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GeometryChapter 3 Resource Book172

LESSON

3.2 Challenge PracticeFor use with pages 157–164

In Exercises 1–3, use the given information to fi nd the measures of the angles in the diagram.

1. GIVEN: l i m, j ⊥ k, m∠ 1 5 428

13 4

911 10

1213 14 17 18

2015 16 19

7 82 5

j k

m

6

2. GIVEN: l i m, m∠ 1 5 358, and m∠ 12 5 1118

12

78

93 4

5 6

15 162019

181713

1011 12

14

m

3. GIVEN: a i b, c i d, e i f, m∠ 7 5 248, and m∠ 20 5 808

1 2 5 6 7 12111413109843

2728

2932

3130

2325 26

24

1516

1718

19 202122

c

feba

d

4. Draw a four-sided fi gure in which }

AB i } DC and }

AD i } BC . Prove ∠ A > ∠ C.

5. In the diagram, m i n. Find the value of x. Explain how you obtained your answer.

358

1488

x8

m

n

LES

SO

N 3

.2

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A31Geometry

Chapter 3 Resource Book

3. l seems to be parallel to n. If two lines are perpendicular to the same line, then they are parallel to each other.

m

n

4. a. ∠ 5, ∠ 11, ∠ 17 b. ∠ 5, ∠ 9, ∠ 17 c. ∠ 8, ∠ 12, ∠ 17 d. ∠ 7, ∠ 9, ∠ 18 e. ∠ 2, ∠ 10, ∠ 14 f. ∠ 4, ∠ 10, ∠ 16 g. ∠ 3, ∠ 11, ∠ 15 h. ∠ 15 5. III; 3 6. L; 50 7. VII; 7 8. XX; 20 9. MIII; 1003

Lesson 3.2Practice Level A

1. Corresponding Angles Postulate

2. Consecutive Interior Angles Theorem

3. Alternate Interior Angles Theorem

4. Alternate Exterior Angles Theorem

5. 1208; 1208 6.1208; 608 7. 1358; 458

8. 1408; 1408 9. 1058; 1058 10. 1108; 708

11. 80; 80 12. 90; 90 13. 60; 120 14. 65; 65

15. 50; 130 16. 90; 90 17. 40 18. 40 19. 35

20. 110 21. 15 22. 33 23. B 24. 135°

25. 45° 26. 140° 27. 130°

Practice Level B

1. 508; Corresponding Angles Postulate

2. 1358; Consecutive Interior Angles Theorem

3. 1308; Alternate Exterior Angles Theorem

4. 1238; Alternate Interior Angles Theorem

5. 608; 1208 6.1208; 1208 7. 1008; 1008

8. 568; 568 9. 1178; 638 10. 1088; 728

11. 110; 110 12. 90; 90 13. 95; 85 14. 75; 75

15. 106; 74 16. 90; 90 17. 65 18. 56 19. 48

20. 23 21. 40 22. 77 23. Given

24. Perpendicular lines form right angles.

25. Defi nition of right angle 26. Given

27. Corresponding Angles Postulate

28. Defi nition of congruent angles

29. Substitution Property of Equality

30. Defi nition of right angle

31. Perpendicular lines form right angles.

Practice Level C

1. 114°; Corresponding Angles Postulate

2. 68°; Alternate Interior Angles Theorem

3. 64°; Alternate Exterior Angles Theorem

4. 1138; 678 5. 468; 1348 6. 798; 1018 7. 43

8. 36 9. 18 10. 90 11. 23 12. 31 13. 19; 98 14. 68; 32 15. 6; 35 16. 32; 64 17. 83; 20

18. 21.5; 23.5 19. Given; Vertical Angles Theorem; Corresponding Angles Postulate; Transitive Property of Congruence

20. Given; Alternate Exterior Angles Theorem; Given; Corresponding Angles Postulate; Transitive Property of Congruence

Review for Mastery

1. Using the Vertical Angles Congruence Theorem, m∠ 8 5 658. By the Corresponding Angles Postulate, m∠ 4 5 658. Because ∠ 8 and ∠ 6 are corresponding angles, by the Corresponding Angles Postulate, you know that m∠ 6 5 658. 2. Using the Vertical Angles Congruence Theorem, m∠ 3 5 1158. By the Corresponding Angles Postulate, m∠ 7 5 1158. Because ∠ 3 and ∠ 1 are corresponding angles, by the Corresponding Angles Postulate, you know that m∠ 1 5 1158. 3. 68 4. 25 5. 12 6. 10

7. 10 8. 5 9. 12 10. 16

Problem Solving Workshop:Using Alternative Methods

1. 1158; by the Alternate Exterior Angles Theorem 2. 308; by the Consecutive Interior Angles Theorem

Challenge Practice

1. m∠ 1 5 428, m∠ 2 5 1388, m∠ 3 5 1388, m∠ 4 5 428, m∠ 5 5 1328, m∠ 6 5 488, m∠ 7 5 488, m∠ 8 5 1328, m∠ 9 5 908, m∠ 10 5 908, m∠ 11 5 908, m∠ 12 5 908, m∠ 13 5 1328, m∠ 14 5 488, m∠ 15 5 488, m∠ 16 5 1328, m∠ 17 5 428, m∠ 18 5 1388, m∠ 19 5 1388, m∠ 20 5 428

2. m∠ 1 5 358, m∠ 2 5 1458, m∠ 3 5 1118, m∠ 4 5 698, m∠ 5 5 1118, m∠ 6 5 698, m∠ 7 5 1458, m∠ 8 5 358, m∠ 9 5 698, m∠ 10 5 1118, m∠ 11 5 698, m∠ 12 5 1118, m∠ 13 5 768, m∠ 14 5 1048, m∠ 15 5 768, m∠ 16 5 1048, m∠ 17 5 1048, m∠ 18 5 768, m∠ 19 5 1048, m∠ 20 5 768

Lesson 3.1, continuedA

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A32GeometryChapter 3 Resource Book

3. m∠ 1 5 1008, m∠ 2 5 808, m∠ 3 5 808, m∠ 4 5 1008, m∠ 5 5 1008, m∠ 6 5 568, m∠ 7 5 248, m∠ 8 5 248, m∠ 9 5 568, m∠ 10 5 1008, m∠ 11 5 1568, m∠ 12 5 248, m∠ 13 5 248, m∠ 14 5 1568, m∠ 15 5 1248, m∠ 16 5 568, m∠ 17 5 1248, m∠ 18 5 568, m∠ 19 5 1008, m∠ 20 5 808, m∠ 21 5 1008, m∠ 22 5 808, m∠ 23 5 1568, m∠ 24 5 248, m∠ 25 5 248, m∠ 26 5 1568, m∠ 27 5 1008, m∠ 28 5 568, m∠ 29 5 248, m∠ 30 5 248, m∠ 31 5 568, m∠ 32 5 1008

4. Sample answer: A D

CB

Statements Reasons

1. }

AB i } DC , }

AD i }

BC 1. Given2. ∠ A and ∠ B and 2. Consecutive

∠ B and ∠ C Interior are supplementary. Angles Theorem

3. m∠ A 1 m∠ B 5 1808, 3. Defi nition of m∠ B 1 m∠ C 5 1808 supplementary angles

4. m∠ A 1 m∠ B 5 4. Substitution m∠ B 1 m∠ C Property of

Equality5. m∠ A 5 m∠ C 5. Subtraction Property of

Equality6. ∠ A > ∠ C 6. Defi nition of

congruent angles

5. x 5 67; Draw a line through the angle x8 that is parallel to both m and n. Then using the Alternate Interior Angles Theorem and the defi nition of supplementary angles, you can determine that x 5 35 1 32 5 67.

Lesson 3.3Practice Level A

1. yes; Corresponding Angles Converse

2. yes; Alternate Interior Angles Converse

3. yes; Alternate Exterior Angles Converse

4. yes; Corresponding Angles Converse

5. no 6. yes; Alternate Interior Angles Converse

7. 40 8. 30 9. 30 10. 14 11. 32 12. 95

13. C 14. m i n 15. p i q 16. p i q

17. neither 18. Given; Corresponding Angles Postulate; Given; Transitive Property of Congruence; Corresponding Angles Converse

19. Each lane is parallel to the one next to it, so l1 i l2, l2 i l3, and l3 i l4. Then l1 i l3 by the Transitive Property of Parallel Lines. By continuing this reasoning, l1 i l4. So, the fi rst lane is parallel to the last lane.

Practice Level B

1. yes; Consecutive Interior Angles Converse

2. yes; Alternate Interior Angles Converse

3. no 4. 40 5. 109 6.115 7. 22 8. 5 9. 80

10. congruent 11. supplementary 12. congruent

13. Each row is parallel to the one next to it, so r1 i r2, r2 i r3, and so on. Then r1 i r3 by the Transitive Property of Parallel Lines. By continuing this reasoning, r1 i r5. So, the fi rst row is parallel to the last row. 14. Given

15. Corresponding Angles Postulate 16. Given

17. Transitive Property of Equality 18. Alternate Exterior Angles Converse

19.

Statements Reasons

1. n i m 1. Given

2. ∠ 1 > ∠ 3 2. Alternate Interior Angles Theorem

3. ∠ 1 > ∠ 2 3. Given

4. ∠ 2 > ∠ 3 4. Transitive Property of Congruence

5. p i r 5. Alternate Interior Angles Converse

It is given that n i m. By the Alternate Interior Angles Theorem, ∠ 1 > ∠ 3. It is also given that ∠ 1 > ∠ 2. So by the Transitive Property of Con-gruence, ∠ 2 > ∠ 3. Therefore, by the Alternate Interior Angles Converse, p i r.

The two proofs use the same given information, theorems, and properties, but the two-column proof uses numbered statements and reasons, while the paragraph proof uses sentences and a conversational style.20. Corresponding Angles Converse

Lesson 3.2, continuedA

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