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4. Contents Lesson 5-1Bisectors, Medians, and Altitudes Lesson 5-2Inequalities and Triangles Lesson 5-3Indirect Proof Lesson 5-4The Triangle Inequality Lesson 5-5Inequalities Involving Two Triangles

5. Lesson 1 Contents Example 1Use Angle Bisectors Example 2Segment Measures Example 3Use a System of Equations to Find a Point

6. Example 1-1a Given: Prove:

7. Example 1-1a Proof: Statements Reasons 1. 1. Given 2. 2. Angle Sum Theorem 3. 3. Substitution 4. 4. Subtraction Property 5. 5. Definition of angle bisector 6. 6. Angle Sum Theorem 7. 7. Substitution 8. 8. Subtraction Property

8. Example 1-1b Prove: Given:.

9. Example 1-1b Proof: Statements. Reasons 1. Given 2. Angle Sum Theorem 3. Substitution 4. Subtraction Property 5. Definition of angle bisector 6. Angle Sum Theorem 7. Substitution 8. Subtraction Property 1. 2. 3. 4. 5. 6. 7. 8. 1. Given

10. Example 1-2a ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c.

11. Example 1-2a Find a. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 14.8 from each side. Divide each side by 4.

12. Example 1-2a Find b. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 6b from each side. Divide each side by 3. Subtract 6 from each side.

13. Example 1-2a Find c. Segment Addition Postulate Centroid Theorem Substitution Multiply each side by 3 and simplify. Subtract 30.4 from each side. Divide each side by 10. Answer:

14. ALGEBRA Points T, H, and G are the midpoints of respectively. Find w, x, and y. Example 1-2b Answer:

15. Example 1-3a COORDINATE GEOMETRY The vertices of  HIJ are H(1, 2), I(–3, –3), and J(–5, 1). Find the coordinates of the orthocenter of  HIJ.

16. Example 1-3a Find an equation of the altitude from The slope of so the slope of an altitude is Point-slope form Distributive Property Add 1 to each side.

17. Example 1-3a Point-slope form Distributive Property Subtract 3 from each side. Next, find an equation of the altitude from I to The slope of so the slope of an altitude is –6.

18. Example 1-3a Equation of altitude from J Multiply each side by 5. Add 105 to each side. Add 4x to each side. Divide each side by –26. Substitution, Then, solve a system of equations to find the point of intersection of the altitudes.

19. Example 1-3a Replace x with in one of the equations to find the y-coordinate. Multiply and simplify. Rename as improper fractions. The coordinates of the orthocenter of Answer:

20. Example 1-3b COORDINATE GEOMETRY The vertices of  ABC are A(–2, 2), B(4, 4), and C(1, –2). Find the coordinates of the orthocenter of  ABC. Answer: (0, 1)

21. End of Lesson 1

22. Lesson 2 Contents Example 1Compare Angle Measures Example 2Exterior Angles Example 3Side-Angle Relationships Example 4Angle-Side Relationships

23. Example 2-1a Determine which angle has the greatest measure. ExploreCompare the measure of  1 to the measures of  2,  3,  4, and  5. PlanUse properties and theorems of real numbers to compare the angle measures.

24. Example 2-1a Solve Compare m  3 to m  1. By the Exterior Angle Theorem, m  1 m  3 m  4. Since angle measures are positive numbers and from the definition of inequality, m  1 > m  3. Compare m  4 to m  1. By the Exterior Angle Theorem, m  1 m  3 m  4. By the definition of inequality, m  1 > m  4. Compare m  5 to m  1. Since all right angles are congruent,  4  5. By the definition of congruent angles, m  4 m  5. By substitution, m  1 > m  5.

25. By the Exterior Angle Theorem, m  5 m  2 m  3. By the definition of inequality, m  5 > m  2. Since we know that m  1 > m  5, by the Transitive Property, m  1 > m  2. Example 2-1a Compare m  2 to m  5. ExamineThe results on the previous slides show that m  1 > m  2, m  1 > m  3, m  1 > m  4, and m  1 > m  5. Therefore,  1 has the greatest measure. Answer:  1 has the greatest measure.

26. Example 2-1b Determine which angle has the greatest measure. Answer:  5 has the greatest measure.

27. Example 2-2a Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m  14. By the Exterior Angle Inequality Theorem, m  14 > m  4, m  14 > m  11, m  14 > m  2, and m  14 > m  4 + m  3. Since  11 and  9 are vertical angles, they have equal measure, so m  14 > m  9. m  9 > m  6 and m  9 > m  7, so m  14 > m  6 and m  14 > m  7. Answer: Thus, the measures of  4,  11,  9,  3,  2,  6, and  7 are all less than m  14.

28. Example 2-2b Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m  5. By the Exterior Angle Inequality Theorem, m  10 > m  5, and m  16 > m  10, so m  16 > m  5, m  17 > m  5 + m  6, m  15 > m  12, and m  12 > m  5, so m  15 > m  5. Answer: Thus, the measures of  10,  16,  12,  15 and  17 are all greater than m  5.

29. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a. all angles whose measures are less than m  4 b. all angles whose measures are greater than m  8 Example 2-2c Answer:  5,  2,  8,  7 Answer:  4,  9,  5

30. Example 2-3a Determine the relationship between the measures of  RSU and  SUR. Answer: The side opposite  RSU is longer than the side opposite  SUR, so m  RSU > m  SUR.

31. Example 2-3b Determine the relationship between the measures of  TSV and  STV. Answer: The side opposite  TSV is shorter than the side opposite  STV, so m  TSV < m  STV.

32. Example 2-3c Determine the relationship between the measures of  RSV and  RUV. Answer: m  RSV > m  RUV m  RSU > m  SUR m  USV > m  SUV m  RSU + m  USV > m  SUR + m  SUV m  RSV > m  RUV

33. Example 2-3d Determine the relationship between the measures of the given angles. a.  ABD,  DAB b.  AED,  EAD c.  EAB,  EDB Answer:  ABD >  DAB Answer:  AED >  EAD Answer:  EAB <  EDB

34. Example 2-4a HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

35. Example 2-4a Theorem 5.10 states that if one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Since  X is opposite the longest side it has the greatest measure. Answer: So, Ebony should tie the ends marked Y and Z.

36. Example 2-4b KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing? Answer:  A and  D

37. End of Lesson 2

38. Lesson 3 Contents Example 1Stating Conclusions Example 2Algebraic Proof Example 3Use Indirect Proof Example 4Geometry Proof

39. Example 3-1a Answer: is a perpendicular bisector. State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector.

40. Example 3-1b Answer: State the assumption you would make to start an indirect proof for the statement

41. Example 3-1c State the assumption you would make to start an indirect proof for the statement m  1 is less than or equal to m  2. Answer: m  1 > m  2 If m  1 m  2 is false, then m  1 > m  2.

42. Example 3-1d State the assumption you would make to start an indirect proof for the statement If B is the midpoint of and then is congruent to The conclusion of the conditional statement is is congruent to The negation of the conclusion is is not congruent to Answer: is not congruent to

43. State the assumption you would make to start an indirect proof of each statement. a. is not an altitude. b. Example 3-1e Answer: Answer: is an altitude.

44. Example 3-1e Answer:  MLH is not congruent to  PLH. d. If is an angle bisector of  MLP, then  MLH is congruent to  PLH. State the assumption you would make to start an indirect proof of each statement. c. m  ABC is greater than or equal to m  XYZ. Answer: m  ABC < m  XYZ

45. Example 3-2a Step 1Assume that. Given: Prove: Indirect Proof: Write an indirect proof.

46. Example 3-2b Step 2Substitute –2 for y in the equation This is a contradiction because the denominator cannot be 0. Substitution Multiply. Add.

47. Example 3-2c Step 3The assumption leads to a contradiction. Therefore, the assumption that must be false, which means that must be true.

48. Step 1Assume that Example 3-2d Indirect Proof: Given: Prove: Write an indirect proof.

49. Example 3-2b Step 2Substitute –3 for a in the inequality This is a contradiction because the denominator cannot be 0. Substitution Multiply. Add.

50. Example 3-2b Step 3The assumption leads to a contradiction. Therefore, the assumption that must be false, which means that must be true.

51. Example 3-3a CLASSES Marta signed up for three classes at a community college for a little under $156. There was an administration fee of $15, but the class costs varied. How can you show that at least one class cost less than $47? Answer: Given: Marta spent less than $156. Prove: At least one of the classes x cost less than $47. That is,

52. Example 3-3a Indirect Proof: Step 1Assume that none of the classes cost less than $47. Step 2 then the minimum total amount Marta spent is However, this is a contradiction since Marta spent less than $156. Step 3The assumption leads to a contradiction of a known fact. Therefore, the assumption that must be false. Thus, at least one of the classes cost less than $47.

53. Example 3-3b SHOPPING David bought four new sweaters for a little under $135. The tax was $7, but the sweater costs varied. How can you show that at least one of the sweaters cost less than $32? Answer: Given: David spent less than $135. Prove: At least one of the sweaters x cost less than $32. That is,

54. Example 3-3b Step 3The assumption leads to a contradiction of a known fact. Therefore, the assumption that must be false. Thus, at least one of the sweaters cost less than $32. Step 1Assume that none of the sweaters cost less than $32. Indirect Proof: Step 2 then the minimum total amount David spent is However, this is a contradiction since David spent less than $135.

55. Example 3-4a Given:  JKL with side lengths 5, 7, and 8 as shown. Prove: m  K < m  L Write an indirect proof.

56. Example 3-4a Step 3Since the assumption leads to a contradiction, the assumption must be false. Therefore, m  K < m  L. Indirect Proof: Step 1Assume that Step 2By angle-side relationships, By substitution,. This inequality is a false statement.

57. Example 3-4b Given:  ABC with side lengths 8, 10, and 12 as shown. Prove: m  C > m  A Write an indirect proof.

58. Example 3-4b Step 3Since the assumption leads to a contradiction, the assumption must be false. Therefore, m  C > m  A. Indirect Proof: Step 1Assume that Step 2By angle-side relationships, By substitution, This inequality is a false statement.

59. End of Lesson 3

60. Lesson 4 Contents Example 1Identify Sides of a Triangle Example 2Determine Possible Side Length Example 3Prove Theorem 5.12

61. Example 4-1a Answer: Because the sum of two measures is not greater than the length of the third side, the sides cannot form a triangle. Determine whether the measures and can be lengths of the sides of a triangle.

62. Example 4-1b Determine whether the measures 6.8, 7.2, and 5.1 can be lengths of the sides of a triangle. Check each inequality. Answer: All of the inequalities are true, so 6.8, 7.2, and 5.1 can be the lengths of the sides of a triangle.

63. Determine whether the given measures can be lengths of the sides of a triangle. a. 6, 9, 16 b. 14, 16, 27 Example 4-1c Answer: no Answer: yes

64. Example 4-2a A 7B 9C 11D 13 Multiple-Choice Test Item In andWhich measure cannot be PR?

65. Example 4-2a Read the Test Item You need to determine which value is not valid. Solve the Test Item Solve each inequality to determine the range of values for PR.

66. Example 4-2a Graph the inequalities on the same number line. The range of values that fit all three inequalities is

67. Example 4-2a Examine the answer choices. The only value that does not satisfy the compound inequality is 13 since 13 is greater than 12.4. Thus, the answer is choice D. Answer: D

68. Example 4-2b A 4B 9C 12D 16 Answer: D Multiple-Choice Test Item Which measure cannot be XZ?

69. Example 4-3a Prove: KJ < KH Given:line through point J Point K lies on t.

70. Example 4-3a Proof: StatementsReasons 1. 1. Given are right angles. 2. 2. Perpendicular lines form right angles. 3. 3. All right angles are congruent. 4. 4. Definition of congruent angles 5. 5. Exterior Angle Inequality Theorem 6. 6. Substitution 7.7. If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

71. Example 4-3b Prove: AB > AD Given:is an altitude in  ABC.

72. Example 4-3b Proof: Statements 1. 2. 3. 4. Reasons 1. Given 2. Definition of altitude 3. Perpendicular lines form right angles. 4. All right angles are congruent. is an altitude in are right angles.

73. Example 4-3b Proof: Statements 5. 6. 7. 8. Reasons 5. Definition of congruent angles 6. Exterior Angle Inequality Theorem 7. Substitution 8.If an angle of a triangle is greater than a second angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

74. End of Lesson 4

75. Lesson 5 Contents Example 1Use SAS Inequality in a Proof Example 2Prove Triangle Relationships Example 3Relationships Between Two Triangles Example 4Use Triangle Inequalities

76. Example 5-1a Write a two-column proof. Given: Prove:

77. Example 5-1a Proof: StatementsReasons 1.1. Given 2.2. Alternate interior angles are congruent. 3. 3. Substitution 4.4. Subtraction Property 5.5. Given 6.6. Reflexive Property 7.7. SAS Inequality

78. Example 5-1b Prove: AD < AB Given:m  1 < m  3 E is the midpoint of Write a two-column proof.

79. Example 5-1b Proof: Statements 1. 2. 3. 4. 5. 6. 7. Reasons 1. Given 2. Definition of midpoint 3. Reflexive Property 4. Given 5. Definition of vertical angles 6. Substitution 7. SAS Inequality E is the midpoint of

80. Example 5-2a Given: Prove:

81. Example 5-2a Proof: StatementsReasons 1.1. Given 2.2. Reflexive Property 3.3. Given 4.4. Given 5.5. Substitution 6. 6. SSS Inequality

82. Example 5-2b Given:X is the midpoint of  MCX is isosceles. CB > CM Prove:

83. Example 5-2b Proof: Statements 1. 2. 3. 4. 5. 6. 7. Reasons 1. Given 2. Definition of midpoint 3. Given 4. Definition of isosceles triangle 5. Given 6. Substitution 7. SSS Inequality X is the midpoint of  MCX is isosceles.

84. Example 5-3a Write an inequality comparing m  LDM and m  MDN using the information in the figure. The SSS Inequality allows us to conclude that Answer:

85. Example 5-3b Write an inequality finding the range of values containing a using the information in the figure. By the SSS Inequality,

86. Example 5-3b SSS Inequality Substitution Subtract 15 from each side. Divide each side by 9. Also, recall that the measure of any angle is always greater than 0. Subtract 15 from each side. Divide each side by 9.

87. Example 5-3b The two inequalities can be written as the compound inequality Answer:

88. Write an inequality using the information in the figure. a. b. Find the range of values containing n. Answer: Example 5-3c Answer: 6 < n < 25

89. Example 5-4a HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which foot can Nitan raise higher above the table? Assume both of Nitan’s legs have the same measurement, the SAS Inequality tells us that the height of the left foot opposite the 65° angle is higher than the height of his right foot opposite the 35° angle. This means that his left foot is raised higher. Answer: his left foot

90. Example 5-4b HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Megan can lift her right foot 18 inches from the table and her left foot 13 inches from the table. Which leg makes the greater angle with the table? Answer: her right leg

91. End of Lesson 5

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