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Study Guide Masters

Copyright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material containedherein on the condition that such material be reproduced only for classroom use; be provided tostudents, teachers, and families without charge; and be used solely in conjunction with GlencoesGeometry: Concepts and Applications. Any other reproduction, for use or sale, is prohibited withoutprior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-02-834825-7 Study Guide Masters

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Lesson Title Page1-1 Patterns and Inductive Reasoning.......11-2 Points, Lines, and Planes ....................21-3 Postulates ............................................31-4 Conditional Statements and Their

Converses .........................................41-5 Tools of the Trade ...............................51-6 A Plan for Problem Solving................62-1 Real Numbers and Number Lines ......72-2 Segments and Properties of Real

Numbers...........................................82-3 Congruent Segments...........................92-4 The Coordinate Plane .......................102-5 Midpoints..........................................113-1 Angles...............................................123-2 Angle Measure..................................133-3 The Angle Addition Postulate ...........143-4 Adjacent Angles and Linear

Pairs of Angles ...............................153-5 Complementary and

Supplementary Angles ...................163-6 Congruent Angles .............................173-7 Perpendicular Lines ..........................184-1 Parallel Lines and Planes ..................194-2 Parallel Lines and Transversals.........204-3 Transversals and Corresponding

Angles ............................................214-4 Proving Lines Parallel.......................224-5 Slope .................................................234-6 Equations of Lines ............................245-1 Classifying Triangles ........................255-2 Angles of a Triangle..........................265-3 Geometry in Motion .........................275-4 Congruent Triangles .........................285-5 SSS and SAS ....................................295-6 ASA and AAS...................................306-1 Medians ............................................316-2 Altitudes and Perpendicular

Bisectors.........................................326-3 Angle Bisectors of Triangles ............33

Lesson Title Page6-4 Isosceles Triangles ............................346-5 Right Triangles..................................356-6 The Pythagorean Theorem................366-7 Distance on the Coordinate Plane .....377-1 Segments, Angles, and

Inequalities.....................................387-2 Exterior Angle Theorem ...................397-3 Inequalities Within a Triangle ............407-4 Triangle Inequality Theorem ............418-1 Quadrilaterals....................................428-2 Parallelograms ..................................438-3 Tests for Parallelograms....................448-4 Rectangles, Rhombi, and Squares.....458-5 Trapezoids.........................................469-1 Using Ratios and Proportions...........479-2 Similar Polygons...............................489-3 Similar Triangles...............................499-4 Proportional Parts and Triangles.......509-5 Triangles and Parallel Lines .............519-6 Proportional Parts and Parallel

Lines...............................................529-7 Perimeters and Similarity .................53

10-1 Naming Polygons..............................5410-2 Diagonals and Angle Measure ...........5510-3 Areas of Polygons.............................5610-4 Areas of Triangles and Trapezoids....5710-5 Areas of Regular Polygons ...............5810-6 Symmetry .........................................5910-7 Tessellations......................................6011-1 Parts of a Circle ................................6111-2 Arcs and Central Angles ...................6211-3 Arcs and Chords ...............................6311-4 Inscribed Polygons............................6411-5 Circumference of a Circle.................6511-6 Area of a Circle.................................6612-1 Solid Figures.....................................6712-2 Surface Areas of Prisms and

Cylinders ........................................6812-3 Volumes of Prisms and Cylinders .....69

Contents

iv

Lesson Title Page12-4 Surface Areas of Pyramids

and Cones.......................................7012-5 Volumes of Pyramids and Cones ......7112-6 Spheres..............................................7212-7 Similarity of Solid Figures ...............7313-1 Simplifying Square Roots.................7413-2 45-45-90 Triangles .......................7513-3 30-60-90 Triangles .......................7613-4 The Tangent Ratio.............................7713-5 Sine and Cosine Ratios .....................7814-1 Inscribed Angles ...............................7914-2 Tangents to a Circle ..........................8014-3 Secant Angles ...................................8114-4 Secant-Tangent Angles .....................8214-5 Segment Measures ............................83

Lesson Title Page14-6 Equations of Circles..........................8415-1 Logic and Truth Tables .....................8515-2 Deductive Reasoning ........................8615-3 Paragraph Proofs ..............................8715-4 Preparing for Two-Column Proofs .....8815-5 Two-Column Proofs..........................8915-6 Coordinate Proofs .............................9016-1 Solving Systems of Equations

by Graphing ...................................9116-2 Solving Systems of Equations

by Using Algebra ...........................9216-3 Translations.......................................9316-4 Reflections ........................................9416-5 Rotations...........................................9516-6 Dilations............................................96

Glencoe/McGraw-Hill 1 Geometry: Concepts and Applications

Patterns and Inductive ReasoningIn daily life, you frequently look at several specific situations and reach a general conclusion based on these cases. For example, you might receive excellent service in a restaurant several times and conclude that the service will be good each time you return.

This type of reasoning, in which you look for a pattern and then make an educated guess based on the pattern, is called inductive reasoning. The educated guess based on these facts is called a conjecture. Not all conjectures are true. When you find an example that shows that the conjecture is false, this example is called a counterexample.

Example: Find the next three terms of the sequence 3, 8, 13, . . . .

Study the pattern in the sequence.3, 8, 13, . . .

5 5

Each term is more than the term before it. Assume that this pattern continues. Then find the next three terms using the pattern of adding 5.3, 8, 13, 18, 23, 28

5 5 5 5 5

Find the next three terms of each sequence.1. 17, 25, 33, . . . 2. 60, 52, 44, . . .

3. 2, 6, 18, . . . 4. 7, 9, 13, . . .

5. 11, 7, 3, . . . 6. 24, 12, 6, . . .

7. Find a counterexample for the statement.All animals have fur.

Study Guide Student EditionPages 49

1-11-1NAME ______________________________________DATE __________PERIOD______

Glencoe/McGraw-Hill T1 Geometry: Concepts and Applications

Patterns and Inductive ReasoningIn daily life, you frequently look at several specific situations and reach a general conclusion based on these cases. For example, you might receive excellent service in a restaurant several times and conclude that the service will be good each time you return.

This type of reasoning, in which you look for a pattern and then make an educated guess based on the pattern, is called inductive reasoning. The educated guess based on these facts is called a conjecture. Not all conjectures are true. When you find an example that shows that the conjecture is false, this example is called a counterexample.

Example: Find the next three terms of the sequence 3, 8, 13, . . . .

Study the pattern in the sequence.3, 8, 13, . . .

5 5

Each term is more than the term before it. Assume that this pattern continues. Then find the next three terms using the pattern of adding 5.3, 8, 13, 18, 23, 28

5 5 5 5 5

Find the next three terms of each sequence.1. 17, 25, 33, . . . 41, 49, 57 2. 60, 52, 44, . . . 36, 28, 20

3. 2, 6, 18, . . . 54, 162, 486 4. 7, 9, 13, . . . 19, 27, 37

5. 11, 7, 3, . . . 1, 5, 9 6. 24, 12, 6, . . . 3, 1.5, 0.75

7. Find a counterexample for the statement.All animals have fur. Sample answer: Dolphins have no fur.


1-11-1NAME ______________________________________DATE __________PERIOD______


Points, Lines, and Planes

Use the figure at the right to name examples of each term.1. ray 2. point

3. line 4. line segment

The diagram at the right represents a baseball field. Name the segment or ray described in Exercises 59.5. The player on third base throws the ball

and the player on first base catches it.

6. The batter hits the ball over thehead of the player on first base.

7. The player on first base throws theball home, but the catcher misses it.

8. The batter hits the ball and it iscaught at second base.

9. Are third base, first base, and thepitchers mound collinear?


1-21-2NAME ______________________________________DATE __________PERIOD______

Term Description Names

point has no size point P

line is an infinite number of points line m or line AB or AB

ray starts with a point called an endpoint ray CD or CD

line segment is part of a line with two endpoints line segment EF or EF

plane is a flat surface that extends without plane G or plane GHIend; has no depth.

R

S

QT

U

V


Points, Lines, and Planes

Use the figure at the right to name examples of each term. Sample answers given.1. ray QR 2. point T

3. line RU 4. line segment SQ

The diagram at the right represents a baseball field. Name the segment or ray described in Exercises 59.5. The player on third base throws the ball

and the player on first base catches it. TF

6. The batter hits the ball over thehead of the player on first base. HF

7. The player on first base throws theball home, but the catcher misses it. FH

8. The batter hits the ball and it iscaught at second base. HS

9. Are third base, first base, and thepitchers mound collinear? no


1-21-2NAME ______________________________________DATE __________PERIOD______

Term Description Names

point has no size point P

line is an infinite number of points line m or line AB or AB

ray starts with a point called an endpoint ray CD or CD

line segment is part of a line with two endpoints line segment EF or EF

plane is a flat surface that extends without plane G or plane GHIend; has no depth.

R

S

QT

U

V


PostulatesPoint, line, and plane are undefined terms in geometry. The postulates describe the fundamental properties of these terms.

A club is divided into committees. The undefined terms are committee and member.Postulate 1: Each pair of committees has exactly one member in common.Postulate 2: Each member is on exactly two committees.Postulate 3: There are exactly four committees.

Use the postulates above to complete Exercises 13.1. Draw a model to illustrate these rules. Use circular shapes to

represent committees and letters to represent members.

2. How many club members are there?

3. How many members are on each committee?

Suppose some companies have collaborated to place severalsatellites in orbit. Let the set of all satellites that a given companyhelped place in orbit be called a network.Postulate A: There are at least two distinct satellites.Postulate B: For each pair of satellites, there is exactly one network

containing them.Postulate C: Each network contains at least two distinct satellites.Postulate D: For each network, there is a satellite not in it.

Use the postulates above to complete Exercises 46.4. Draw a model to illustrate these rules. Use rectangles to

represent networks and letters to represent satellites.

5. What is the least number of satellites?

6. What is the least number of networks?


1-31-3NAME ______________________________________DATE __________PERIOD______


PostulatesPoint, line, and plane are undefined terms in geometry. The postulates describe the fundamental properties of these terms.

A club is divided into committees. The undefined terms are committee and member.Postulate 1: Each pair of committees has exactly one member in common.Postulate 2: Each member is on exactly two committees.Postulate 3: There are exactly four committees.

Use the postulates above to complete Exercises 13.1. Draw a model to illustrate these rules. Use circular shapes to

represent committees and letters to represent members.A sample answer is given.

2. How many club members are there? 6

3. How many members are on each committee? 3

Suppose some companies have collaborated to place severalsatellites in orbit. Let the set of all satellites that a given companyhelped place in orbit be called a network.Postulate A: There are at least two distinct satellites.Postulate B: For each pair of satellites, there is exactly one network

containing them.Postulate C: Each network contains at least two distinct satellites.Postulate D: For each network, there is a satellite not in it.

Use the postulates above to complete Exercises 46.4. Draw a model to illustrate these rules. Use rectangles to

represent networks and letters to represent satellites.A sample answer is given.

5. What is the least number of satellites? 3

6. What is the least number of networks? 3


1-31-3NAME ______________________________________DATE __________PERIOD______


Conditional Statements and Their ConversesIf-then statements are commonly used in everyday life. Forexample, an advertisement might say, If you buy our product,then you will be happy. Notice that an if-then statement hastwo parts, a hypothesis (the part following if) and aconclusion (the part following then).

New statements can be formed from the original statement.Statement p qConverse q p

Example: Rewrite the following statement in if-then form. Thenwrite the converse, inverse, and contrapositive.

All elephants are mammals.

If-then form: If an animal is an elephant, then it is a mammal.Converse: If an animal is a mammal, then it is an elephant.

Identify the hypothesis and conclusion of each conditionalstatement.

1. If today is Monday, then tomorrow is Tuesday.

2. If a truck weighs 2 tons, then it weighs 4000 pounds.

Write each conditional statement in if-then form.3. All chimpanzees love bananas.

4. Collinear points lie on the same line.

Write the converse, of each conditional.5. If an animal is a fish, then it can swim.

6. All right angles are congruent.


1-41-4NAME ______________________________________DATE __________PERIOD______


Conditional Statements and Their ConversesIf-then statements are commonly used in everyday life. Forexample, an advertisement might say, If you buy our product,then you will be happy. Notice that an if-then statement hastwo parts, a hypothesis (the part following if) and aconclusion (the part following then).

New statements can be formed from the original statement.Statement p qConverse q p

Example: Rewrite the following statement in if-then form. Thenwrite the converse, inverse, and contrapositive.

All elephants are mammals.

If-then form: If an animal is an elephant, then it is a mammal.Converse: If an animal is a mammal, then it is an elephant.

Identify the hypothesis and conclusion of each conditionalstatement.

1. If today is Monday, then tomorrow is Tuesday.H: today is Monday; C: tomorrow is Tuesday

2. If a truck weighs 2 tons, then it weighs 4000 pounds.H: a truck weighs 2 tons; C: it weighs 4000 pounds

Write each conditional statement in if-then form.3. All chimpanzees love bananas.

If an animal is a chimpanzee, then it loves bananas.

4. Collinear points lie on the same line.If points are collinear, then they lie on the same line.

Write the converse, of each conditional.5. If an animal is a fish, then it can swim.

If an animal can swim, then it is a fish.

6. All right angles are congruent.If angles are congruent, then they are right angles.


1-41-4NAME ______________________________________DATE __________PERIOD______


Tools of the Trade

A straightedge is any object that can be used to draw a straightline. A compass is any object that can be used to draw circles.

The figures in Exercises 47 below were constructed using astraightedge. A figure can be traced if it has no more than twopoints where an odd number of segments meet.

Use a straightedge or compass to determine whether eachstatement is true or false.1. Circle P is larger than circle Q.

2. Circle P is smaller than circle Q.

3. The two circles are the same size.

Determine whether each figure can be traced. If so, name thestarting point and number the sides in the order they should betraced.4. 5.

6. 7.


1-51-5NAME ______________________________________DATE __________PERIOD______

Q

P

T U

V

RS

PLQ

M

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7

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K


Tools of the Trade

A straightedge is any object that can be used to draw a straightline. A compass is any object that can be used to draw circles.

The figures in Exercises 47 below were constructed using astraightedge. A figure can be traced if it has no more than twopoints where an odd number of segments meet.

Use a straightedge or compass to determine whether eachstatement is true or false.1. Circle P is larger than circle Q. false

2. Circle P is smaller than circle Q. false

3. The two circles are the same size. true

Determine whether each figure can be traced. If so, name thestarting point and number the sides in the order they should betraced. Sample answers given.4. 5.

6. 7.

This figure cannot be traced.


1-51-5NAME ______________________________________DATE __________PERIOD______

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P

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V

RS

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6

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NAME ______________________________________DATE __________PERIOD______

A Plan for Problem Solving

The following four-step plan can be used to solve any problem.

When finding a solution, it may be necessary to use a formula.Two useful formulas are the area formula and perimeter formulafor a rectangle.

Examples1 Find the perimeter and area

of the rectangle at the right.

P 2 2w 2(7) 2(2) 14 4 or 18

The perimeter is 18 inches.

A w 7 2 or 14

The area is 14 in2.

Find the perimeter and area of each rectangle.1. 2. 3.

Find the missing measure in each formula.4. 3, w 7, P 5. w = 5.2, = 6.5, A = 6. w = 4, A = 36, = 7. P = 65, = 18, w = ? ?

? ?

Problem-Solving Plan1. Explore the problem. Identify what you want to know.2. Plan the solution. Choose a strategy.3. Solve the problem. Use the strategy to solve the problem.4. Examine the solution. Check your answer.

The formula for the area of a rectangle is A w, where Area of a Rectangle A represents the area expressed in square units,

represents the length, and w represents the width.The formula for the perimeter of a rectangle is

Perimeter of a Rectangle P 2 2w, where P represents the perimeter, represents the length and w represents the width.

2 Find the width of a rectanglewhose area is 52 cm2 andwhose length is 13 cm.

A w51

23

1133w

4 wThe width is 4 cm.

1-61-6



NAME ______________________________________DATE __________PERIOD______

A Plan for Problem Solving

The following four-step plan can be used to solve any problem.

When finding a solution, it may be necessary to use a formula.Two useful formulas are the area formula and perimeter formulafor a rectangle.

Examples1 Find the perimeter and area

of the rectangle at the right.

P 2 2w 2(7) 2(2) 14 4 or 18

The perimeter is 18 inches.

A w 7 2 or 14

The area is 14 in2.

Find the perimeter and area of each rectangle.1. 2. 3.

20 cm, 24 cm2 20.4 in., 22.4 in2 32 yd, 64 yd2

Find the missing measure in each formula.4. 3, w 7, P 20 5. w = 5.2, = 6.5, A = 33.86. w = 4, A = 36, = 9 7. P = 65, = 18, w = 14.5? ?

? ?

Problem-Solving Plan1. Explore the problem. Identify what you want to know.2. Plan the solution. Choose a strategy.3. Solve the problem. Use the strategy to solve the problem.4. Examine the solution. Check your answer.

The formula for the area of a rectangle is A w, where Area of a Rectangle A represents the area expressed in square units,

represents the length, and w represents the width.The formula for the perimeter of a rectangle is

Perimeter of a Rectangle P 2 2w, where P represents the perimeter, represents the length and w represents the width.

2 Find the width of a rectanglewhose area is 52 cm2 andwhose length is 13 cm.

A w51

23

1133w

4 wThe width is 4 cm.

1-61-6


Real Numbers and Number Lines

Numbers can be grouped into sets with identifying characteristics.

Sets of Numbers

Name Definition Examples

whole numbers 0 and the natural, or counting 0, 1, 2, 3, . . . numbers

integers 0, the positive integers, and . . . 3, 2, 1, 0, 1, 2, 3, . . . the negative integers

rational numbers any number of the form ab

, where 13

, 7.9, 258, 9.3686868 . . .a and b are integers and b 0

irrational numbers decimals that neither terminate 0.513947836 . . . , nor repeat 1.010010001 . . .

real numbers rational and irrational numbers 4.68, 191, 21.494994999 . . .

Each real number corresponds to exactly one point on a number line. The distance between two points on a number line is the positive difference of their coordinates.

For each situation, write a real number with ten digits to theright of the decimal point.1. a rational number between 5 and 6 that terminates

2. an irrational number between 1 and 2

3. a rational number between 3 and 2 with a 3-digit repeatingpattern

Use the number line to find each measure.

4. HI 5. AD 6. BH 7. AJ

8. BC 9. CG 10. CJ 11. FC


2-12-1NAME ______________________________________DATE __________PERIOD______


Real Numbers and Number Lines

Numbers can be grouped into sets with identifying characteristics.

Sets of Numbers

Name Definition Examples

whole numbers 0 and the natural, or counting 0, 1, 2, 3, . . . numbers

integers 0, the positive integers, and . . . 3, 2, 1, 0, 1, 2, 3, . . . the negative integers

rational numbers any number of the form ab

, where 13

, 7.9, 258, 9.3686868 . . .a and b are integers and b 0

irrational numbers decimals that neither terminate 0.513947836 . . . , nor repeat 1.010010001 . . .

real numbers rational and irrational numbers 4.68, 191, 21.494994999 . . .

Each real number corresponds to exactly one point on a number line. The distance between two points on a number line is the positive difference of their coordinates.

For each situation, write a real number with ten digits to theright of the decimal point.1. a rational number between 5 and 6 that terminates

5.2958105483

2. an irrational number between 1 and 21.9393393339. . .

3. a rational number between 3 and 2 with a 3-digit repeatingpattern2.3113113113. . .

Use the number line to find each measure.

4. HI 3 5. AD 7 6. BH 10 7. AJ 18

8. BC 2.5 9. CG 5 10. CJ 12.5 11. FC 3.5


2-12-1NAME ______________________________________DATE __________PERIOD______


Segments and Properties of Real Numbers

Point B is between points A and C if A, B, and C are collinear andAB BC AC.

Three segment measures are given. The three pointsnamed are collinear. Determine which point is betweenthe other two.1. NQ 17, NK 6, QK 11 2. JB 9.8, BP 3.2, JP 6.6

For each of the following, draw the missing point. Then usethe information given to write the coordinates of all threepoints.3. CD DF CF, CD 3, DF 5

The coordinate of C is 20.

4. MK KP MP, MK 6, KP 4The coordinate of M is 25.

5. QB BA QA, AB 10, QB 8The coordinate of B is 1.

Draw a figure for each situation described.6. Points M, A, N, and Q are collinear. 7. Points N, Q, W, C, and E are

Point N is between points A and Q. collinear. Point N bisects CE. PointPoint M is not between points A W bisects NE. Point Q is theand Q. midpoint of WE.

8. Points J, Y, G, and K are collinear. 9. Points B, Z, T, and R are collinear.Point G is the midpoint of KJ and Point B is the midpoint of RT.KJ JY. Point T is between points B and Z.


2-22-2NAME ______________________________________DATE __________PERIOD______


Segments and Properties of Real Numbers

Point B is between points A and C if A, B, and C are collinear andAB BC AC.

Three segment measures are given. The three pointsnamed are collinear. Determine which point is betweenthe other two.1. NQ 17, NK 6, QK 11 K 2. JB 9.8, BP 3.2, JP 6.6 P

For each of the following, draw the missing point. Then usethe information given to write the coordinates of all threepoints.3. CD DF CF, CD 3, DF 5

The coordinate of C is 20.

4. MK KP MP, MK 6, KP 4The coordinate of M is 25.

5. QB BA QA, AB 10, QB 8The coordinate of B is 1.

Draw a figure for each situation described. Sample answers given.6. Points M, A, N, and Q are collinear. 7. Points N, Q, W, C, and E are

Point N is between points A and Q. collinear. Point N bisects CE. PointPoint M is not between points A W bisects NE. Point Q is theand Q. midpoint of WE.

8. Points J, Y, G, and K are collinear. 9. Points B, Z, T, and R are collinear.Point G is the midpoint of KJ and Point B is the midpoint of RT.KJ JY. Point T is between points B and Z.

M A N Q C EQN W

K G J Y R B T Z


2-22-2NAME ______________________________________DATE __________PERIOD______

Study GuideNAME ______________________________________DATE __________PERIOD______


Congruent Segments

Two segments are congruent if they have the same length. Themidpoint of a segment separates the segment into two congruentsegments. To bisect a segment means to separate it into twocongruent parts. The midpoint always bisects a segment.

Use the line to name all segments congruent to each given segment.

1. HM 2. JL

3. NJ 4. HI

Use the number line to name the midpoint of each segment.

5. EI 6. IL 7. AK 8. CF

9. AC 10. DL 11. CG 12. IK

For each exercise below, the coordinates of points P and Q,respectively, are given. Graph P and Q. Then draw and labelthe coordinate of the midpoint of PQ .13. 4 and 2 14. 9 and 5 15. 3 and 4

Student EditionPages 6267

2-32-3

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Study GuideNAME ______________________________________DATE __________PERIOD______


Congruent Segments

Two segments are congruent if they have the same length. Themidpoint of a segment separates the segment into two congruentsegments. To bisect a segment means to separate it into twocongruent parts. The midpoint always bisects a segment.

Use the line to name all segments congruent to each given segment.

1. HM HM, IN 2. JL JL, HJ, IK, KM, LN

3. NJ NJ, MI, LH 4. HI HI, IJ, JK, KL, LM, MN

Use the number line to name the midpoint of each segment.

5. EI G 6. IL K 7. AK E 8. CF D

9. AC B 10. DL H 11. CG E 12. IK J

For each exercise below, the coordinates of points P and Q,respectively, are given. Graph P and Q. Then draw and labelthe coordinate of the midpoint of PQ .13. 4 and 2 14. 9 and 5 15. 3 and 4

Student EditionPages 6267

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2-42-4NAME ______________________________________DATE __________PERIOD______

The Coordinate PlaneEvery point in the coordinate plane can be denoted by an ordered pair consisting of two numbers. The first number is thex-coordinate, and the second number is the y-coordinate.

Example: Write the ordered pair for each point shown at theright.

The ordered pair for R is (2, 4).The ordered pair for S is (3, 3).The ordered pair for T is (4, 2).The ordered pair for U is (1, 4).The ordered pair for W is (0, 2).The ordered pair for X is (2, 0).

Write the ordered pair for each point shown at the right.1. A 2. B 3. C

4. D 5. E 6. F

7. G 8. H 9. I

Graph each point on the coordinate plane.10. M(6, 4) 11. N(5, 4)

12. P(3, 5) 13. Q(6, 0)

14. J(0, 4) 15. K(7, 5)

16. Y(9, 3) 17. Z(8, 5)

To determine the coordinates for a point, follow these steps.1. Start at the origin and count the number of units to the right or left of the origin.

The positive direction is to the right, and the negative direction is to the left.2. Then count the number of units up or down. The positive direction is up, and the

negative direction is down.Note: If you do not move either right or left, the x-coordinate is 0. If you do not

move up or down, the y-coordinate is 0.

O

y

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2-42-4NAME ______________________________________DATE __________PERIOD______

The Coordinate PlaneEvery point in the coordinate plane can be denoted by an ordered pair consisting of two numbers. The first number is thex-coordinate, and the second number is the y-coordinate.

Example: Write the ordered pair for each point shown at theright.

The ordered pair for R is (2, 4).The ordered pair for S is (3, 3).The ordered pair for T is (4, 2).The ordered pair for U is (1, 4).The ordered pair for W is (0, 2).The ordered pair for X is (2, 0).

Write the ordered pair for each point shown at the right.1. A (2, 3) 2. B (3, 0) 3. C (4, 3)4. D (4, 4) 5. E (4, 3) 6. F (2, 1)7. G (0, 3) 8. H (0, 4) 9. I (3, 4)

Graph each point on the coordinate plane.10. M(6, 4) 11. N(5, 4)

12. P(3, 5) 13. Q(6, 0)

14. J(0, 4) 15. K(7, 5)

16. Y(9, 3) 17. Z(8, 5)

To determine the coordinates for a point, follow these steps.1. Start at the origin and count the number of units to the right or left of the origin.

The positive direction is to the right, and the negative direction is to the left.2. Then count the number of units up or down. The positive direction is up, and the

negative direction is down.Note: If you do not move either right or left, the x-coordinate is 0. If you do not

move up or down, the y-coordinate is 0.

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Z



2-52-5NAME ______________________________________DATE __________PERIOD______

MidpointsThere are two situations in which you may need to find themidpoint of a segment.

Use the number line to find the coordinate of the midpoint ofeach segment.

1. AB 2. BC 3. CE 4. DE

5. AE 6. FC 7. GE 8. BF

Refer to the coordinate plane at the right to find thecoordinates of the midpoint of each segment.9. JK 10. KL

11. LM 12. MN

13. NT 14. MT

Midpoint on a Number Line Midpoint in the Coordinate Plane

The coordinate of the midpoint of asegment whose endpoints have coordinates a and b is a

2b

.

Example 1:

The coordinate of the midpoint of RS is 3

2 9 or 3.

The coordinates of the midpoint of asegment whose endpoints have coordinates (x1, y1) and (x2, y2) are

x1 2 x2, y1 2 y2.

Example 2:

The coordinates of the midpoint of EF are

2 2 3, 4 2 1 or 12, 52.



2-52-5NAME ______________________________________DATE __________PERIOD______

MidpointsThere are two situations in which you may need to find themidpoint of a segment.

Use the number line to find the coordinate of the midpoint ofeach segment.

1. AB 129 2. BC 92 3. CE 4 4. DE 8

5. AE 1 6. FC 5 7. GE 123 8. BF 52

Refer to the coordinate plane at the right to find thecoordinates of the midpoint of each segment.9. JK (1, 4) 10. KL (3, 3)

11. LM 3, 12 12. MN 12, 213. NT 72, 52 14. MT 0, 72

Midpoint on a Number Line Midpoint in the Coordinate Plane

The coordinate of the midpoint of asegment whose endpoints have coordinates a and b is a

2b

.

Example 1:

The coordinate of the midpoint of RS is 3

2 9 or 3.

The coordinates of the midpoint of asegment whose endpoints have coordinates (x1, y1) and (x2, y2) are

x1 2 x2, y1 2 y2.

Example 2:

The coordinates of the midpoint of EF are

2 2 3, 4 2 1 or 12, 52.


Angles

An angle is formed by two noncollinear rays with a commonendpoint called a vertex. You could name the angle at theright as S, RST, TSR, or 1.

When two or more angles have a common vertex, you need touse either three letters or a number to name the angles.Make sure there is no doubt which angle your namedescribes.

Solve.1. Label the three points K, N, and Q on 2. Name all angles having N as their

the angle below so that the angle has vertex. sides KQ and KN.

3. Draw BMA with sides MB and MA that has point P in its interior.

4. Using the figure from Exercise 3, draw MP.Name the two new angles formed.

5. In the figure at the right, label angles 1, 2, 3, and 4 using the information below.RMT is 1.MTS is 2.RTM is 3.TRM is 4.


3-13-1NAME ______________________________________DATE __________PERIOD______

R

ST

1

M

R

TS

G

AE

NF

1 23


Angles

An angle is formed by two noncollinear rays with a commonendpoint called a vertex. You could name the angle at theright as S, RST, TSR, or 1.

When two or more angles have a common vertex, you need touse either three letters or a number to name the angles.Make sure there is no doubt which angle your namedescribes.

Solve.1. Label the three points K, N, and Q on 2. Name all angles having N as their

the angle below so that the angle has vertex. 1, 2, 3, GNE,sides KQ and KN. GNF, ANF

3. Draw BMA with sides MB and MA that Sample answer:has point P in its interior.

4. Using the figure from Exercise 3, draw MP.Name the two new angles formed.BMP, PMA

5. In the figure at the right, label angles 1, 2, 3, and 4 using the information below.RMT is 1.MTS is 2.RTM is 3.TRM is 4.


3-13-1NAME ______________________________________DATE __________PERIOD______

R

ST

1

K

Q

N

B

M

P

A

M

R

TS

1

2

4

3

G

AE

NF

1 23


Angle Measure

Angles are measured in degrees.

A right angle has An acute angle has An obtuse angle hasa measure of 90. a measure between a measure between

0 and 90. 90 and 180.

A straight angle has a measure of 180.

Classify each angle as acute, obtuse, or right.1. FBH 2. CBD

3. ABC 4. ABG

5. ABE 6. EBH

7. DBH 8. FBG

Use a protractor to draw an angle having each measurement.Then classify each angle as acute, obtuse, or right.9. 110 10. 28 11. 90


3-23-2NAME ______________________________________DATE __________PERIOD______


Angle Measure

Angles are measured in degrees.

A right angle has An acute angle has An obtuse angle hasa measure of 90. a measure between a measure between

0 and 90. 90 and 180.

A straight angle has a measure of 180.

Classify each angle as acute, obtuse, or right.1. FBH acute 2. CBD acute

3. ABC acute 4. ABG obtuse

5. ABE right 6. EBH right

7. DBH obtuse 8. FBG acute

Use a protractor to draw an angle having each measurement.Then classify each angle as acute, obtuse, or right.9. 110 obtuse 10. 28 acute 11. 90 right


3-23-2NAME ______________________________________DATE __________PERIOD______


The Angle Addition Postulate

According to the Angle Addition Postulate, if D is in the interior ofABC, then mABD mDBC mABC.

Example: In the figure at the right, mABC 160, m1 x 14, and m2 3x 10. Find the value of x.

m1 m2 mABC(x 14) (3x 10) 160

4x 4 1604x 156

x 39

Solve.1. Find m1 if mARC 78. 2. Find m2 if mYJK 160.

3. Find x if mALY 71. 4. Find n if mQRS 12n.

5. If mWYV 4x 2, mVYZ 2x 5, 6. If QS bisects RQP, mRQS and mWYZ 77, find the measures 2x 10, and mSQP 3x 18,of WYV and VYZ. find mRQS.


3-33-3NAME ______________________________________DATE __________PERIOD______

A

B

D

C

1 2

A

B C

R1

M

N

Y J

K2

AU

Y

L

QR

HS

PQ

R ST


The Angle Addition Postulate

According to the Angle Addition Postulate, if D is in the interior ofABC, then mABD mDBC mABC.

Example: In the figure at the right, mABC 160, m1 x 14, and m2 3x 10. Find the value of x.

m1 m2 mABC(x 14) (3x 10) 160

4x 4 1604x 156

x 39

Solve.1. Find m1 if mARC 78. 30 2. Find m2 if mYJK 160. 14

3. Find x if mALY 71. 9 4. Find n if mQRS 12n. 11

5. If mWYV 4x 2, mVYZ 2x 5, 6. If QS bisects RQP, mRQS and mWYZ 77, find the measures 2x 10, and mSQP 3x 18,of WYV and VYZ. 54, 23 find mRQS. 66


3-33-3NAME ______________________________________DATE __________PERIOD______

A

B

D

C

1 2

A

B C

R1 M

N

Y J

K2

AU

Y

L

QR

HS

PQ

R ST


Adjacent Angles and Linear Pairs of Angles

Pairs of Angles

Special Name Definition Examples

angles in the same plane that

adjacent angles have a common vertex and a common side, but no common interior points 3 and 4 are adjacent angles.

adjacent angles whose linear pair noncommon sides are opposite

rays5 and 6 form a linear pair.

m1 45, m2 135, m3 125, m4 45, m5 135,m6 35, and CAT is a right angle. Determine whether each statement is true or false.1. 1 and 2 form a linear pair.

2. 4 and 5 form a linear pair.

3. 6 and 3 are adjacent angles.

4. 7 and 8 are adjacent angles.

5. CAT and 7 are adjacent angles.

Use the terms adjacent angles, linear pair, or neither todescribe angles 1 and 2 in as many ways as possible.6. 7. 8.


3-43-4NAME ______________________________________DATE __________PERIOD______

5 6

1 26 3

54

21

21 21


Adjacent Angles and Linear Pairs of Angles

Pairs of Angles


angles in the same plane that

adjacent angles have a common vertex and a common side, but no common interior points 3 and 4 are adjacent angles.

adjacent angles whose linear pair noncommon sides are opposite

rays5 and 6 form a linear pair.

m1 45, m2 135, m3 125, m4 45, m5 135,m6 35, and CAT is a right angle. Determine whether each statement is true or false.1. 1 and 2 form a linear pair.

false2. 4 and 5 form a linear pair.

true3. 6 and 3 are adjacent angles.

false4. 7 and 8 are adjacent angles.

true5. CAT and 7 are adjacent angles.

false

Use the terms adjacent angles, linear pair, or neither todescribe angles 1 and 2 in as many ways as possible.6. 7. 8.

adjacent angles, adjacent angles adjacent angles, linear pair linear pair


3-43-4NAME ______________________________________DATE __________PERIOD______

5 6

1 26 3

54

21

21 21



3-53-5NAME ______________________________________DATE __________PERIOD______

Complementary and Supplementary Angles

The table identifies several different types of angles that occur in pairs.

Pairs of Angles


complementary two angles whose measures have angles a sum of 90

supplementary two angles whose measures have angles a sum of 180

Each pair of angles is either complementary orsupplementary. Find the value of x in each figure.1. 2. 3.

4. 5. 6.

7. If mP 28, R and P are 8. If S and G are supplementary,supplementary, T and P are mS 6x 10, and mG complementary, and Z and T 15x 23, find x and the measurecomplementary, find mR, of each angle. mT, and mZ.

30 60

16020

15

x x 20 a

b(6x 7)

(3x 38)



3-53-5NAME ______________________________________DATE __________PERIOD______

30 60

16020

15

x x 20 a

b(6x 7)

(3x 38)

Complementary and Supplementary Angles

The table identifies several different types of angles that occur in pairs.

Pairs of Angles


complementary two angles whose measures have angles a sum of 90

supplementary two angles whose measures have angles a sum of 180

Each pair of angles is either complementary orsupplementary. Find the value of x in each figure.1. 75 2. 160 3. 115

4. 4 5. 17 6. 19

7. If mP 28, R and P are 8. If S and G are supplementary,supplementary, T and P are mS 6x 10, and mG complementary, and Z and T 15x 23, find x and the measurecomplementary, find mR, of each angle. mT, and mZ. 152, 62, 28 x 7, mS 52, mG 128


Congruent Angles

Opposite angles formed by intersecting lines arecalled vertical angles. Vertical angles are alwayscongruent. 1 and 3, and 2 and 4 are pairsof vertical angles.

Identify each pair of angles in Exercises 14 as adjacent, vertical,complementary, supplementary, and/or as a linear pair.1. 1 and 2 2. 1 and 4

3. 3 and 4 4. 1 and 5

5. Find x, y, and z. 6. Find x and y if CBD FDG.

Use the figure shown to find each of the following.7. x

8. mLAT

9. mTAO

10. mPAO


3-63-6NAME ______________________________________DATE __________PERIOD______

34

12

3 451

2

x51

zy


Congruent Angles

Opposite angles formed by intersecting lines arecalled vertical angles. Vertical angles are alwayscongruent. 1 and 3, and 2 and 4 are pairsof vertical angles.

Identify each pair of angles in Exercises 14 as adjacent, vertical,complementary, supplementary, and/or as a linear pair.1. 1 and 2 2. 1 and 4

adjacent vertical

3. 3 and 4 4. 1 and 5adjacent, adjacent, complementary supplementary, linear pair

5. Find x, y, and z. 6. Find x and y if CBD FDG.129, 51, 129 63, 63

Use the figure shown to find each of the following.7. x 8

8. mLAT 155

9. mTAO 25

10. mPAO 155


3-63-6NAME ______________________________________DATE __________PERIOD______

34

12

3 451

2

x51

zy


Perpendicular Lines

Perpendicular lines are lines that intersect to form four right angles.

AB FE, AE GC and C is the midpoint ofAE. Determine whether each of the followingis true or false.1. GV AE

2. AE FV

3. 4 1

4. 3 4

5. m1 + m2 = 90 6. m3 + m4 = 180

7. m1 + m5 = 90 8. m4 = m1 + m2

9. mAVF = 90 10. mBVE = 90

11. GVA is a right angle. 12. 3, 4, and 8 are right angles.

13. 6 and 3 are supplementary. 14. 2 and 6 are complementary.

15. FVB and 4 are complementary. 16. AVE and BVF are supplementary.

17. AE is the only line perpendicular to GC at C.


3-73-7NAME ______________________________________DATE __________PERIOD______


Perpendicular Lines

Perpendicular lines are lines that intersect to form four right angles.

AB FE, AE GC and C is the midpoint ofAE. Determine whether each of the followingis true or false.1. GV AE true

2. AE FV false

3. 4 1 false

4. 3 4 true

5. m1 + m2 = 90 true 6. m3 + m4 = 180 true

7. m1 + m5 = 90 true 8. m4 = m1 + m2 true

9. mAVF = 90 true 10. mBVE = 90 true

11. GVA is a right angle. 12. 3, 4, and 8 are right angles.false true

13. 6 and 3 are supplementary. 14. 2 and 6 are complementary.false true

15. FVB and 4 are complementary. 16. AVE and BVF are supplementary.false true

17. AE is the only line perpendicular to GC at C. true


3-73-7NAME ______________________________________DATE __________PERIOD______


Parallel Lines and Planes

When planes do not intersect, they are said to be parallel. Also,when lines in the same plane do not intersect, they are parallel.But when lines are not in the same plane and do not intersect, they are skew.

Example: Name the parts of the triangular prism shown at the right. Sample answers are given.

parallel planes: planes PQR and NOMparallel segments: MO and RQskew segments: MN and RQ

Refer to the figure in the example.1. Name two more pairs of parallel segments.

2. Name two more segments skew to NM.

3. Name a segment that is parallel to plane MRQ.

Name the parts of the hexagonal prism shown at the right.4. three segments that are parallel to BC

5. three segments that are parallel to JK

6. a segment that is skew to QP

7. the plane that is parallel to plane AJQ


4-14-1NAME ______________________________________DATE __________PERIOD______


Parallel Lines and Planes

When planes do not intersect, they are said to be parallel. Also,when lines in the same plane do not intersect, they are parallel.But when lines are not in the same plane and do not intersect, they are skew.

Example: Name the parts of the triangular prism shown at the right. Sample answers are given.

parallel planes: planes PQR and NOMparallel segments: MO and RQskew segments: MN and RQ

Refer to the figure in the example.1. Name two more pairs of parallel segments.

Sample answers: MN and RP, NO and PQ, NP and MR,NP and OQ, MR and OQ

2. Name two more segments skew to NM.PQ, OQ

3. Name a segment that is parallel to plane MRQ. NP

Name the parts of the hexagonal prism shown at the right.4. three segments that are parallel to BC FE, QP, KL

5. three segments that are parallel to JK AB, MP, DE

6. a segment that is skew to QPSample answers: AJ, AF, and DE

7. the plane that is parallel to plane AJQ plane CDM


4-14-1NAME ______________________________________DATE __________PERIOD______


Parallel Lines and Transversals

A line that intersects two or more lines in a plane atdifferent points is called a transversal. Eight angles areformed by a transversal and two lines.

Types of Angles

Angle Definition Examples

interior lie between the two lines 1, 2, 5, 6

alternate interior on opposite sides of the transversal 1 and 6, 2 and 5

consecutive interior on the same side of the transversal 1 and 5, 2 and 6

exterior lie outside the two lines 3, 4, 7, 8

alternate exterior on opposite sides of the transversal 3 and 7, 4 and 8

Identify each pair of angles as alternate interior, alternateexterior, consecutive interior, or vertical.1. 6 and 10 2. 14 and 13

3. 14 and 6 4. 1 and 5

5. 12 and 15 6. 2 and 16

In the figure, AB DC and BC AD.7. For which pair of parallel lines are 1 and 4

alternate interior angles?

8. For which pair of parallel lines are 2 and 3alternate interior angles?


4-24-2NAME ______________________________________DATE __________PERIOD______


Parallel Lines and Transversals

A line that intersects two or more lines in a plane atdifferent points is called a transversal. Eight angles areformed by a transversal and two lines.

Types of Angles

Angle Definition Examples

interior lie between the two lines 1, 2, 5, 6

alternate interior on opposite sides of the transversal 1 and 6, 2 and 5

consecutive interior on the same side of the transversal 1 and 5, 2 and 6

exterior lie outside the two lines 3, 4, 7, 8

alternate exterior on opposite sides of the transversal 3 and 7, 4 and 8

Identify each pair of angles as alternate interior, alternateexterior, consecutive interior, or vertical.1. 6 and 10 2. 14 and 13

alternate exterior consecutive interior

3. 14 and 6 4. 1 and 5alternate interior alternate exterior

5. 12 and 15 6. 2 and 16consecutive interior vertical

In the figure, AB DC and BC AD.7. For which pair of parallel lines are 1 and 4

alternate interior angles? BC and AD

8. For which pair of parallel lines are 2 and 3alternate interior angles? AB and DC


4-24-2NAME ______________________________________DATE __________PERIOD______



4-34-3NAME ______________________________________DATE __________PERIOD______

Transversals and Corresponding AnglesIf two parallel lines are cut by a transversal, then the followingpairs of angles are congruent.

corresponding anglesalternate interior anglesalternate exterior angles

If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

Example: In the figure m n and p is a transversal. If m2 35,find the measures of the remaining angles.

Since m2 35, m8 35 (corresponding angles).Since m2 35, m6 35 (alternate interior angles).Since m8 35, m4 35 (alternate exterior angles).

m2 m5 180. Since consecutive interior angles are supplementary, m5 145, which implies that m3, m7, and m1 equal 145.

In the figure at the right p q, m1 78, and m2 47. Find the measure of each angle.1. 3 2. 4 3. 5

4. 6 5. 7 6. 8 7. 9

Find the values of x and y, in each figure.8. 9. 10.

Find the values of x, y, and z in each figure.11. 12.

(y 8)

(3x 5)

(6x 14)5x 9y (3y 10)

7y

6x

(8x 40)

(3z 18)

3y72

x

zx

(y 12)

(y 18)



4-34-3NAME ______________________________________DATE __________PERIOD______

Transversals and Corresponding AnglesIf two parallel lines are cut by a transversal, then the followingpairs of angles are congruent.

corresponding anglesalternate interior anglesalternate exterior angles

If two parallel lines are cut by a transversal, then consecutive interior angles are supplementary.

Example: In the figure m n and p is a transversal. If m2 35,find the measures of the remaining angles.

Since m2 35, m8 35 (corresponding angles).Since m2 35, m6 35 (alternate interior angles).Since m8 35, m4 35 (alternate exterior angles).

m2 m5 180. Since consecutive interior angles are supplementary, m5 145, which implies that m3, m7, and m1 equal 145.

In the figure at the right p q, m1 78, and m2 47. Find the measure of each angle.1. 3 2. 4 3. 5

102 102 78

4. 6 5. 7 6. 8 7. 9133 47 47 133

Find the values of x and y, in each figure.8. 9. 10.

21, 60 18, 10 10, 19

Find the values of x, y, and z in each figure.11. 12.

108, 36, 30 90, 93, 15

(y 8)

(3x 5)

(6x 14)5x 9y (3y 10)

7y

6x

(8x 40)

(3z 18)

3y72

x

zx

(y 12)

(y 18)


Proving Lines ParallelSuppose two lines in a plane are cut by a transversal. Withenough information about the angles that are formed, you can decide whether the two lines are parallel.

Example: If 1 2, which lines must be parallel? Explain.

AC BD because a pair of corresponding angles are congruent.

Find x so that a b.1. 2. 3.

4. 5. 6.

Given the following information, determine which lines, if any,are parallel. State the postulate or theorem that justifies youranswer.

7. 1 8 8. 4 9

9. m7 m13 180 10. 9 13


4-44-4NAME ______________________________________DATE __________PERIOD______

IF THEN

corresponding angles are congruent,alternate interior angles are congruent,alternate exterior angles are congruent, the lines are parallel.consecutive interior angles are supplementary,the lines are perpendicular to the same line,

b

a

(4x 10)

110

a b

(3x 50)(2x 5)

a

b(6x 12)

2x

a b

(4x 22) a

b

57

(3x 9)

a

b(6x 7)

(3x 38)

ec d

f

1 2 5 63 4 7 8

9 10 13 1411 12 15 16


Proving Lines ParallelSuppose two lines in a plane are cut by a transversal. Withenough information about the angles that are formed, you can decide whether the two lines are parallel.

Example: If 1 2, which lines must be parallel? Explain.

AC BD because a pair of corresponding angles are congruent.

Find x so that a b.1. 2. 3.

15 45 214. 5. 6.

17 44 15

Given the following information, determine which lines, if any,are parallel. State the postulate or theorem that justifies youranswer.

7. 1 8 8. 4 9c d, alternate exterior e f, alternateangles congruent interior angles

congruent9. m7 m13 180 10. 9 13

e f, consecutive c d, interior angles correspondingsupplementary angles congruent


4-44-4NAME ______________________________________DATE __________PERIOD______

IF THEN

corresponding angles are congruent,alternate interior angles are congruent,alternate exterior angles are congruent, the lines are parallel.consecutive interior angles are supplementary,the lines are perpendicular to the same line,

b

a

(4x 10)

110

a b

(3x 50)(2x 5)

a

b(6x 12)

2x

a b

(4x 22) a

b

57

(3x 9)

a

b(6x 7)

(3x 38)

ec d

f

1 2 5 63 4 7 8

9 10 13 1411 12 15 16



4-54-5NAME ______________________________________DATE __________PERIOD______

Slope

Example: Find the slope of the line passing through A(2, 5)and B(1, 3). State the slope of a line parallel to .Then state the slope of a line perpendicular to .

Let (x1, y1) (2, 5) and (x2, y2) (1, 3). Then m 3

1 (5

2)

83

.

Any line in the coordinate plane parallel to hasslope 8

3.

Since 83

38

1, the slope of a line perpendicular to

the line is 38

.

Find the slope of the line passing through the given points.1. C(2, 4), D(8, 12) 2. J(4, 6), K(3, 10) 3. P(0, 12), R(12, 0)

4. S(15, 15), T(15, 0) 5. F(21, 12), G(6, 4) 6. L(7, 0), M(17, 10)

Find the slope of the line parallel to the line passing througheach pair of points. Then state the slope of the lineperpendicular to the line passing through each pair of points.7. I(9, 3), J(6, 10) 8. G(8, 12), H(4, 1) 9. M(5, 2), T(9, 6)

To find the slope of a line containing two points withcoordinates (x1, y1) and (x2, y2), use the following formula.

m xy2

2

yx

1

1

, where x1 x2

The slope of a vertical line, where x1 x2, is undefined.

Two lines have the same slope if and only if they are paralleland nonvertical.

Two nonvertical lines are perpendicular if and only if theproduct of their slopes is 1.



4-54-5NAME ______________________________________DATE __________PERIOD______

Slope

Example: Find the slope of the line passing through A(2, 5)and B(1, 3). State the slope of a line parallel to .Then state the slope of a line perpendicular to .

Let (x1, y1) (2, 5) and (x2, y2) (1, 3). Then m 3

1 (5

2)

83

.

Any line in the coordinate plane parallel to hasslope 8

3.

Since 83

38

1, the slope of a line perpendicular to

the line is 38

.

Find the slope of the line passing through the given points.1. C(2, 4), D(8, 12) 2. J(4, 6), K(3, 10) 3. P(0, 12), R(12, 0)

85

176 1

4. S(15, 15), T(15, 0) 5. F(21, 12), G(6, 4) 6. L(7, 0), M(17, 10)

12

1267 1

52

Find the slope of the line parallel to the line passing througheach pair of points. Then state the slope of the lineperpendicular to the line passing through each pair of points.7. I(9, 3), J(6, 10) 8. G(8, 12), H(4, 1) 9. M(5, 2), T(9, 6)

73,

37

1112,

1121

1, 1

To find the slope of a line containing two points withcoordinates (x1, y1) and (x2, y2), use the following formula.

m xy2

2

yx

1

1

, where x1 x2

The slope of a vertical line, where x1 x2, is undefined.

Two lines have the same slope if and only if they are paralleland nonvertical.

Two nonvertical lines are perpendicular if and only if theproduct of their slopes is 1.



4-64-6NAME ______________________________________DATE __________PERIOD______

Equations of LinesYou can write an equation of a line if you are given

the slope and the coordinates of a point on the line, or the coordinates of two points on the line.

Example: Write the equation in slope-intercept form of the linethat has slope 5 and an x-intercept of 3.

Since the slope is 5, you can substitute 5 for m in y mx b.y 5x b

Since the x-intercept is 3, the point (3, 0) is on the line.y 5x b0 5(3) b y 0 and x 30 15 bb 15 Solve for b.So the equation is y 5x 15.

If you know two points on a line, you will need to find the slopeof the line passing through the points and then write theequation.

Write the equation in slope-intercept form of the line thatsatisfies the given conditions.1. m 3, y-intercept 4 2. m 2

5, x-intercept 6

3. passes through (5, 10) and (2, 4) 4. passes through (8, 6) and (3, 3)

5. perpendicular to the y-axis, 6. parallel to the y-axis, passes through (6, 4) passes through (7, 3)

7. m 3 and passes through (4, 6) 8. perpendicular to the graph ofy 4x 1 and passes through (6, 3)



4-64-6NAME ______________________________________DATE __________PERIOD______

Equations of LinesYou can write an equation of a line if you are given

the slope and the coordinates of a point on the line, or the coordinates of two points on the line.

Example: Write the equation in slope-intercept form of the linethat has slope 5 and an x-intercept of 3.

Since the slope is 5, you can substitute 5 for m in y mx b.y 5x b

Since the x-intercept is 3, the point (3, 0) is on the line.y 5x b0 5(3) b y 0 and x 30 15 bb 15 Solve for b.So the equation is y 5x 15.

If you know two points on a line, you will need to find the slopeof the line passing through the points and then write theequation.

Write the equation in slope-intercept form of the line thatsatisfies the given conditions.1. m 3, y-intercept 4 2. m 2

5, x-intercept 6

y 3x 4 y 25x 152

3. passes through (5, 10) and (2, 4) 4. passes through (8, 6) and (3, 3)

y 67x 470 y 1

91x 1

61

5. perpendicular to the y-axis, 6. parallel to the y-axis, passes through (6, 4) passes through (7, 3)y 4 x 7

7. m 3 and passes through (4, 6) 8. perpendicular to the graph ofy 4x 1 and passes through (6, 3)

y 3x 18 y 14x 32



5-15-1NAME ______________________________________DATE __________PERIOD______

Classifying TrianglesTriangles are classified in two different ways, either by their angles or by their sides.

Classification of TrianglesAngles Sides

Examples: Classify each triangle by its angles and by its sides.1 2

DEF is obtuse and scalene. MNO is acute and equilateral.

Use a protractor and ruler to draw triangles using the givenconditions. If possible, classify each triangle by the measuresof its angles and sides.1. KLM, m K 90, 2. XYZ, m X 60, 3. DEF, m D 150,

KL 2.5 cm, KM 3 cm XY YZ ZX 3 cm DE DF 1 inch

4. GHI, m G 30, 5. NOP, m N 90, 6. QRS, m Q 100,m H 45, GH 4 cm NO NP 2.5 cm QS 1 inch,

QR 112

inches

scalene no two sides congruentisosceles at least two sides congrentequilateral all sides congruent

acute all acute anglesobtuse one obtuse angleright one right angle

D E

F

2.3 cm5 cm

4 cm27

51

102 NM

O

60 60

60



5-15-1NAME ______________________________________DATE __________PERIOD______

Classifying TrianglesTriangles are classified in two different ways, either by their angles or by their sides.

Classification of TrianglesAngles Sides

Examples: Classify each triangle by its angles and by its sides.1 2

DEF is obtuse and scalene. MNO is acute and equilateral.

Use a protractor and ruler to draw triangles using the givenconditions. If possible, classify each triangle by the measuresof its angles and sides.1. KLM, m K 90, 2. XYZ, m X 60, 3. DEF, m D 150,

KL 2.5 cm, KM 3 cm XY YZ ZX 3 cm DE DF 1 inch

right, scalene acute, equilateral obtuse, isoceles

4. GHI, m G 30, 5. NOP, m N 90, 6. QRS, m Q 100,m H 45, GH 4 cm NO NP 2.5 cm QS 1 inch,

QR 112

inches

obtuse, scalene right, isosceles

scalene no two sides congruentisosceles at least two sides congrentequilateral all sides congruent

acute all acute anglesobtuse one obtuse angleright one right angle

D E

F

2.3 cm5 cm

4 cm27

51

102 NM

O

60 60

60

MK

L

YX

Z

D E

F

G H

I

N O

P

S

R

Q

obtuse,scalene



5-25-2NAME ______________________________________DATE __________PERIOD______

Angles of a TriangleOn a separate sheet of paper, draw a triangle of any size. Labelthe three angles D, E, and F. Then tear off the three corners andrearrange them so that the three vertices meet at one point, with D and F each adjacent to E. What do you notice?

Examples: Find the value of x in each figure.1 2

28 41 x 180 x 41 6369 x 180 x 22

x 111

Find the value of x.1. 2. 3.

4. 5. 6.

Find the measure of each angle.7. 1 8. 2 9. 3

10. 4 11. 5 12. 6

The sum of the measures of the angles of a triangle is 180.

28 41

x

47

58

x

2196

x(3x 1)

31

2134

x(2x 3)

100

512x60

94

43

56 2 168

41

63

x

D E

F



5-25-2NAME ______________________________________DATE __________PERIOD______

Angles of a TriangleOn a separate sheet of paper, draw a triangle of any size. Labelthe three angles D, E, and F. Then tear off the three corners andrearrange them so that the three vertices meet at one point, with D and F each adjacent to E. What do you notice?

Examples: Find the value of x in each figure.1 2

28 41 x 180 x 41 6369 x 180 x 22

x 111

Find the value of x.1. 2. 3.

75 63 20

4. 5. 6.

5523 17

Find the measure of each angle.7. 1 8. 2 9. 3

112 68 90

10. 4 11. 5 12. 690 22 158

The sum of the measures of the angles of a triangle is 180.

28 41

x

47

58

x

2196

x(3x 1)

31

2134

x(2x 3)

100

512x60

94

43

56 2 168

41

63

x

D E

F



5-35-3NAME ______________________________________DATE __________PERIOD______

Congruent TrianglesWhen two figures have exactly the same shape and size, they aresaid to be congruent. For two congruent triangles there are threepairs of corresponding (matching) sides and three pairs ofcorresponding angles. To write a correspondence statementabout congruent triangles, you should name correspondingangles in the same order. Remember that congruent parts aremarked by identical markings.Example: Write a correspondence statement for the triangles in

the diagram.

LMO XYZ

Complete each correspondence statement.1. 2. 3.

SAT BCD GHK

Write a congruence statement for each pair of congruenttriangles.4. 5. 6.

Draw triangles EDG and QRS. Label the corresponding parts if EDG QRS. Then complete each statement.7. E 8. DG

9. EDG 10. GE

11. ED 12. EGD ? ?

? ?

? ?

???

A T

S

B

F

C D

B

CM

N

G

T

H

K

W I

NT

C A

R S

OV

E

GK

M

F

L M

O X Y

Z



5-35-3NAME ______________________________________DATE __________PERIOD______

Congruent TrianglesWhen two figures have exactly the same shape and size, they aresaid to be congruent. For two congruent triangles there are threepairs of corresponding (matching) sides and three pairs ofcorresponding angles. To write a correspondence statementabout congruent triangles, you should name correspondingangles in the same order. Remember that congruent parts aremarked by identical markings.Example: Write a correspondence statement for the triangles in

the diagram.

LMO XYZ

Complete each correspondence statement.1. 2. 3.

SAT FCB BCD NMD GHK GTKWrite a congruence statement for each pair of congruenttriangles.4. 5. 6.

TWN INW CAR SOV FEM GKMDraw triangles EDG and QRS. Label the corresponding parts if EDG QRS. Then complete each statement.7. E Q 8. DG RS9. EDG QRS 10. GE SQ

11. ED QR 12. EGD QSR? ? ? ?

? ?

???

A T

S

B

F

C D

B

CM

N

G

T

H

K

W I

NT

C A

R S

OV

E

GK

M

F

E D

G

Q R

S

L M

O X Y

Z

Drawings may vary.


Geometry in Motion

Transformations

Term Definition Examples

translation slide a figure from one position to another without turning it

reflection flip a figure over a line

rotation turn a figure around a fixed point

Determine whether each figure is a translation, reflection, orrotation of the given figure.1. a. b. c.

2. a. b. c.

3. a. b. c.


5-45-4NAME ______________________________________DATE __________PERIOD______


Geometry in Motion

Transformations

Term Definition Examples

translation slide a figure from one position to another without turning it

reflection flip a figure over a line

rotation turn a figure around a fixed point

Determine whether each figure is a translation, reflection, orrotation of the given figure.1. a. b. c.

reflection translation rotation

2. a. b. c.

translation rotation reflection

3. a. b. c.

reflection rotation translation


5-45-4NAME ______________________________________DATE __________PERIOD______


SSS and SAS

You can show that two triangles are congruent with the following.

SSS Postulate If three sides of one triangle are congruent to three

(side-side-side) corresponding sides of another triangle, the trianglesare congruent.

SAS Postulate If two sides and the included angle of one triangle are

(side-angle-side) congruent to the corresponding sides and includedangle of another triangle, the triangles are congruent.

Determine whether each pair of triangles is congruent. If so,write a congruence statement and explain why the trianglesare congruent.1. 2.

3. 4.

5. 6.


5-55-5NAME ______________________________________DATE __________PERIOD______


SSS and SAS


SSS Postulate If three sides of one triangle are congruent to three

(side-side-side) corresponding sides of another triangle, the trianglesare congruent.

SAS Postulate If two sides and the included angle of one triangle are

(side-angle-side) congruent to the corresponding sides and includedangle of another triangle, the triangles are congruent.

Determine whether each pair of triangles is congruent. If so,write a congruence statement and explain why the trianglesare congruent. Sample answers given.1. 2.

PQS PRS; SAS JMK LKM; SAS3. 4.

PRQ STU; SSS BAE BCD; SAS5. 6.

WXV YXZ; SSS LMN NOL; SAS


5-55-5NAME ______________________________________DATE __________PERIOD______


ASA and AAS


If two angles and the included side of one triangle ASA Postulate are congruent to the corresponding angles and (angle-side-angle) included side of another triangle, the triangles are

congruent.

If two angles and a nonincluded side of one triangle AAS Theorem are congruent to the corresponding two angles and (angle-angle-side) nonincluded side of another triangle, the triangles

are congruent.

Determine whether each pair of triangles is congruent. If so,write a congruence statement and explain why the trianglesare congruent. If it is not possible to prove that they arecongruent, write not possible.1. 2.

3. 4.

5. 6.


5-65-6NAME ______________________________________DATE __________PERIOD______


ASA and AAS


If two angles and the included side of one triangle ASA Postulate are congruent to the corresponding angles and (angle-side-angle) included side of another triangle, the triangles are

congruent.

If two angles and a nonincluded side of one triangle AAS Theorem are congruent to the corresponding two angles and (angle-angle-side) nonincluded side of another triangle, the triangles

are congruent.

Determine whether each pair of triangles is congruent. If so,write a congruence statement and explain why the trianglesare congruent. If it is not possible to prove that they arecongruent, write not possible. Sample answers given.1. 2.

GIH JIK; AAS ABC QPR; ASA3. 4.

not possible ABC EDC; ASA5. 6.

LHI JKI; ASA TUV FGE; AAS


5-65-6NAME ______________________________________DATE __________PERIOD______


MediansA median is a segment that joins a vertex of a triangle and themidpoint of the side opposite that vertex. The medians of a triangleintersect at a common point called the centroid. An importanttheorem about medians and centroids is as follows.

The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

Example:

DF is a median of DEC.

In ABC, AN, BP, and CM are medians.1. If BP 10, find BE.

2. If EM 3, find EC.

3. If EN 12, find AN.

4. If CM 3x 6 and CE x 12, what is x?

5. If EN x 5 and AE x 17, find AN.

Draw and label a figure to illustrate each situation.6. NRW is a right triangle with right angle at N.

NX is a median of NRW.

7. OQ is a median of POM.


6-16-1NAME ______________________________________DATE __________PERIOD______


MediansA median is a segment that joins a vertex of a triangle and themidpoint of the side opposite that vertex. The medians of a triangleintersect at a common point called the centroid. An importanttheorem about medians and centroids is as follows.

The length of the segment from the vertex to the centroid is twice the length of the segment from the centroid to the midpoint.

Example:

DF is a median of DEC.

In ABC, AN, BP, and CM are medians.1. If BP 10, find BE. 63

2

2. If EM 3, find EC. 6

3. If EN 12, find AN. 36

4. If CM 3x 6 and CE x 12, what is x? 8

5. If EN x 5 and AE x 17, find AN. 66

6-7. Sample answers are given.Draw and label a figure to illustrate each situation.6. NRW is a right triangle with right angle at N.

NX is a median of NRW.

7. OQ is a median of POM.


6-16-1NAME ______________________________________DATE __________PERIOD______


Altitudes and Perpendicular BisectorsAn altitude of a triangle is a perpendicular segment that has oneendpoint at a vertex of the triangle and the other endpoint on theside opposite that vertex. A perpendicular bisector of a side of atriangle is a segment or line that contains the midpoint of that sideand is perpendicular to that side.

Example:

XW is an altitude of XYZ.UV is a perpendicular bisector of XYZ.

In FGH, GC CH, FE EH, and FGD HGD.1. Which segment is a median of FGH?

2. Which segment is an altitude of FGH?

3. Which segment is a perpendicular bisector of FGH?

Draw and label a figure to illustrate each situation.4. BV is a median and a perpendicular bisector of BST.

5. KT is an altitude of KLM, and L is between T and M.


6-26-2NAME ______________________________________DATE __________PERIOD______


Altitudes and Perpendicular BisectorsAn altitude of a triangle is a perpendicular segment that has oneendpoint at a vertex of the triangle and the other endpoint on theside opposite that vertex. A perpendicular bisector of a side of atriangle is a segment or line that contains the midpoint of that sideand is perpendicular to that side.

Example:

XW is an altitude of XYZ.UV is a perpendicular bisector of XYZ.

In FGH, GC CH, FE EH, and FGD HGD.1. Which segment is a median of FGH? CF

2. Which segment is an altitude of FGH? AH

3. Which segment is a perpendicular bisector of FGH?BE

4-5. Sample answers are given.Draw and label a figure to illustrate each situation.4. BV is a median and a perpendicular bisector of BST.

5. KT is an altitude of KLM, and L is between T and M.


6-26-2NAME ______________________________________DATE __________PERIOD______


Angle Bisectors of TrianglesAn angle bisector of a triangle is a segment that bisects an angleof the triangle and has one endpoint at the vertex of that angle andthe other endpoint on the side opposite that vertex.

Example:

RV is an angle bisector of RST.

In ACE, CF, EB, and AD are angle bisectors.1. m3 ________

2. m2 ________

3. ________ bisects CAE.

4. ________ bisects ACE.

5. m6 ________(mCEA)

6. mACE ________(m3)

7. Draw and label a figure to illustrate this situation. HS is an angle bisector of GHI, and S is between G and I.


6-36-3NAME ______________________________________DATE __________PERIOD______


Angle Bisectors of TrianglesAn angle bisector of a triangle is a segment that bisects an angleof the triangle and has one endpoint at the vertex of that angle andthe other endpoint on the side opposite that vertex.

Example:

RV is an angle bisector of RST.

In ACE, CF, EB, and AD are angle bisectors.1. m3 ________ m4

2. m2 ________ m1

3. ________ bisects CAE. AD

4. ________ bisects ACE. CF

5. m6 ________(mCEA) 21

6. mACE ________(m3) 2

7. Draw and label a figure to illustrate this situation. HS is an angle bisector of GHI, and S is between G and I. A sample answer is given.


6-36-3NAME ______________________________________DATE __________PERIOD______


Isosceles TrianglesRemember that two sides of an isosceles triangle are congruent.Two important theorems about isosceles triangles are as follows.

Example: Find the value of x.Since AB BC, the angles opposite AB and BC arecongruent. So m A m C.Therefore, 3x 10 2x 6.

3x 10 2x 63x 10 10 2x 6 10 Add 10 to each side.

3x 2x 163x 2x 2x 16 2x Subtract 2x from each side.

x 16

Find the value of x.1. 2.

3. 4.

5. 6.

If two sides of a triangle are congruent, then the angles opposite thosesides are congruent.If two angles of a triangle are congruent, then the sides opposite thoseangles are congruent.


6-46-4NAME ______________________________________DATE __________PERIOD______


Isosceles TrianglesRemember that two sides of an isosceles triangle are congruent.Two important theorems about isosceles triangles are as follows.

Example: Find the value of x.Since AB BC, the angles opposite AB and BC arecongruent. So m A m C.Therefore, 3x 10 2x 6.

3x 10 2x 63x 10 10 2x 6 10 Add 10 to each side.

3x 2x 163x 2x 2x 16 2x Subtract 2x from each side.

x 16

Find the value of x.1. 2.

20 8

3. 4.

76 15

5. 6.

16 10

If two sides of a triangle are congruent, then the angles opposite thosesides are congruent.If two angles of a triangle are congruent, then the sides opposite thoseangles are congruent.


6-46-4NAME ______________________________________DATE __________PERIOD______



6-56-5NAME ______________________________________DATE __________PERIOD______

Right TrianglesTwo right triangles are congruent if one of the followingconditions exist.

State the additional information needed to prove each pair oftriangles congruent by the given theorem or postulate.1. HL 2. HA 3. LL

4. LA 5. HA 6. LA

Theorem 6-6 If two legs of one right triangle are congruent to the corresponding legs LL of another right triangle, then the triangles are congruent.

Theorem 6-7 If the hypotenuse and an acute angle of one right triangle are congruentHA to the hypotenuse and corresponding angle of another right triangle,then the two triangles are congruent.

Theorem 6-8 If one leg and an acute angle of a right triangle are congruent to the LA corresponding leg and angle of another right triangle, then the triangles

are congruent.

Postulate 6-1 If the hypotenuse and a leg of one right triangle are congruent to theHL hypotenuse and corresponding leg of another right triangle, then thetriangles are congruent.



6-56-5NAME ______________________________________DATE __________PERIOD______

Right TrianglesTwo right triangles are congruent if one of the followingconditions exist.

State the additional information needed to prove each pair oftriangles congruent by the given theorem or postulate.1. HL 2. HA 3. LL

EG FG AB DB NP VX4. LA 5. HA 6. LA

M K or no extra information BC DF or MJL KJL needed AB AD, and

BAC DAF or C F

Theorem 6-6 If two legs of one right triangle are congruent to the corresponding legs LL of another right triangle, then the triangles are congruent.

Theorem 6-7 If the hypotenuse and an acute angle of one right triangle are congruentHA to the hypotenuse and corresponding angle of another right triangle,then the two triangles are congruent.

Theorem 6-8 If one leg and an acute angle of a right triangle are congruent to the LA corresponding leg and angle of another right triangle, then the triangles

are congruent.

Postulate 6-1 If the hypotenuse and a leg of one right triangle are congruent to theHL hypotenuse and corresponding leg of another right triangle, then thetriangles are congruent.


The Pythagorean TheoremIn a right triangle, the square of the hypotenuse, c, is equal to thesum of the squares of the lengths of the other two sides, a and b.

c2 a2 b2

52 32 4252 9 1625 25

Example: How long must a ladder be to reach a window13 feet above ground? For the sake of stability,the ladder must be placed 5 feet away from thebase of the wall.

c2 (13)2 (5)2c2 169 25c2 194c2 194 or about 13.9 ft

Solve. Round decimal answers to the nearest tenth.1. In a softball game, how far must the

catcher throw to second base?

3. If Briny is 32 miles due east of Oxfordand Myers is 21 miles due south ofOxford, how far is the shortestdistance from Myers to Briny?

2. How long must the brace be on acloset rod holder if the vertical side is17 cm and the horizontal side must beattached 30 cm from the wall?

4. In a baseball game, how far must theshortstop (halfway between secondbase and third base) throw to make anout at first base?


6-66-6NAME ______________________________________DATE __________PERIOD______


The Pythagorean TheoremIn a right triangle, the square of the hypotenuse, c, is equal to thesum of the squares of the lengths of the other two sides, a and b.

c2 a2 b2

52 32 4252 9 1625 25

Example: How long must a ladder be to reach a window13 feet above ground? For the sake of stability,the ladder must be placed 5 feet

study guides

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parallel lines

congruent angles

perpendicular lines

adjacent angles

number lines

supplementary angles

equations of lines

congruent triangles