proof, parallel and 1 perpendicular lines - wikispacestristanbates.wikispaces.com/file/view/geometry...

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Unit

1

1

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Proof, Parallel and Perpendicular Lines

Essential Questions

Why are properties, postulates, and theorems important in mathematics?

How are angles and parallel and perpendicular lines used in real-world settings?

Unit OverviewIn this unit you will begin the study of an axiomatic system, Geometry. You will investigate the concept of proof and discover the importance of proof in mathematics. You will extend your knowledge of the characteristics of angles and parallel and perpendicular lines and explore practical applications involving angles and lines.

Academic VocabularyAdd these words and others you encounter in this unit to your Math Notebook.

angle bisector complementary angles conditional statement congruent conjecture counterexample deductive reasoning inductive reasoning

midpoint of a segment parallel perpendicular postulate proof supplementary angles theorem

These assessments, following Activities 1.4, 1.7, and 1.9, will give you an opportunity to demonstrate what you have learned about reasoning, proof, and some basic geometric fi gures.

Embedded Assessment 1

Conditional Statements and Logicp. 37


Angles and Parallel Lines p. 65


Slope, Distance, and Midpoint p. 81

EMBEDDED ASSESSMENTS

??

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2 SpringBoard® Mathematics with Meaning™ Geometry

1. Solve each equation.a. 5x - 2 = 8b. 4x - 3 = 2x + 9c. 6x + 3 = 2x + 8

2. Graph y = 3x - 3 and label the x- and y-intercepts.

3. Tell the slope of a line that contains the points (5, -3) and (7, 3).

4. Write the equation of a line that has slope 1 __ 3 and y-intercept 4.

5. Write the equation of the line graphed below.

x

y

8

10

6

4

2

–8–10 –6 –4 –2 2 4 6 8 10–2

–4

–6

–8

–10

6. Give the characteristics of a rectangular prism.

7. Draw a right triangle and label the hypotenuse and legs.

8. Use a protractor to ! nd the measure of each angle.a.

b.

c.

UNIT 1

Getting Ready

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Unit 1 • Proof, Parallel and Perpendicular Lines 3

ACTIVITY

1.1Geometric FiguresWhat’s My Name?SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Group Presentation

Below are some types of ! gures you have seen in earlier mathematics courses. Describe each ! gure. Using geometric terms and symbols, list as many names as possible for each ! gure.

1. Q 2.

3. X Y Z 4. D E

5. ! m 6.

N

"

P

KT

J

7.

D B

A

C

TRY THESE AIdentify each geometric ! gure. " en use symbols to write two di# erent names for each.

a. Q P b. N C

F G HF G H

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My Notes

ACTIVITY 1.1continued

Geometric FiguresWhat’s My Name?What’s My Name?

8. Draw four angles with di! erent characteristics. Describe each angle. Name the angles using numbers and letters.

9. Compare and contrast each pair of angles.

a. b.

1 21 2

CD

BA

CD

BA

SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Interactive Word Wall, Activating Prior Knowledge, Discussion Group, Self/Peer Revision

When you compare and contrast two fi gures, you describe how they are alike or different.

MATH TERMS


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My Notes c.

d.

10. a. ! e " gure below shows two intersecting lines. Name two angles that are supplementary to !4.

b. Explain why the angles you named in part a must have the same measure.

F

D

E30°

T

R

S

60°

F

D

E30°

T

R

S

60°

F

D

E30°

Z

X

Y

150°

F

D

E30°

Z

X

Y

150°

SUGGESTED LEARNING STRATEGIES: Discussion Group, Peer/Self Revision



436 5


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My NotesMy Notes



TRY THESE BComplete the chart by naming all the listed angle types in each ! gure.

A B

F

E

C D

Acute angles

Obtuse angles

Angles with the same measure

Supplementary angles

Complementary angles

11. In the circle below, draw and label each geometric term.

a. Radius OA

b. Chord BA

c. Diameter CA

12. Refer to your drawings in circle above. What is the geometric term for point O?

SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge

7810 9


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My Notes 13. In the space below, draw a circle with center P and radius PQ = 1 in. Locate a point A so that PA = 1 1 __ 2 in. Locate a point B so that PB = 3 __ 4 in.

14. Use your diagram to complete these statements.

a. A lies _________________ the circle because ___________________________________.

b. B lies _________________ the circle because ___________________________________.

15. In your diagram above, draw circles with radii PA and PB.! ese three circles are called _______________________.

SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Activating Prior Knowledge




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My Notes



Write your answers on notebook paper. Show your work.

CHECK YOUR UNDERSTANDING

Write your answers on notebook paper. Show your work.

1. Describe all acceptable ways to name a line. 2. Describe all acceptable ways to name a plane. 3. Compare and contrast collinear and coplanar

points. 4. What are some acceptable ways to name an

angle? 5. a. Why is there a problem with using !B to

name the angle below?b. How many angles are in the ! gure?

6. Compare and contrast the terms acute angle, obtuse angle, right angle, and straight angle.

7. " e measure of !A is 42°.a. What is the measure of an angle that is

complementary to !A?b. What is the measure of an angle that is

supplementary to !A? 8. Draw a circle P.

a. Draw a segment that has one endpoint on the circle but is not a chord.

b. Draw a segment that intersects the circle in two points, contains the center, but is not a radius, diameter, or chord.

9. MATHEMATICAL R E F L E C T I O N

How can you use the ! gures in this activity to

describe real-world objects and situations? Give examples.

TRY THESE CClassify each segment in circle O. Use all terms that apply.___

AF : ___________________________

BO : ___________________________

CO : ___________________________

DO : ___________________________

EO : ___________________________

CE : ________________________

B

AC

D

O

CB

A

FE D


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UNIT 1Practice


ACTIVITY 1.1

1. Which is the correct name for this line?

G E M

a. !" # G b. ____

GM c. ! !!

MG d. !" # ME

2. Use the diagram to name each of the following.

L NM

P RQ

a. parallel lines b. perpendicular lines

3. In this diagram, m$SUT = 25°.

P S

R

U

T

Q

a. Name another angle that has measure 25°. b. Name a pair of complementary angles. c. Name a pair of supplementary angles.

Use this circle Q for Items 4–7.

WR

J

Q

T

U

4. Name the radii of circle Q. 5. Name the diameter(s) in circle Q. 6. Name the chord(s) in circle Q. 7. Which statement below must be true about

circle Q? a. The distance from U to W is the same as the

distance from R to T. b. The distance from U to W is the same as the

distance from Q to J. c. The distance from R to T is half the distance

from Q to R. d. The distance from R to T is twice the

distance from Q to J.

ACTIVITY 1.2

8. Use inductive reasoning to determine the next two terms in the sequence.

a. 1, 3, 7, 15, 31, … b. 3, -6, 12, -24, 48, …

9. Write the ! rst ! ve terms of two di" erent sequences for which 24 is the third term.

10. Generate a sequence using the description: the ! rst term in the sequence is 2 and the terms increased by consecutive odd numbers beginning with 3.

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UNIT 1 Practice


11. Use this picture pattern.

a. Draw the next shape in the pattern, b. Write a sequence of numbers that could be

used to express the pattern, c. Verbally describe the pattern of the

sequence.

12. Use expressions for odd integers to con! rm the conjecture that the product of two odd integers is an odd integer.

ACTIVITY 1.3

13. Identify the property that justi! es the statement: 5(x – 3) = 5x – 15

a. multiplication b. transitive c. subtraction d. distributive

14. Complete the prove statement and write a two column proof for the equation:

Given: 5(x – 2) = 2x – 4 Prove: 15. Write the statement in if-then form. ! e sum of two even numbers is even.

Use this statement for Items 16–19.

If today is ! ursday, then tomorrow is Friday. 16. State the conclusion of the statement. 17. Write the converse of the statement. 18. Write the inverse of the statement. 19. Write the contrapositive of the statement. 20. Given the false conditional statement, “If a

vehicle is built to " y, then it is an airplane”, write a counter example.

21. Give an example of a statement that is false and logically equivalent to its inverse.

ACTIVITY 1.4

Construct a truth table for each of the following compound statements.

22. p ! "q 23. p # "q 24. "("p $ q) 25. p $ (q ! p)

ACTIVITY 1.5

26. If Q is between A and M and MQ = 7.3 and AM = 8.5, then QA =

a. 5.8 b. 1.2 c. 7.3 d. 14.6

27. Given: K is between H and J, HK = 2x - 5, KJ = 3x + 4, and HJ = 24. What is the value of x?

a. 9 b. 5 c. 19 d. 3

28. If K is the midpoint of __

HJ , HK = x + 6, and HJ = 4x - 6, then KJ = ? .

a. 15 b. 9 c. 4 d. 10

29. P lies in the interior of %RST. m%RSP = 40° and m%TSP = 10°. m%RST = ?

a. 100° b. 50° c. 30° d. 10°

30. && ' QS bisects %PQR. If m%PQS = 3x and m%RQS = 2x + 6, then m%PQR = ? .

a. 18° b. 36° c. 30° d. 6°

31. %P and %Q are supplementary. m%P = 5x + 3 and m%Q = x + 3. x = ?

a. 14 b. 0 c. 29 d. 30


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UNIT 1Practice


In this fi gure, ! " # AE ! ! " # CG , m"BHC = 20°, and m"DHE = 60°. Use this fi gure determine the angle measures for Items 32–35.

G C

A

H

D

E

BF

32. m$CHD 33. m$AHF 34. m$GHF 35. m$DHA 36. In this ! gure, m$3 = x + 18, m$4 = x + 15,

and m$5 = 4x + 3. Show all work for parts a, b, and c.

1 5

43

a. What is the value of x? b. What is the measure of $1 c. Is $3 complementary to $1? Explain.

37. In this ! gure, m$1 = 4x + 50 and m$3 = 2x + 66. Show all work for parts a, b, and c.

12

3

a. What is the value of x? b. What is the measure of m$3? c. What is the measure of m$2?

38. Given: " # AE % " # AB , "" # AD bisects $EAC, m$CAB = 2x - 4 and m$CAE = 3x + 14. Show all work for parts a and b.

B

E

A

C

D

a. What is the value of x? b. What is the measure of m$DAC?


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UNIT 1 Practice


ACTIVITY 1.6

Use this diagram for Items 39–50.

l

t

m

54

3

12

9

86

7

14

1312

11

16

15

10

n

39. List all angles in the diagram that form a corresponding angle pair with !5.

40. List all angles in the diagram that form an alternate interior angle pair with !14.

41. List all angles in the diagram that form a same side interior angle pair with !14.

42. Given l " n. If m!6 = 120°, then m!16 = ? . 43. Given l " n. If m!12 = 5x +10 and

m!14 = 6x - 4, then x = ? and m!4 = ? .

44. Given l " n. If m!14 = 75°, then m!2 = ? . 45. Given l " n. If m!9 = 3x + 12 and

m!15 = 2x + 27, then x = ? and m!10 = ? .

46. Given l " n. If m!3 = 100°, then m!2 = ? . 47. Given l " n. If m!16 = 5x + 18 and

m!15 = 3x+ 2, then x = ? and m!8 = ? .

48. If m!10 = 135°, m!12 = 75°, and l " n determine the measure of each angle. Justify each answer.

a. m!16 b. m!15 c. m!8 d. m!14 e. m!3

49. If l " n and m # n, then m!3 = ? 50. If l " n and m # l, then m!13 = ? .

ACTIVITY 1.7

Use this diagram to identify the property, postulate, or theorem that justifi es each statement in Items 51–55.

A E

PD

Q1

23

4

R

51. PQ + QR = PR a. Angle addition postulate b. Addition property c . Definition of congruent segments d. Segment addition postulate

52. If Q is the midpoint of ___

PR , then ___

PQ $ ___

QR . a. Definition of midpoint b. Definition of congruent segments c. Definition of segment bisector d. Segment addition postulate

53. !3 $ !4 a. Definition of linear pair b. Definition of congruent angles c. Vertical angles are congruent. d. Definition of angle bisector

54. If !1 is complementary to !2, then m!1 + m!2 = 90

a. Angle Addition Postulate b. Addition property c. Definition of perpendicular d. Definition of complementary

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UNIT 1Practice


55. If m!3 = 90, then !3 is a right angle. a. Definition of perpendicular b. Definition of complementary angles c. Definition of right angle d. Vertical angles are congruent. 56. Write a two-column proof. Given: a " b, c " d Prove: !5 # !12

a

c

d

b

12

34

912

1110

1514

1316

67

85

ACTIVITY 1.8

57. Determine the slope of the line that contains the points with coordinates (1, 5) and (-2, 7).

58. Which of the following is NOT an equation for a line parallel to y = 1 __ 2 x - 6?

a. y = 2 __ 4 x + 6 b. y = 0.5x - 3

c. y = 1 __ 2 x + 1 d. y = 2x - 4

59. Determine the slope of a line perpendicular to the line with equation 6x - 4y = 30.

60. Line m contains the points with coordinates (-4, 1) and (5, 8), and line n contains the points with coordinates (6, -2) and (10, 7). Are the lines parallel, perpendicular, or neither? Justify your answer.

ACTIVITY 1.9

61. Calculate the distance between the points A(-4, 2) and B(15, 6).

62. Calculate the distance between the points R(1.5, 7) and S(-2.3, -8).

63. Kevin and Clarice both live on a street that runs through the center of town. If the police station marks the midpoint between their houses, at what point is the police station on the number line?

Kevin’sHouse

-11 1

Clarice’sHouse

64. Determine the coordinates of the midpoint of the segment with endpoints R(3, 16) and S(7, -6).

65. Determine the coordinates of the midpoint of the segment with endpoints W(-5, 10.2) and X(12, 4.5).

66. Two explorers on an expedition to the Arctic Circle have radioed their coordinates to base camp. Explorer A is at coordinates (-26, -15). Explorer B is at coordinates (13, 21). ! e base camp is located at the origin.

a. Determine the linear distance between the two explorers.

b. Determine the midpoint between the two explorers.

c. Determine the distance between the midpoint of the explorers and the base camp.


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UNIT 1 Reflection


An important aspect of growing as a learner is to take the time to refl ect on your learning. It is important to think about where you started, what you have accomplished, what helped you learn, and how you will apply your new knowledge in the future. Use notebook paper to record your thinking on the following topics and to identify evidence of your learning.

Essential Questions

1. Review the mathematical concepts and your work in this unit before you write thoughtful responses to the questions below. Support your responses with specifi c examples from concepts and activities in the unit.

Why are properties, postulates, and theorems important in mathematics? How are angles and parallel and perpendicular lines used in real-world settings?

Academic Vocabulary

2. Look at the following academic vocabulary words: angle bisector counterexample perpendicular complementary angles deductive reasoning postulate conditional statement inductive reasoning proof congruent midpoint of a segment supplementary angles conjecture parallel theorem

Choose three words and explain your understanding of each word and why each is important in your study of math.

Self-Evaluation

3. Look through the activities and Embedded Assessments in this unit. Use a table similar to the one below to list three major concepts in this unit and to rate your understanding of each.

Unit Concepts

Is Your Understanding Strong (S) or Weak (W)?

Concept 1

Concept 2

Concept 3

a. What will you do to address each weakness?b. What strategies or class activities were particularly helpful in learning the

concepts you identifi ed as strengths? Give examples to explain.

4. How do the concepts you learned in this unit relate to other math concepts and to the use of mathematics in the real world?

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Unit 1

Math Standards Review

1. Keisha walked from her house to Doris’ house. ! is can best be represented by which of the following: A. a ray C. a lineB. a segment D. a point

2. Sam wanted to go " shing. He took his " shing pole and walked 6 blocks due south to pick up his friend. Together, they walked 8 blocks due east before they reached the lake. ! ere is a path that goes directly from Sam’s house to the lake. If he had taken that path, how many blocks would he have walked?

3. To amuse her brother, Shirka was making triangular formations out of rocks. She made the following pattern:

How many rocks will be in the next triangular formation in the pattern?

1. Ⓐ Ⓑ Ⓒ Ⓓ

2.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

3.

○‒ ⊘⊘⊘⊘○• ○• ○• ○• ○• ○•⓪⓪⓪⓪⓪⓪①①①①①①②②②②②②③③③③③③④④④④④④⑤⑤⑤⑤⑤⑤⑥⑥⑥⑥⑥⑥⑦⑦⑦⑦⑦⑦⑧⑧⑧⑧⑧⑧⑨⑨⑨⑨⑨⑨

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Unit 1 (continued)

Math Standards Review

4. When camping, Sudi and Raul did not want to get lost. ! ey made a coordinate grid for their camp site. Camp was at the point (0, 0). On their grid, Sudi went to the point (-3, 10) and Raul went to the point (0, 6).

Part A: On the grid below, draw a diagram of the camp site and label the campsite location, Sudi’s location, and Raul’s location.

–12

–10

–8

–6

–4

4

10864–2

2

12

2 12–10 –8 –6 –4–12 –2

y

x

10

8

6

Part B: Find the distance between Sudi and Raul.

Part C: Find the midpoint between Sudi and Raul.

Read

Explain

Solve

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