesson 8-6
CO
OR
DIN
ATE
GEO
MET
RYA
BC
Dis
a q
uad
rila
tera
l w
ith
ver
tice
s A
(�4,
�3)
,B
(3,�
3),C
(6,4
),D
(�7,
4).
1.V
erif
y th
at A
BC
Dis
a t
rape
zoid
.
A�B�
|| C�D�
2.D
eter
min
e w
het
her
AB
CD
is a
n i
sosc
eles
tra
pezo
id.E
xpla
in.
iso
scel
es;
AD
��
58�an
d B
C�
�58�
CO
OR
DIN
ATE
GEO
MET
RYE
FG
His
a q
uad
rila
tera
l w
ith
ver
tice
s E
(1,3
),F
(5,0
),G
(8,�
5),H
(�4,
4).
3.V
erif
y th
at E
FG
His
a t
rape
zoid
.
E�F�
|| G�H�
4.D
eter
min
e w
het
her
EF
GH
is a
n i
sosc
eles
tra
pezo
id.E
xpla
in.
no
t is
osc
eles
;E
H�
�26�
and
FG
��
34�
CO
OR
DIN
ATE
GEO
MET
RYL
MN
Pis
a q
uad
rila
tera
l w
ith
ver
tice
s L
(�1,
3),M
(�4,
1),
N(�
6,3)
,P(0
,7).
5.V
erif
y th
at L
MN
Pis
a t
rape
zoid
.L�M�
|| N�P�
6.D
eter
min
e w
het
her
LM
NP
is a
n i
sosc
eles
tra
pezo
id.E
xpla
in.
no
t is
osc
eles
;L
P�
�17�
and
MN
��
8�
ALG
EBR
AF
ind
th
e m
issi
ng
mea
sure
(s)
for
the
give
n t
rap
ezoi
d.
7.F
or t
rape
zoid
HJ
KL
,San
d T
are
8.F
or t
rape
zoid
WX
YZ
,Pan
d Q
are
mid
poin
ts o
f th
e le
gs.F
ind
HJ
.58
mid
poin
ts o
f th
e le
gs.F
ind
WX
.5
9.F
or t
rape
zoid
DE
FG
,Tan
d U
are
10.F
or i
sosc
eles
tra
pezo
id Q
RS
T,f
ind
the
mid
poin
ts o
f th
e le
gs.F
ind
TU
,m�
E,
len
gth
of
the
med
ian
,m�
Q,a
nd
m�
S.
and
m�
G.
28,9
5,14
542
.5,1
25,5
5
ST
RQ
125�
60 25
DE F
G
TU
85�
35�
42
14
WX
YZ
PQ
12 19
HJ
KLS
T72 86
©G
lenc
oe/M
cGra
w-H
ill45
0G
lenc
oe G
eom
etry
CO
OR
DIN
ATE
GEO
MET
RYR
ST
Uis
a q
uad
rila
tera
l w
ith
ver
tice
s R
(�3,
�3)
,S(5
,1),
T(1
0,�
2),U
(�4,
�9)
.
1.V
erif
y th
at R
ST
Uis
a t
rape
zoid
.
R�S�
|| T�U�
2.D
eter
min
e w
het
her
RS
TU
is a
n i
sosc
eles
tra
pezo
id.E
xpla
in.
no
t is
osc
eles
;R
U�
�37�
and
ST
��
34�
CO
OR
DIN
ATE
GEO
MET
RYB
GH
Jis
a q
uad
rila
tera
l w
ith
ver
tice
s B
(�9,
1),G
(2,3
),H
(12,
�2)
,J(�
10,�
6).
3.V
erif
y th
at B
GH
Jis
a t
rape
zoid
.
B�G�
|| H�J�
4.D
eter
min
e w
het
her
BG
HJ
is a
n i
sosc
eles
tra
pezo
id.E
xpla
in.
no
t is
osc
eles
;B
J�
�50�
and
GH
��
125
�
ALG
EBR
AF
ind
th
e m
issi
ng
mea
sure
(s)
for
the
give
n t
rap
ezoi
d.
5.F
or t
rape
zoid
CD
EF
,Van
d Y
are
6.F
or t
rape
zoid
WR
LP
,Ban
d C
are
mid
poin
ts o
f th
e le
gs.F
ind
CD
.38
mid
poin
ts o
f th
e le
gs.F
ind
LP
.18
7.F
or t
rape
zoid
FG
HI,
Kan
d M
are
8.F
or i
sosc
eles
tra
pezo
id T
VZ
Y,f
ind
the
mid
poin
ts o
f th
e le
gs.F
ind
FI,
m�
F,
len
gth
of
the
med
ian
,m�
T,a
nd
m�
Z.
and
m�
I.51
,40,
5519
.5,6
0,12
0
9.C
ON
STR
UC
TIO
NA
set
of
stai
rs l
eadi
ng
to t
he
entr
ance
of
a bu
ildi
ng
is d
esig
ned
in
th
esh
ape
of a
n i
sosc
eles
tra
pezo
id w
ith
th
e lo
nge
r ba
se a
t th
e bo
ttom
of
the
stai
rs a
nd
the
shor
ter
base
at
the
top.
If t
he
bott
om o
f th
e st
airs
is
21 f
eet
wid
e an
d th
e to
p is
14,
fin
dth
e w
idth
of
the
stai
rs h
alfw
ay t
o th
e to
p.17
.5 f
t
10.D
ESK
TO
PSA
car
pen
ter
nee
ds t
o re
plac
e se
vera
l tr
apez
oid-
shap
ed d
eskt
ops
in a
clas
sroo
m.T
he
carp
ente
r kn
ows
the
len
gth
s of
bot
h b
ases
of
the
desk
top.
Wh
at o
ther
mea
sure
men
ts,i
f an
y,do
es t
he
carp
ente
r n
eed?
Sam
ple
an
swer
:th
e m
easu
res
of
the
bas
e an
gle
sV
YZ
T60
�34 5
H
KM
G
IF
140�
125�
21
36
WR
LP
BC
4266F
E
DC
VY
2818Pra
ctic
e (
Ave
rag
e)
Trap
ezo
ids
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csTr
apez
oid
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
©G
lenc
oe/M
cGra
w-H
ill45
1G
lenc
oe G
eom
etry
Lesson 8-6
Pre-
Act
ivit
yH
ow a
re t
rap
ezoi
ds
use
d i
n a
rch
itec
ture
?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-6
at
the
top
of p
age
439
in y
our
text
book
.
How
mig
ht
trap
ezoi
ds b
e u
sed
in t
he
inte
rior
des
ign
of
a h
ome?
Sam
ple
an
swer
:fl
oo
r ti
les
for
a ki
tch
en o
r b
ath
roo
m
Rea
din
g t
he
Less
on
1.In
th
e fi
gure
at
the
righ
t,E
FG
His
a t
rape
zoid
,Iis
th
e
mid
poin
t of
F �E�
,an
d J
is t
he
mid
poin
t of
G�H�
.Ide
nti
fy e
ach
of
th
e fo
llow
ing
segm
ents
or
angl
es i
n t
he
figu
re.
a.th
e ba
ses
of t
rape
zoid
EF
GH
F�G�,E�
H�b
.th
e tw
o pa
irs
of b
ase
angl
es o
f tr
apez
oid
EF
GH
�E
and
�H
;�
Fan
d �
Gc.
the
legs
of
trap
ezoi
d E
FG
HF�E�
,G�H�
d.
the
med
ian
of
trap
ezoi
d E
FG
HI�J�
2.D
eter
min
e w
het
her
eac
h s
tate
men
t is
tru
eor
fal
se.I
f th
e st
atem
ent
is f
alse
,exp
lain
wh
y.
a.A
tra
pezo
id i
s a
spec
ial
kin
d of
par
alle
logr
am.
Fals
e;sa
mp
le a
nsw
er:
Ap
aral
lelo
gra
m h
as t
wo
pai
rs o
f p
aral
lel s
ides
,bu
t a
trap
ezo
id h
as o
nly
on
e p
air
of
par
alle
l sid
es.
b.
Th
e di
agon
als
of a
tra
pezo
id a
re c
ongr
uen
t.Fa
lse;
sam
ple
an
swer
:Th
is is
on
lytr
ue
for
iso
scel
es t
rap
ezo
ids.
c.T
he
med
ian
of
a tr
apez
oid
is p
aral
lel
to t
he
legs
.Fa
lse;
sam
ple
an
swer
:Th
em
edia
n is
par
alle
l to
th
e b
ases
.d
.T
he
len
gth
of
the
med
ian
of
a tr
apez
oid
is t
he
aver
age
of t
he
len
gth
of
the
base
s.tr
ue
e.A
tra
pezo
id h
as t
hre
e m
edia
ns.
Fals
e;sa
mp
le a
nsw
er:
A t
rap
ezo
id h
as o
nly
on
e m
edia
n.
f.T
he
base
s of
an
iso
scel
es t
rape
zoid
are
con
gru
ent.
Fals
e:sa
mp
le a
nsw
er:T
he
leg
s ar
e co
ng
ruen
t.g.
An
iso
scel
es t
rape
zoid
has
tw
o pa
irs
of c
ongr
uen
t an
gles
.tr
ue
h.
Th
e m
edia
n o
f an
iso
scel
es t
rape
zoid
div
ides
th
e tr
apez
oid
into
tw
o sm
alle
r is
osce
les
trap
ezoi
ds.
tru
e
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a n
ew g
eom
etri
c th
eore
m i
s to
rel
ate
it t
o on
e yo
u a
lrea
dykn
ow.N
ame
and
stat
e in
wor
ds a
th
eore
m a
bou
t tr
ian
gles
th
at i
s si
mil
ar t
o th
e th
eore
min
th
is l
esso
n a
bou
t th
e m
edia
n o
f a
trap
ezoi
d.Tr
ian
gle
Mid
seg
men
t Th
eore
m;
am
idse
gm
ent
of
a tr
ian
gle
is p
aral
lel t
o o
ne
sid
e o
f th
e tr
ian
gle
,an
d it
sle
ng
th is
on
e-h
alf
the
len
gth
of
that
sid
e.
HE
JI
GF
©G
lenc
oe/M
cGra
w-H
ill45
2G
lenc
oe G
eom
etry
Qu
adri
late
rals
in C
on
stru
ctio
n
Qu
adri
late
rals
are
oft
en u
sed
in c
onst
ruct
ion
wor
k.
1.T
he
diag
ram
at
the
righ
t re
pres
ents
a r
oof
fram
e an
d sh
ows
man
y qu
adri
late
rals
.Fin
d th
e fo
llow
ing
shap
es i
n t
he
diag
ram
an
d sh
ade
in t
hei
r ed
ges.
See
stu
den
ts’w
ork
.
a.is
osce
les
tria
ngl
e
b.
scal
ene
tria
ngl
e
c.re
ctan
gle
d.
rhom
bus
e.tr
apez
oid
(not
iso
scel
es)
f.is
osce
les
trap
ezoi
d
2.T
he
figu
re a
t th
e ri
ght
repr
esen
ts a
win
dow
.Th
e w
oode
n p
art
betw
een
th
e pa
nes
of
glas
s is
3 i
nch
es
wid
e.T
he
fram
e ar
oun
d th
e ou
ter
edge
is
9 in
ches
w
ide.
Th
e ou
tsid
e m
easu
rem
ents
of
the
fram
e ar
e 60
in
ches
by
81 i
nch
es.T
he
hei
ght
of t
he
top
and
bott
om p
anes
is
the
sam
e.T
he
top
thre
e pa
nes
are
th
e sa
me
size
.
a.H
ow w
ide
is t
he
bott
om p
ane
of g
lass
?42
in.
b.
How
wid
e is
eac
h t
op p
ane
of g
lass
?12
in.
c.H
ow h
igh
is
each
pan
e of
gla
ss?
30 in
.
3.E
ach
edg
e of
th
is b
ox h
as b
een
rei
nfo
rced
w
ith
a p
iece
of
tape
.Th
e bo
x is
10
inch
es
hig
h,2
0 in
ches
wid
e,an
d 12
in
ches
dee
p.W
hat
is
the
len
gth
of
the
tape
th
at h
as
been
use
d?16
8 in
.10
in.
12 in
.
20 in
.60 in
.
81 in
.
9 in
.
3 in
.
3 in
.
3 in
.
9 in
.
9 in
.
jack
rafte
rs
hip
rafte
r
com
mon
rafte
rs
Ro
of
Fram
e
ridge
boa
rdva
lley
rafte
r
plat
e
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-6
8-6
Answers (Lesson 8-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Co
ord
inat
e P
roo
f w
ith
Qu
adri
late
rals
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill45
3G
lenc
oe G
eom
etry
Lesson 8-7
Posi
tio
n F
igu
res
Coo
rdin
ate
proo
fs u
se p
rope
rtie
s of
lin
es a
nd
segm
ents
to
prov
ege
omet
ric
prop
erti
es.T
he
firs
t st
ep i
n w
riti
ng
a co
ordi
nat
e pr
oof
is t
o pl
ace
the
figu
re o
n t
he
coor
din
ate
plan
e in
a c
onve
nie
nt
way
.Use
th
e fo
llow
ing
guid
elin
es f
or p
laci
ng
a fi
gure
on
the
coor
din
ate
plan
e.
1.U
se t
he o
rigin
as
a ve
rtex
, so
one
set
of
coor
dina
tes
is (
0, 0
), o
r us
e th
e or
igin
as
the
cent
er o
f th
e fig
ure.
2.P
lace
at
leas
t on
e si
de o
f th
e qu
adril
ater
al o
n an
axi
s so
you
will
hav
e so
me
zero
coo
rdin
ates
.
3.Tr
y to
kee
p th
e qu
adril
ater
al in
the
firs
t qu
adra
nt s
o yo
u w
ill h
ave
posi
tive
coor
dina
tes.
4.U
se c
oord
inat
es t
hat
mak
e th
e co
mpu
tatio
ns a
s ea
sy a
s po
ssib
le.
For
exa
mpl
e, u
se e
ven
num
bers
if y
ou
are
goin
g to
be
findi
ng m
idpo
ints
.
Pos
itio
n a
nd
lab
el a
rec
tan
gle
wit
h s
ides
aan
d b
un
its
lon
g on
th
e co
ord
inat
e p
lan
e.
•P
lace
on
e ve
rtex
at
the
orig
in f
or R
,so
one
vert
ex i
s R
(0,0
).•
Pla
ce s
ide
R �U�
alon
g th
e x-
axis
an
d si
de R�
S�al
ong
the
y-ax
is,
wit
h t
he
rect
angl
e in
th
e fi
rst
quad
ran
t.•
Th
e si
des
are
aan
d b
un
its,
so l
abel
tw
o ve
rtic
es S
(0,a
) an
d U
(b,0
).•
Ver
tex
Tis
bu
nit
s ri
ght
and
au
nit
s u
p,so
th
e fo
urt
h v
erte
x is
T(b
,a).
Nam
e th
e m
issi
ng
coor
din
ates
for
eac
h q
uad
rila
tera
l.
1.2.
3.
C(c
,c)
N(b
,c)
B(b
,c);
C(a
�b
,c)
Pos
itio
n a
nd
lab
el e
ach
qu
adri
late
ral
on t
he
coor
din
ate
pla
ne.
Sam
ple
an
swer
sar
e g
iven
.4.
squ
are
ST
UV
wit
h
5.pa
rall
elog
ram
PQ
RS
6.re
ctan
gle
AB
CD
wit
hsi
de s
un
its
wit
h c
ongr
uen
t di
agon
als
len
gth
tw
ice
the
wid
th
x
y
D( 2
a, 0
)
C( 2
a, a
)B
( 0, a
)
A( 0
, 0)
x
y
S( a
, 0)
R( a
, b)
Q( 0
, b)
P( 0
, 0)
x
y
V( s
, 0)
U( s
, s)
T( 0
, s)
S( 0
, 0)
x
y
D( a
, 0)C
( ?, c
)B
( b, ?
)
A( 0
, 0)
x
y
Q( a
, 0)
P( a
� b
, c)
N( ?
, ?)
M( 0
, 0)
x
y
B( c
, 0)
C( ?
, ?)
D( 0
, c)
A( 0
, 0)
x
y
U( b
, 0)
T( b
, a)
S( 0
, a)
R( 0
, 0)
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill45
4G
lenc
oe G
eom
etry
Pro
ve T
heo
rem
sA
fter
a f
igu
re h
as b
een
pla
ced
on t
he
coor
din
ate
plan
e an
d la
bele
d,a
coor
din
ate
proo
f ca
n b
e u
sed
to p
rove
a t
heo
rem
or
veri
fy a
pro
pert
y.T
he
Dis
tan
ce F
orm
ula
,th
e S
lope
For
mu
la,a
nd
the
Mid
poin
t T
heo
rem
are
oft
en u
sed
in a
coo
rdin
ate
proo
f.
Wri
te a
coo
rdin
ate
pro
of t
o sh
ow t
hat
th
e d
iago
nal
s of
a s
qu
are
are
per
pen
dic
ula
r.T
he
firs
t st
ep i
s to
pos
itio
n a
nd
labe
l a
squ
are
on t
he
coor
din
ate
plan
e.P
lace
it
in t
he
firs
t qu
adra
nt,
wit
h o
ne
side
on
eac
h a
xis.
Lab
el t
he
vert
ices
an
d dr
aw t
he
diag
onal
s.
Giv
en:s
quar
e R
ST
UP
rove
:S �U�
⊥R�
T�
Pro
of:T
he
slop
e of
S�U�
is �0 a
� �a 0
��
�1,
and
the
slop
e of
R�T�
is �a a
� �0 0
��
1.
Th
e pr
odu
ct o
f th
e tw
o sl
opes
is
�1,
so S�
U�⊥
R�T�
.
Wri
te a
coo
rdin
ate
pro
of t
o sh
ow t
hat
th
e le
ngt
h o
f th
e m
edia
n o
f a
trap
ezoi
d i
sh
alf
the
sum
of
the
len
gth
s of
th
e b
ases
.
Po
siti
on
A�D�
on
th
e x-
axis
an
d B�
C�p
aral
lel t
o A�
D�.
Lab
el t
wo
ver
tice
s A
(0,0
) an
d D
(2a,
0).L
abel
an
oth
erve
rtex
B(2
b,2
c).F
or
vert
ex C
,th
e y
valu
e is
th
e sa
me
as f
or
vert
ex B
,so
th
e la
st v
erte
x is
C(2
d,2
c).
Th
e co
ord
inat
es o
f m
idp
oin
t M
are
��2b2�
0�
,�0
� 22c �
��(b
,c);
the
coo
rdin
ates
of
mid
po
int
Nar
e
��2a� 2
2d�
,�2c
2�0
���
(a�
d,c
).B�
C�,M�
N�,a
nd
A�D�
are
ho
rizo
nta
l seg
men
ts,
so f
ind
th
eir
len
gth
s by
su
btr
acti
ng
th
e x-
coo
rdin
ates
.
AD
�2a
�0
�2a
,BC
�2d
�2b
,MN
�a
�d
�b
Th
en �1 2� (
AD
�B
C)
��1 2� (
2a�
2d�
2b)
�a
�d
�b
�M
N.
Th
eref
ore
,th
e le
ng
th o
f th
e m
edia
n o
f a
trap
ezo
id is
hal
f th
e su
m o
f th
ele
ng
ths
of
the
bas
es.
x
y
D( 2
a, 0
)
B( 2
b, 2
c)
MN
C( 2
d, 2
c)
A( 0
, 0)
x
y
U( a
, 0)
T( a
, a)
S( 0
, a)
R( 0
, 0)
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Co
ord
inat
e P
roo
f Wit
h Q
uad
rila
tera
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
Exam
ple
Exam
ple
Exer
cise
Exer
cise
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A21 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Co
ord
inat
e P
roo
f w
ith
Qu
adri
late
rals
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill45
5G
lenc
oe G
eom
etry
Lesson 8-7
Pos
itio
n a
nd
lab
el e
ach
qu
adri
late
ral
on t
he
coor
din
ate
pla
ne.
1.re
ctan
gle
wit
h l
engt
h 2
au
nit
s an
d 2.
isos
cele
s tr
apez
oid
wit
h h
eigh
t a
un
its,
hei
ght
au
nit
sba
ses
c�
bu
nit
s an
d b
�c
un
its
Nam
e th
e m
issi
ng
coor
din
ates
for
eac
h q
uad
rila
tera
l.
3.re
ctan
gle
C(c
,a)
4.re
ctan
gle
G(0
,2b
)
5.pa
rall
elog
ram
R(a
�b
,c),
S(a
,0)
6.is
osce
les
trap
ezoi
dW
(a�
b,c
),Y
(0,c
)
Pos
itio
n a
nd
lab
el t
he
figu
re o
n t
he
coor
din
ate
pla
ne.
Th
en w
rite
a c
oord
inat
ep
roof
for
th
e fo
llow
ing.
7.T
he
segm
ents
join
ing
the
mid
poin
ts o
f th
e si
des
of a
rh
ombu
s fo
rm a
rec
tan
gle.
Giv
en:
AB
CD
is a
rh
om
bus.
M,N
,P,a
nd
Qar
e th
e m
idp
oin
ts
of
A�B�
,B�C�
,C�D�
,an
d D�
A�,r
esp
ecti
vely
.P
rove
:M
NP
Qis
a r
ecta
ng
le.
Pro
of:
M�
(�a,
b),
N�
(a,b
),P
�(a
,�b
),an
d Q
�(�
a,�
b).
slo
pe
of
M�N�
�� a
b �� (�
b a)�
or
0sl
op
e o
f Q�
P����
�ba�
�(�ab
)�
or
0
slo
pe
of
M�Q�
�� �
� ab �
� (�b a)
�o
r u
nd
efin
edsl
op
e o
f N�
P��
�ba�
�(�ab
)�
or
un
def
ined
M�N�
and
Q�P�
hav
e th
e sa
me
slo
pe.
M�Q�
and
N�P�
also
hav
e th
e sa
me
slo
pe.
M�N�
is p
erp
end
icu
lar
to M�
Q�.T
her
efo
re,b
oth
pai
rs o
f o
pp
osi
te s
ides
are
par
alle
l an
d c
on
secu
tive
sid
es a
re p
erp
end
icu
lar.
Th
is m
ean
s th
at M
NP
Qis
a r
ecta
ng
le.
xO
y B( 0
, 2b)
D( 0
, �2b
)PN
M Q
C( 2
a, 0
)
A( �
2a, 0
)
x
y
T( b
, 0)
W( ?
, c)
Y( ?
, ?)
S( a
, 0)
Ox
y
S( a
, ?)
R( ?
, ?)
P( b
, c)
T( 0
, 0)
O
x
y
E( 5
b, 0
)
F( 5
b, 2
b)G
( ?, ?
)
D( 0
, 0)
Ox
y
U( c
, 0)
C( ?
, ?)
D( 0
, a)
A( 0
, 0)
O
xO
y
B( b
� c
, 0)
B( b
, a)
C( c
, a)
A( 0
, 0)
xO
y
B( 2
a, 0
)
D( 0
, a)
C( 2
a, a
)
A( 0
, 0)
©G
lenc
oe/M
cGra
w-H
ill45
6G
lenc
oe G
eom
etry
Pos
itio
n a
nd
lab
el e
ach
qu
adri
late
ral
on t
he
coor
din
ate
pla
ne.
1.pa
rall
elog
ram
wit
h s
ide
len
gth
bu
nit
s 2.
isos
cele
s tr
apez
oid
wit
h h
eigh
t b
un
its,
and
hei
ght
au
nit
sba
ses
2c�
au
nit
s an
d 2c
�a
un
its
Nam
e th
e m
issi
ng
coor
din
ates
for
eac
h q
uad
rila
tera
l.
3.pa
rall
elog
ram
4.is
osce
les
trap
ezoi
d
K(2
b�
c,a)
,L(c
,a)
X(�
a,0)
,Y(�
b,c
)
Pos
itio
n a
nd
lab
el t
he
figu
re o
n t
he
coor
din
ate
pla
ne.
Th
en w
rite
a c
oord
inat
ep
roof
for
th
e fo
llow
ing.
5.T
he
oppo
site
sid
es o
f a
para
llel
ogra
m a
re c
ongr
uen
t.
Giv
en:
AB
CD
is a
par
alle
log
ram
.P
rove
:A�
B��
C�D�
,A�D�
�B�
C�P
roo
f:
AB
��
(a�
0�
)2�
(�
0 �
0�
)2 ��
�a
2 �o
r a
CD
��
[(a
��
b)
�b
�]2
�(
�c
�c
�)2 �
��
a2 �
or
a
AD
��
(b�
0�
)2�
(�
c�
0�
)2 ��
�b
2�
�c
2 �A
B�
�[(
a�
�b
) �
a�
]2�
(c�
�0)
2�
��
b2
��
c2 �
AB
�C
Dan
d A
D�
BC
,so
A�B�
�C�
D�an
d A�
D��
B�C�
6.TH
EATE
RA
sta
ge i
s in
th
e sh
ape
of a
tra
pezo
id.W
rite
a
coor
din
ate
proo
f to
pro
ve t
hat
T �R�
and
S�F�
are
para
llel
.
Giv
en:
T(1
0,0)
,R(2
0,0)
,S(3
0,25
),F
(0,2
5)P
rove
:T�R�
|| S�F�
Pro
of:
Th
e sl
op
e o
f T�R�
�� 20 0
� �0 10
��
� 10 0�an
d
the
slo
pe
of
SF
��2 35 0� �
2 05�
�� 30 0�
.Sin
ce T�
R�an
d
S�F�
bo
th h
ave
a sl
op
e o
f 0,
T�R�|| S�
F�.
x
y T( 1
0, 0
)
S( 3
0, 2
5)F
( 0, 2
5)
R( 2
0, 0
)O
xO
y
B( a
, 0)
D( b
, c)
C( a
� b
, c)
A( 0
, 0)
x
y
W( a
, 0)
Z( b
, c)
Y( ?
, ?)
X( ?
, ?)
Ox
y
J(2b
, 0)
K( ?
, a)
L(c,
?)
H( 0
, 0)
O
xO
y
B( a
� 2
c, 0
)
D( a
, b)
C( 2
c, b
)
A( 0
, 0)
xO
y
B( b
, 0)
D( c
, a)
C( b
� c
, a)
A( 0
, 0)P
ract
ice (
Ave
rag
e)
Co
ord
inat
e P
roo
f w
ith
Qu
adri
late
rals
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csC
oo
rdin
ate
Pro
of
wit
h Q
uad
rila
tera
ls
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
©G
lenc
oe/M
cGra
w-H
ill45
7G
lenc
oe G
eom
etry
Lesson 8-7
Pre-
Act
ivit
yH
ow c
an y
ou u
se a
coo
rdin
ate
pla
ne
to p
rove
th
eore
ms
abou
tq
uad
rila
tera
ls?
Rea
d th
e in
trod
uct
ion
to
Les
son
8-7
at
the
top
of p
age
447
in y
our
text
book
.
Wha
t sp
ecia
l ki
nds
of q
uadr
ilat
eral
s ca
n be
pla
ced
on a
coo
rdin
ate
syst
em s
oth
at t
wo
side
s of
th
e qu
adri
late
ral
lie
alon
g th
e ax
es?
rect
ang
les
and
sq
uar
es
Rea
din
g t
he
Less
on
1.F
ind
the
mis
sin
g co
ordi
nat
es i
n e
ach
fig
ure
.Th
en w
rite
th
e co
ordi
nat
es o
f th
e fo
ur
vert
ices
of
the
quad
rila
tera
l.
a.is
osce
les
trap
ezoi
db
.par
alle
logr
am
R(�
a,0)
,S(�
b,c
),T
(b,c
),U
(a,0
)M
(0,0
),N
(0,a
),P
(b,c
�a)
,Q(b
,c)
2.R
efer
to
quad
rila
tera
l E
FG
H.
a.F
ind
the
slop
e of
eac
h s
ide.
slo
pe
E�F�
�3,
slo
pe
F�G��
�1 3� ,sl
op
e G�
H��
3,sl
op
e H�
E��
�1 3�
b.
Fin
d th
e le
ngt
h o
f ea
ch s
ide.
EF
��
10�,F
G�
�10�
,GH
��
10�,H
E�
�10�
c.F
ind
the
slop
e of
eac
h d
iago
nal
.sl
op
e E�
G��
1,sl
op
e F�H�
��
1d
.F
ind
the
len
gth
of
each
dia
gon
al.
EG
��
32�,F
H�
�8�
e.W
hat
can
you
con
clu
de a
bou
t th
e si
des
of E
FG
H?
All
fou
r si
des
are
co
ng
ruen
tan
d b
oth
pai
rs o
f o
pp
osi
te s
ides
are
par
alle
l.f.
Wh
at c
an y
ou c
oncl
ude
abo
ut
the
diag
onal
s of
EF
GH
?T
he
dia
go
nal
s ar
ep
erp
end
icu
lar
and
th
ey a
re n
ot
con
gru
ent.
g.C
lass
ify
EF
GH
as a
par
alle
logr
am,a
rh
ombu
s,or
a s
quar
e.C
hoo
se t
he
mos
t sp
ecif
icte
rm.E
xpla
in h
ow y
our
resu
lts
from
par
ts a
-f s
upp
ort
you
r co
ncl
usi
on.
EF
GH
is a
rh
om
bus.
All
fou
r si
des
are
co
ng
ruen
t an
d t
he
dia
go
nal
s ar
e p
erp
end
icu
lar.
Sin
ce t
he
dia
go
nal
s ar
e n
ot
con
gru
ent,
EF
GH
is n
ot
a sq
uar
e.
Hel
pin
g Y
ou
Rem
emb
er3.
Wh
at i
s an
eas
y w
ay t
o re
mem
ber
how
bes
t to
dra
w a
dia
gram
th
at w
ill
hel
p yo
u d
evis
ea
coor
din
ate
proo
f?S
amp
le a
nsw
er:
A k
ey p
oin
t in
th
e co
ord
inat
e p
lan
e is
the
ori
gin
.Th
e ev
eryd
ay m
ean
ing
of
ori
gin
is p
lace
wh
ere
som
eth
ing
beg
ins.
So
loo
k to
see
if t
her
e is
a g
oo
d w
ay t
o b
egin
by
pla
cin
g a
ver
tex
of
the
fig
ure
at
the
ori
gin
.
x
y
O
E
FG
H
x
y
Q( b
, ?)
P( ?
, c �
a)
N( ?
, a)
M( ?
, ?)
Ox
y
U( a
, ?)
T( b
, ?)
S( ?
, c)
R( ?
, ?)
O
©G
lenc
oe/M
cGra
w-H
ill45
8G
lenc
oe G
eom
etry
Co
ord
inat
e P
roo
fsA
n i
mpo
rtan
t pa
rt o
f pl
ann
ing
a co
ordi
nat
e pr
oof
is c
orre
ctly
pla
cin
g an
d la
beli
ng
the
geom
etri
c fi
gure
on
th
e co
ordi
nat
e pl
ane.
Dra
w a
dia
gram
you
wou
ld u
se i
n a
coo
rdin
ate
pro
of o
f th
efo
llow
ing
theo
rem
.
Th
e m
edia
n o
f a
trap
ezoi
d i
s pa
rall
el t
o th
e ba
ses
of t
he
trap
ezoi
d.
In t
he
diag
ram
,not
e th
at o
ne
vert
ex,A
,is
plac
ed a
t th
e or
igin
.Als
o,th
e co
ordi
nat
es o
f B
,C,a
nd
Du
se 2
a,2b
,an
d 2c
in o
rder
to
avoi
d th
e u
se o
f fr
acti
ons
wh
en f
indi
ng
the
coor
din
ates
of
the
mid
poin
ts,
Man
d N
.
Wh
en d
oin
g co
ordi
nat
e pr
oofs
,th
e fo
llow
ing
stra
tegi
es m
ay b
e h
elpf
ul.
1.If
you
are
ask
ed t
o pr
ove
that
seg
men
ts a
re p
aral
lel
or p
erpe
ndi
cula
r,u
se s
lope
s.
2.If
you
are
ask
ed t
o pr
ove
that
seg
men
ts a
re c
ongr
uen
t or
hav
e re
late
d m
easu
res,
use
th
e di
stan
ce f
orm
ula
.
3.If
you
are
ask
ed t
o pr
ove
that
a s
egm
ent
is b
isec
ted,
use
th
e m
idpo
int
form
ula
.
For
eac
h o
f th
e fo
llow
ing
theo
rem
s,a
dia
gram
has
bee
n p
rovi
ded
to
be
use
d i
n a
form
al p
roof
.Nam
e th
e m
issi
ng
coor
din
ates
in
th
e d
iagr
am.T
hen
,usi
ng
the
Giv
enan
d t
he
Pro
vest
atem
ents
,pro
ve t
he
theo
rem
.
1.T
he
med
ian
of
a tr
apez
oid
is p
aral
lel
to t
he
base
s of
th
e tr
apez
oid.
(Use
th
e di
agra
mgi
ven
in
th
e ex
ampl
e ab
ove.
)
Coo
rdin
ates
of
Man
d N
:M
(b,c
);N
(d�
a,c)
Giv
en:T
rape
zoid
AB
CD
has
med
ian
M�N�
.P
rove
:B �C�
|| M�N�
and
A�D�
|| M�N�
Pro
of:
Th
e sl
op
e o
f A�
D�is
� 20 a� �0 0
�o
r 0.
Th
e sl
op
e o
f B�
C�is
� 22 dc� �
2 2c b�
or
0.
Th
e sl
op
e o
f M�
N�is
� (a�c
� b)c �
b�
or
0.S
ince
all
thre
e sl
op
es a
re
equ
al,B�
C�|| M�
N�an
d A�
D�|| M�
N�.
2.T
he
med
ian
s to
th
e le
gs o
f an
iso
scel
es t
rian
gle
are
con
gru
ent.
Coo
rdin
ates
of
Tan
d K
:T(a
,b);
K(3
a,b
)
Giv
en:�
AB
Cis
iso
scel
es w
ith
med
ian
s T�
B�an
d K�
A�.
Pro
ve:T �
B��
K�A�
Pro
of:
TB
��
(4a
��
a)2
��
(0 �
�b
)2 �o
r �
9a2
��
b2
�K
A�
�(3
a�
�0)
2�
�(b
��
0)2
�o
r �
9a2
��
b2
�S
ince
TB
�K
A,T�
B��
K�A�
.
x
y
O
TK
A( 0
, 0)
B( 4
a, 0
)
C( 2
a, 2
b)
x
y
O
MN
A( 0
, 0)
D( 2
a, 0
)
C( 2
d, 2
c)B
( 2b,
2c)
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
8-7
8-7
Exam
ple
Exam
ple
Answers (Lesson 8-7)
© Glencoe/McGraw-Hill A23 Glencoe Geometry
Chapter 8 Assessment Answer Key Form 1 Form 2APage 459 Page 460 Page 461
(continued on the next page)
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
C
B
C
D
A
B
A
D
A
C
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
C
B
D
A
B
A
D
B
B
x � 7,m�WYZ � 41
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
B
D
A
C
A
A
C
B
A
D
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Chapter 8 Assessment Answer KeyForm 2A (continued) Form 2BPage 462 Page 463 Page 464
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
D
C
C
A
B
D
B
D
B
7 or �4
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
B
D
D
A
B
B
A
C
C
D
B
12.
13.
14.
15.
16.
17.
18.
19.
20.
B:
B
D
B
D
C
C
A
A
C
22
© Glencoe/McGraw-Hill A25 Glencoe Geometry
Chapter 8 Assessment Answer KeyForm 2CPage 465 Page 466
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
10,440
19
40
18
8
122
(6, 4)
Yes; A�B� and C�D� are|| and �.
No; the slopes are
�94
�, �17
�, 1, and �23
�.
Thus, ABCD doesnot have || sides.
��23
�
22
Yes; if the diagonalsof a � are �, then
the � is a rectangle.
67.5
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
(4, 0)
31
Yes; ABCD has onlyone pair of oppositesides ||, B�C� and A�D�.
6
A(a, 0), A�C� ⊥ B�D�
true
true
true
false
true
true
false
x � 9, y � 2
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 8 Assessment Answer KeyForm 2DPage 467 Page 468
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
6120
38
90
50
3.6
117
��12
�, 3�
Yes; Both pairs ofopp. sides are �.
ABCD has two pairs
of || sides, A�B� || C�D�and B�D� || D�A�; it is a �.
�4
8
One rt. � means thatthe other � will be rt.�. If all 4 � are rt. �,the � is a rectangle.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
B:
72
(�3, 1)
16
Yes; ABCD has onlyone pair of opp.
sides ||, A�D� and B�C�.
8
B(a, a); Themidpoint of bothdiagonals is at
��a2
�, �a2
��.
false
true
false
false
true
false
false
90
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 8 Assessment Answer KeyForm 3Page 469 Page 470
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
3960
30; 30, 47, 120,179, 174, and 170
�18x0
�
65
8 or 32
Yes; the diagonalsbisect each other.
slope of C�D� � �23
�;
slope of D�A� � �2.
28
72
8�2�
4
9
13.
14.
15.
16.
17.
18.
19.
20.
B:
13
Sample answer:(b � d, c)
Yes; A�B� ⊥ B�C�,
B�C� ⊥ C�D�, C�D� ⊥ A�D�,so all � are rt. �.
ABCD has 2 pairs ofopp. sides �, A�B� �C�D� and B�C� � D�A�,so ABCD is a �.
C(�2a, 0); thesegment joining the
midpoints of B�C�and A�B� is || to A�C�.
Opp. sides of a �are �.
If both pairs of opp.sides of a quad.are �, then thequad. is a �.
27
54
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 8 Assessment Answer KeyPage 471, Open-Ended Assessment
Scoring Rubric
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of anglesof polygons, properties of parallelograms, rectangles,rhombi, squares, and trapezoids, and coordinate proofs.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Graphs and figures are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of angles ofpolygons, properties of parallelograms, rectangles, rhombi,squares, and trapezoids, and coordinate proofs.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Graphs and figures are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts ofangles of polygons, properties of parallelograms,rectangles, rhombi, squares, and trapezoids, andcoordinate proofs.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Graphs and figures are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work shown to substantiate the
final computation.• Graphs and figures may be accurate but lack detail or
explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the conceptsof angles of polygons, properties of parallelograms,rectangles, rhombi, squares, and trapezoids, andcoordinate proofs.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Graphs and figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
Chapter 8 Assessment Answer KeyPage 471, Open-Ended Assessment
Sample Answers
© Glencoe/McGraw-Hill A29 Glencoe Geometry
1. a. Any type of convex polygon can bedrawn as long as one is regular andone is not regular and both have thesame number of sides.
b. Check to be sure that the exteriorangles have been properly drawn andaccurately measured.
c. 4(90) � 360; 120 � 70 � 60 � 110 �360; The sum of the exterior angles ofeach figure should be 360°. The sum ofthe exterior angles of both the regularconvex polygon and the irregularconvex polygon is 360°.
2. The student should draw a rectangle andjoin the midpoints of consecutive sides.The figure formed inside is a rhombus.Since all four small triangles can beproved to be congruent by SAS, the foursides of the interior quadrilateral arecongruent by CPCTC, making it arhombus.
3. The student should draw an isoscelestrapezoid with one pair of opposite sidesparallel and the other pair of oppositesides congruent, as in the figure below.
4. a. Possible properties:A square has four congruent sides anda rectangle may not.A square has perpendicular diagonalsand a rectangle may not.The diagonals of a square bisect theangles and those in a rectangle maynot.
b. Possible properties:A square has four right angles and arhombus may not.The diagonals of a square are congruentand those of a rhombus may not be.
c. Possible properties:A rectangle has four right angles anda parallelogram may not.The diagonals of a rectangle arecongruent and those of aparallelogram may not be.
5. The slope of diagonal A�C� is �a �c
b�, while
the slope of diagonal B�D� is �b �c
a�. These slopes are not necessarily oppositereciprocals of each other. Therefore John’sfigure could not be used to prove that thediagonals of a rhombus are perpendicular.The figure John drew on the coordinateplane has vertices that make it aparallelogram but not always a rhombus.
90�70�
60�
110�120�
90�
90�90�
In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.
An
swer
s
© Glencoe/McGraw-Hill A30 Glencoe Geometry
Chapter 8 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 472 Page 473 Page 474
Quiz 2Page 473
Quiz 4Page 474
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
isoscelestrapezoid
parallelogram
trapezoid
square
rhombus
rectangle
kite
diagonals
median
bases
angles formed by thebase and one of thelegs of a trapezoid
the nonparallel sidesof a trapezoid
1.
2.
3.
4.
5.
12,240
45
20
(�1, 10)
A
1.
2.
3.
4.
5.
No; none of thetests for �s are
fulfilled.
false
false
true
m�1 � 68,m�2 � 56,m�3 � 34
1.
2.
3.
4.
5.
Rhombus; allsides are �.
108
50
true
12
1.
2.
3.
4.
5.
C(�a, 0)
D(0, �a)
A(�d, c)
X(a � b, c),Y(�e � d, c)
C(a � b, c),AB � CD �
�b2 ��c2�,BC � AD � a
© Glencoe/McGraw-Hill A31 Glencoe Geometry
Chapter 8 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 475 Page 476
An
swer
s
Part I
Part II
6.
7.
8.
9.
10.
50
42
yes
m�1 � 12, m�2 �78, m�3 � 12,
m�4 � 156
If one pair of opp.sides is || and �,then the quad.
is a �.
1.
2.
3.
4.
5.
B
D
A
B
C
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
BR��� and BT���
y � ��32
�x � 9
11, 2
Assume thatneither bell costmore than $45.
20.5 in., 33 in.,18.75 in., and 30 in.
A(3, 0.5) andB(�0.5, �1)
� of elevation:�XYZ; � of
depression: �WXY
13
55; 125; 5
8.5
LMNP is a trapezoid
since P�L� || M�N� and
P�N� ⁄|| L�M�.
© Glencoe/McGraw-Hill A32 Glencoe Geometry
Chapter 8 Assessment Answer KeyStandardized Test Practice
Page 477 Page 478
1.
2.
3.
4.
5.
6.
7.
8. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D9. 11.
11. 12.
13.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
14.
15.
16.
hexagon;concave;irregular
88.9 cm
D�L�
1 / 7 7
1 1 . 2
9 0 0
1 0