Proving Lines Parallel GEOMETRY LESSON 3-2 Pages 125-129 Exercises 1. BE || CG; Conv. of Corr. Post. 2. CA || HR; Conv. of Corr. Post. 3. JO || LM; if two lines and a transversal form same-side int. that are suppl., then the lines are ||. s 4. a || b; if two lines and a transversal form same-side int. that are suppl., then the lines are ||. 5. a || b; if two lines and a transversal form same-side int. that are suppl., then the lines are ||. 6. none 7. none 8. a || b; Conv. of Corr. Post. 9. none 10. a || b; Conv. of Alt. Int. Thm. 11. || m; Conv. of Corr. Post. 12. none 13. a || b; Conv. of Corr. Post. 14. none 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 24. When the frame is put together, each of the frame is a right . Two right are suppl. By the Conv. of the Same-Side Int. Thm., opp. sides of the frame are ||. 25. The corr. are , so the lines are || by the Conv. of Corr. Post. 26. a. Corr. b–c. 1, 3 (any order) d. Conv. of Corr. s 15. || m; Conv. of Alt. Int. Thm. 16. a. Def. of b. Given c. All right are . d. Conv. of Corr. Post. 17. a. 1 b. 1 c. 2 d. 3 e. Conv. of Corr. s 18. 30 19. 50 20. 59 21. 31 22. 5 23. 20 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 27. 10; m 1 = m 2 = 70 28. 5; m 1 = m 2 = 50 29. 2.5; m 1 = m 2 = 30 30. 1.25; m 1 = m 2 = 10 31. The corr. he draws are . 32. PL || NA and PN || LA by Conv. of Same-Side Int. Thm. s 33. PL || NA by Conv. of Same-Side Int. Thm. 34. none 35. PN || LA by Conv. of Same-Side Int. Thm. 36. Answers may vary. Sample: In the diagram, AB BH and AB BD, but BH || BD. They intersect. 37. Reflexive: a || a; false; any line intersects itself. Symmetric: If a || b, then b || a; true; b and a are coplanar and do not intersect. Transitive: In general, if a || b, and b || c, then a || c; true; however, when a || b, and b || a, it does not follow that a || a. s 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 38. Reflexive: a a; false; lines are two lines that intersect to form right . Symmetric: If a b, then b a; true; b and a intersect to form right . Transitive: If a b, and b c, then a c; false; in a plane, two lines to the same line are ||. 39. The corr. are , and the oars are || by the Conv. of Corr. Post. 40. Answers may vary. Sample: 3 9; j || k by Conv. of the Alt. Int. Thm. 41. Answers may vary. Sample: 3 9; j || k by Conv. of the Alt. Int. Thm. and || m by Conv. of Same-Side Int. Thm. s 42. Answers may vary. Sample: 3 11; || m by Conv. of the Alt. Int. Thm. and j || k by Conv. of Corr. Post. 43. Answers may vary. Sample: 3 and 12 are suppl.; j || k by the Conv. of Corr. Post. 44. Vert. Thm. and Conv. of Corr. Post. 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 45. It is given that || m, so 4 8 by Corr. Post. It is also given that 12 8, so 4 12 by Trans. Prop. of . So, j || k by the Conv. of Corr. Post. s 46. 47. 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 48. 49. 50. a. Answers may vary. Sample: b. Given: a || b with transversal e, c bisects AOB, d bisects AXZ. c. Prove: c || d 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 50. (continued) d. To prove that c || d, show that 1 3. 1 3 if AOB OXZ. AOB OXZ by the Corr. Post. s e. 1. a || b (Given) 2. AOB AXZ (Corr. Post.) 3. m AOB = m AXZ (Def. of ) 4. m AOB = m 1 + m 2; m AXZ = m 3 + m 4 ( Add. Post.) 5. c bisects AOB; d bisects AXZ. (Given) 6. m 1 = m 2; m 3 = m 4 (Def. of bisector) 7. m 1 + m 2 = m 3 + m 4 (Trans. Prop. of ) 8. m 1 + m 1 = m 3 + m 3 (Subst.) 9. 2m 1 = 2m 3 (Add. Prop.) 10. m 1 = m 3 (Div. Prop.) 11. c || d (Conv. of Corr. Post.) s 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 54. (continued) b. x + 21 = 2x so x = 21. Lines c and d are not || because x cannot = both 21 and 23 (OR equivalent explanation). [1] incorrect equations OR incorrect solutions 55. [4] a. 51. C 52. F 53. B 54.  [2] a. 136 + (x + 21) = 180 so x = 23 (OR equivalent equation resulting in x = 23).     55. (continued) b. m 1 + m 3 = 180. If 2x – 38 + 6x + 18 = 180, then x = 25. The measures are 2x – 38 = 12 and 25, but 12 25. So a can’t be || to b. [3]  appropriate methods, but with one computational error [2]  incorrect diagram solved correctly OR correct diagram solved incorrectly = / 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 55. (continued)  [1] correct answer (lines a and b are not ||), without work shown 56. m 1 = 70 since it is a suppl. of the 110° . m 2 = 110 since same-side int. are suppl. 57. m 1 = 66 because alt. int. are . m 2 = 180 – 94 = 86 because same-side int. are suppl. 58. If you are west of the Mississippi River, then you are in Nebraska. Original statement is true, converse is false. 59. If a circle has a radius of 4 cm, then it has a diameter of 8 cm. Both are true. 60. If same-side int. are suppl., then a line intersects a pair of || lines. Both are true. 61. If you form the past tense of a verb, then you add ed to the verb. Original statement is false, converse is false. 62. If there are clouds in the sky, then it is raining. Original statement is true, converse is false. 63. 201.1 in.2 64. 28.3 cm2 s s s s 3-2

Proving Lines Parallel GEOMETRY LESSON 3-2 65. 63.6 ft2 66. 78.5 in.2 67. 6.2 m2 68. 4.5 m2 69. 17.7 ft2 70. 0.3 m2 3-2