Inequalities in Triangles

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Geometry 5-5 Inequalities in Triangles Within a triangle: – the biggest side is opposite the biggest angle. – the smallest side is opposite the smallest.

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Presentation transcript:

Inequalities in Triangles GEOMETRY LESSON 5-5 1. 3 2 because they are vertical and m 1 > m 3 by Corollary to the Ext. Thm. So, m 1 > m 2 by subst. 2. An ext. of a is larger than either remote int. . 3. m 1 > m 4 by Corollary to the Ext. Thm. and 4 2 because if Pages 276-279 Exercises 3. (continued) lines, then alt. int. are . 4. M, L, K 5. D, C, E 6. G, H, I 7. A, B, C 8. E, F, D 9. Z, X, Y 10. MN, ON, MO 11. FH, GF, GH 12. TU, UV, TV 13. AC, AB, CB 14. EF, DE, DF 15. ZY, XZ, XY 16. No; 2 + 3 6. 17. Yes; 11 + 12 > 15; 12 + 15 > 11; 11 + 15 > 12 s > 5-5

Inequalities in Triangles GEOMETRY LESSON 5-5 23. 11 < s < 21 24. 0 < s < 12 25. 15 < s < 41 26. 3 < s < 11 27. 15 < s < 55 28. Answers may vary. Sample: If Y is the distance between Wichita and Topeka, then 20 < Y < 200. 29. Let the distance between the peaks be d and the distances from the hiker to each of the peaks be a and b. Then d + a > b and d + b > a. Thus, d > b – a and d > a –b. 18. No; 8 + 10 19. 19. Yes; 1 + 15 > 15; 15 + 15 > 11. 20. Yes; 2 + 9 > 10; 9 + 10 > 2; 2 + 10 > 9. 21. No; 4 + 5 9. 22. 4 < s < 20 > 5-5

Inequalities in Triangles GEOMETRY LESSON 5-5 30. a. b. the third side of the 1st is longer than the third side of the 2nd . c. See diagram in part (a). d. The included of the first is greater than the included of the second . 31. Answers may vary. Sample: The shortcut across the grass is shorter than the sum of the two paths. 32. AB 33. a. m OTY b. m 3 c. Base of an isosc. are . d. Add. Post. e. Comparison Prop. of Ineq. f. Subst. (step 2) g. An ext. of a is greater than either remote int. s . 5-5

Inequalities in Triangles GEOMETRY LESSON 5-5 33. (continued) h. Trans. Prop. of Ineq. 34. RS 35. CD 36. XY 37. 38. (2, 4), (2, 5), (2, 6), (3, 3), (3, 4), (3, 6), (3, 7), (4, 3), (4, 4), (4, 5), (4, 6), (4, 7), (4, 8) 1 2 39. 40. CD = AC was given so ACD is isosc. By def. of isos. . This means m D = m CAD. Then m DAB > m CAD by the Comparison Prop. of Ineq. So by subst., m DAB > m D and by Thm. 5-11 DB > AB. Since DC + CB = DB, by subst. DC + CB > AB. Using subst. again, AC + CB > AB. 41. T is the largest in PTA. Thus PA > PT because the longest side of a is opp. The largest . 5 18 5-5

Inequalities in Triangles GEOMETRY LESSON 5-5 42. D 43. A 44. B 45. C 46. B 47. [2] a. Since m A > m C > m B, the sides opp. them are related in the same way: BC > AB > AC. Of AB and AC, AB is longer than AC. Since 9 in. > 5 in., AB = 9 in. and AC = 5 in. b. BC is the longest side, so 9 < BC < 14. The possible whole number measures for BC are 10 in., 11 in., 12 in., and 13 in. [1] part (a) OR part (b) incorrect 48. m A > m B 49. m X ≤ m B 50. The angle is not a right angle. 5-5

Inequalities in Triangles GEOMETRY LESSON 5-5 51. The triangle is obtuse. 52. 90 53. 35 54. 145 55. 55 56. 8.0 ft2 57. 962.1 mm2 58. 0.8 m2 59. 314.2 mi2 5-5