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LESSON 3.3 Answers for the lesson “Prove Lines are Parallel” Skill Practice 1. Sample: 12. no 13. yes; Corresponding Angles n Converse 12 3 4 5 6 7 8 14. no 15. yes; Alternate Exterior Angles m Converse 16. Sample answer: 1 and 8, 2 and 7 4 1 3 2 50 Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 2. Given two lines cut by a transversal, alternate interior angles are congruent if and only if the lines are parallel; given two lines cut by a transversal, alternate exterior angles are congruent if and only if the lines are parallel; given two lines cut by a transversal, consecutive interior angles are supplementary if and only if the lines are parallel. 3. 40 4. 60 5. 15 6. 90 7. 60 8. 20 9. The student believes that x 5 y but there is no indication that they are equal. 8 5 7 6 m3 5 508, Vertical Angles Congruence Theorem; m4 5 1308, Linear Pair Postulate; m2 5 1308, Vertical Angles Congruence Theorem; m8 5 1308, Alternate Interior Angles Theorem; m6 5 1308, Vertical Angles Congruence Theorem; m5 5 508, Linear Pair Postulate; m7 5 508, Vertical Angles Congruence Theorem 10. yes; Alternate Interior Angles Converse 11. yes; Alternate Exterior Angles Converse Geometry Answer Transparencies for Checking Homework 68 17. a. mDCG 5 1158, 24. 1 angle. Sample answer: Using mCGH 5 658 the Vertical Angles Congruence Theorem, the Linear Pair Postulate, and the Alternate Interior Angles Theorem the other angle measures can be found. b. They are consecutive interior angles and they are supplementary. c. yes; Consecutive Interior 25. Sample answer: 1 > 4 Angles Converse 18. a. Sample: p 1 2 q Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. b. Given: 1 and 2 are supplementary, Prove: p i q 19. yes; Consecutive Interior Angles Converse 20. yes; Alternate Exterior Angles Converse 21. no 22. The student assumed the congruent angles were alternate interior angles ‹]› ‹]› between AD and BC . By the Alternate Interior ‹]› ‹]› Angles Converse; AB i DC . 23. D therefore 4 and 7 are supplementary. Lines j and k are parallel by the Consecutive Interior Angles Converse. ]› ]› ]› 26. EA i HC ; EB is not parallel ]›, GHC >HEA, GHD to HD is not congruent to HEB. 27. a. 1 line b. an infinite number of lines c. 1 plane 28. a. 54 b. 47.5 c. No, Sample answer: For p to be parallel to q, x 5 54, then y 5 63 because of the linear pair formed, but in order for r and s to be parallel, y must equal 47.5. Problem Solving 29. Alternate Interior Angles Converse 30. Corresponding Angles Converse Geometry Answer Transparencies for Checking Homework 69 31. 3. Substitution 35. Statements (Reasons) 4. Definition of supplementary angles 5. Consecutive Interior Angles Converse 32. Alternate Exterior Angles Converse Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 33. Yes. Sample answer: E 20th is parallel to E 19th by the Corresponding Angles Converse. E 19th is parallel to E 18th by the Alternate Exterior Angles Converse. E 18th is parallel to E 17th by the Alternate Interior Angles Converse. They are all parallel by the Transitive Property of Parallel Lines. 34. Statements (Reasons) 1. 1 > 2, 3 > 4 (Given) 2. 2 > 3 (Vertical Angles Congruence Theorem) 3. 1 > 4 (Transitive Property of Angle Congruence) 4. } AB i } CD (Alternate Interior Angles Converse) 1. a i b, 2 > 3 (Given) 2. 2 and 4 are supplementary. (Consecutive Interior Angles Theorem) 3. 3 and 4 are supplementary. (Substitution) 4. c i d (Consecutive Interior Angles Converse) 36. Statements (Reasons) 1. 2 > 7 (Given) 2. 7 > 6 (Vertical Angles Congruence Theorem) 3. 2 > 6 (Transitive Property of Congruence) 4. m i n (Corresponding Angles Converse) 37. You are given that 3 and 5 are supplementary. By the Linear Pair Postulate, 5 and 6 are also supplementary. So 3 > 6 by the Congruent Supplements Theorem. By the Alternate Interior Angles Converse, m i n. Geometry Answer Transparencies for Checking Homework 70 38. a. 1 p q 2 4 r 40–44. Sample answers are given. 40. Consecutive Interior Angles t 3 Converse 41. Corresponding Angles Converse 42. Corresponding Angles Converse b. Given: p i q and q i r, 43. Vertical Angles Congruence Prove: p i r Theorem followed by the Corresponding Angles Converse c. Statements (Reasons) 1. p i q andq i r (Given) 2. 1 > 2 (Alternate Interior Angles Theorem) Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 3. 2 > 3 (Vertical Angles Congruence Theorem) 4. 3 > 4 (Alternate Interior Angles Theorem) 5. 1 > 4 (Transitive Property of Angle Congruence) 6. p i r (Alternate Interior Angles Converse) 44. Consecutive Interior Angles Converse 45. a. Q A 1 C 2 3 4 P D B n b. If two parallel lines are cut by a transversal, the angle bisectors of alternate interior angle pairs are parallel. 39. a. Sample answer: Corresponding Angles Converse Theorem b. Slide the triangle along a fixed horizontal line and use the edge that forms the 908 angle to draw vertical lines. Geometry Answer Transparencies for Checking Homework 71 Mixed Review of Problem Solving 45. b. (cont.) Statements (Reasons) 1. * i n (Given) 2. AQP > BPQ (Alternate Interior Angles Theorem) 3. m1 1 2 5 mAQP, m4 1 3 5 mBPQ (Angle Addition Postulate) Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. 4. m1 5 m2, m3 5 m4 (Definition of angle bisector) 1. a. Sample answer: q and p, k and m b. Sample answer: q and m c. Sample answer: n and m, n and k 2. a. 2: supplementary, 3: supplementary, 4: vertical, 5: corresponding, 6: supplementary, 7: alternate exterior, 8: supplementary 5. m2 1 m2 5 mAQP, m3 1 m3 5 mBPQ (Subtitution) 3. 538; Alternate Exterior Angles 6. 2m2 5 2m3 (Transitive Property of Equality) 4. yes; Alternate Interior Angles 7. m2 5 m3 (Division Property of Equality) 5. a. 11 8. 2 > 3 (Definition of Congruent Angles) ]› i PD ]› 9. QC (Alternate Interior Angles Converse) b. 2, 6, 8 Theorem Converse b. 238; Transitive Property of Parallel Lines and Alternate Interior Angles Theorem Geometry Answer Transparencies for Checking Homework 72 6. 1508; 1 5 0 7. 92, supplementary to 888; Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved. x 5 116, c i d by the Alternate Interior Angles Converse followed by the Consecutive Interior Angles Theorem. Geometry Answer Transparencies for Checking Homework 73