1. Concept

3. Example 1 Identifying Postulates ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. A. Points F and G lie in plane Q and on line m. Line m lies entirely in plane Q. Answer: Points F and G lie on line m, and the line lies in plane Q. Postulate 2.5, which states that if two points lie in a plane, the entire line containing the points lies in that plane, shows that this is true.

4. Example 1 ARCHITECTURE Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the statement is true. B. Points A and C determine a line. Answer: Points A and C lie along an edge, the line that they determine. Postulate 2.1, which says through any two points there is exactly one line, shows that this is true. Identifying Postulates

5. Example 1 A.Through any two points there is exactly one line. B.A line contains at least two points. C.A plane contains at least three noncollinear points. D.A plane contains at least two noncollinear points. ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. A. Plane P contains points E, B, and G.

6. Example 1 A.Through any two points there is exactly one line. B.A line contains at least two points. C.If two lines intersect, then their intersection is exactly one point. D.If two planes intersect, then their intersection is a line. ARCHITECTURE Refer to the picture. State the postulate that can be used to show the statement is true. B. Line AB and line BC intersect at point B.

7. Example 2 Analyze Statements Using Postulates Answer:Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. A. Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G.

8. Example 2 Analyze Statements Using Postulates Answer: Never; noncollinear points do not lie on the same line by definition. B. Determine whether the following statement is always, sometimes, or never true. Explain. contains three noncollinear points.

9. Example 2 A. Determine whether the statement is always, sometimes, or never true. Plane A and plane B intersect in exactly one point. A.always B.sometimes C.never

10. Example 2 B. Determine whether the statement is always, sometimes, or never true. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. A.always B.sometimes C.never

11. Concept

12. Example 3 Write a Paragraph Proof Given: Prove:ACD is a plane. Proof: and must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on. Points A, C, and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line.

13. Example 3

14. Proof: Example 3 ?

15. A.Definition of midpoint B.Segment Addition Postulate C.Definition of congruent segments D.Substitution

16. Concept

18. Example 1 Justify Each Step When Solving an Equation Solve 2(5 – 3a) – 4(a + 7) = 92. Write a justification for each step. Algebraic StepsProperties 2(5 – 3a) – 4(a + 7)=92Original equation 10 – 6a – 4a – 28=92Distributive Property –18 – 10a=92Substitution Property –18 – 10a + 18 =92 + 18Addition Property

19. Example 1 Justify Each Step When Solving an Equation Answer: a = –11 –10a=110Substitution Property Division Property a=–11Substitution Property

20. Example 1 A.a = 12 B.a = –37 C.a = –7 D.a = 7 Solve –3(a + 3) + 5(3 – a) = –50.

21. Example 2 Write an Algebraic Proof Begin by stating what is given and what you are to prove.

22. Example 2 Write an Algebraic Proof 2. d – 5 = 20t2. Addition Property of Equality StatementsReasons Proof: 1. Given 1. d = 20t + 5 4.4. Symmetric Property of Equality 3.3. Division Property of Equality = t

23. Example 2 Which of the following statements would complete the proof of this conjecture? If the formula for the area of a trapezoid is, then the height h of the trapezoid is given by.

24. Example 2 StatementsReasons Proof: 3.3. Division Property of Equality 4.4. Symmetric Property of Equality 1. Given 1. 2._____________2. Multiplication Property of Equality ?

25. Example 2 A.2A = (b 1 + b 2 )h B. C. D.

26. Example 3 Write a Geometric Proof If  A  B, m  B = 2m  C, and m  C = 45, then m  A = 90. Write a two-column proof to verify this conjecture.

27. Example 3 5. m  A = 90 5. Substitution StatementsReasons Proof: 4. Substitution 4. m  A = 2(45) Write a Geometric Proof 2. m  A = m  B 2. Definition of angles 1. Given 1.  A  B; m  B = 2m  C; m  C = 45 3. Transitive Property of Equality 3. m  A = 2m  C

28. Example 3

29. StatementsReasons Proof: 1. Given 1. 2.2. _______________ ? 3. AB = RS3. Definition of congruent segments 4. AB = 124. Given 5. RS = 125. Substitution

30. Example 3 A. Reflexive Property of Equality B. Symmetric Property of Equality C.Transitive Property of Equality D. Substitution Property of Equality

31. Concept

33. Example 1 Use the Segment Addition Postulate 2. Definition of congruent segments AB = CD 2. 3. Reflexive Property of Equality BC = BC 3. 4. Segment Addition Postulate AB + BC = AC 4. Proof: StatementsReasons 1. 1. Given AB  CD ___

34. Example 1 6. Segment Addition Postulate CD + BC = BD 6. 7. Transitive Property of Equality AC = BD 7. Proof: StatementsReasons 5. Substitution Property of Equality 5. CD + BC = AC Use the Segment Addition Postulate 8. Definition of congruent segments 8. AC  BD ___

35. Example 1 Prove the following. Given:AC = AB AB = BX CY = XD Prove:AY = BD

36. Example 1 1. Given AC = AB, AB = BX 1. 2. Transitive Property AC = BX 2. 3. Given CY = XD 3. 4. Addition PropertyAC + CY = BX + XD4. AY = BD 6. Substitution6. Proof: StatementsReasons Which reason correctly completes the proof? 5. ________________ AC + CY = AY; BX + XD = BD 5. ?

37. Example 1 A.Addition Property B.Substitution C.Definition of congruent segments D.Segment Addition Postulate

38. Concept

40. Example 2 Proof Using Segment Congruence BADGE Jamie is designing a badge for her club. The length of the top edge of the badge is equal to the length of the left edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge. Given: Prove:

41. Example 2 Proof Using Segment Congruence 5. Substitution 5. Proof: Statements Reasons 1. Given 1. 2. Definition of congruent segments 2. 3. Given 3. 4. Transitive Property 4. YZ ___

42. Example 2 Prove the following. Given: Prove:

43. Example 2 Which choice correctly completes the proof? Proof: Statements Reasons 1. Given 1. 2. Transitive Property 2. 3. Given 3. 4. Transitive Property 4. 5. _______________ 5. ?

44. Example 2 A.Substitution B.Symmetric Property C.Segment Addition Postulate D.Reflexive Property

45. Concept

47. Example 1 Use the Angle Addition Postulate CONSTRUCTION Using a protractor, a construction worker measures that the angle a beam makes with a ceiling is 42°. What is the measure of the angle the beam makes with the wall? The ceiling and the wall make a 90  angle. Let  1 be the angle between the beam and the ceiling. Let  2 be the angle between the beam and the wall. m  1 + m  2= 90Angle Addition Postulate 42 + m  2= 90m  1 = 42 42 – 42 + m  2= 90 – 42Subtraction Property of Equality m  2= 48Substitution

48. Example 1 Use the Angle Addition Postulate Answer:The beam makes a 48° angle with the wall.

49. Example 1 A.32 B.94 C.104 D.116 Find m  1 if m  2 = 58 and m  JKL = 162.

50. Concept

51. Example 2 Use Supplement or Complement TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? AnalyzeMake a sketch of the situation. The time is 4 o’clock and the second hand bisects the angle between the hour and minute hands.

52. 60 + 60 = 120 Example 2 Use Supplement or Complement Formulate Use the Angle Addition Postulate and the definition of angle bisector. Determine Since the angles are congruent by the definition of angle bisector, each angle is 60°. Answer:Both angles are 60°. JustifyUse the Angle Addition Postulate to check your answer. m  1 + m  2 = 120 120 = 120

53. Example 2 Use Supplement or Complement Evaluate The sketch we drew helps us determine an appropriate solution method. Our answer is reasonable.

54. Example 2 A.20 B.30 C.40 D.50 QUILTING The diagram shows one square for a particular quilt pattern. If m  BAC = m  DAE = 20, and  BAE is a right angle, find m  CAD.

55. Concept

59. Example 3 Proofs Using Congruent Comp. or Suppl. Theorems Given: Prove:

60. Example 3 Proofs Using Congruent Comp. or Suppl. Theorems 1. Given 1.m  3 + m  1 = 180;  1 and  4 form a linear pair. 4.  s suppl. to same  are . 4.  3   4 Proof: StatementsReasons 2. Linear pairs are supplementary. 2.  1 and  4 are supplementary. 3. Definition of supplementary angles 3.  3 and  1 are supplementary.

61. Example 3 In the figure,  NYR and  RYA form a linear pair,  AXY and  AXZ form a linear pair, and  RYA and  AXZ are congruent. Prove that  NYR and  AXY are congruent.

62. Example 3 Which choice correctly completes the proof? Proof: StatementsReasons 1. Given 1.  NYR and  RYA,  AXY and  AXZ form linear pairs. 2.If two  s form a linear pair, then they are suppl.  s. 2.  NYR and  RYA are supplementary.  AXY and  AXZ are supplementary. 3. Given 3.  RYA   AXZ 4.  NYR   AXY 4. ____________ ?

63. Example 3 A.Substitution B.Definition of linear pair C.  s supp. to the same  or to   s are . D.Definition of supplementary  s

64. Concept

65. Example 4 Use Vertical Angles If  1 and  2 are vertical angles and m  1 = d – 32 and m  2 = 175 – 2d, find m  1 and m  2. Justify each step. 1.Given 1.  1 and  2 are vertical  s. 2.Vertical Angles Theorem 3. Definition of congruent angles 4. Substitution 2.  1   2 3. m  1 = m  2 4. d – 32 = 175 – 2d StatementsReasons Proof:

66. Example 4 Use Vertical Angles 5.Addition Property 5. 3d – 32 = 175 6.Addition Property 7. Division Property 6. 3d = 207 7. d = 69 StatementsReasons Answer: m  1 = 37 and m  2 = 37 m  1=d – 32m  2 = 175 – 2d =69 – 32 or 37= 175 – 2(69) or 37

67. Example 4 A. B. C. D.

68. Concept