LESSON Challenge Practice 3.2 For use with pages 157–164 Exterior Angles Theorem ... by the Corresponding Angles Postulate, you know that m∠ 1 5 1158 ... by the Consecutive Interior
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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
∠ 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.
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
NS
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