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# Geometry Section 2-6

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Algebraic Proof

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### Geometry Section 2-6

1. 1. Section 2-6 Algebraic Proof Wednesday, November 6, 13
2. 2. Essential Questions How do you use Algebra to write two-column proofs? How do you use properties of equality to write geometric proofs? Wednesday, November 6, 13
3. 3. Vocabulary 1. Algebraic Proof: 2. Two-column Proof: 3. Formal Proof: Wednesday, November 6, 13
4. 4. Vocabulary 1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps. 2. Two-column Proof: 3. Formal Proof: Wednesday, November 6, 13
5. 5. Vocabulary 1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps. 2. Two-column Proof: A format for a proof where one column provides statements of what is being done, and the second is to justify each statement 3. Formal Proof: Wednesday, November 6, 13
6. 6. Vocabulary 1. Algebraic Proof: When a series of algebraic steps are used to solve problems and justify steps. 2. Two-column Proof: A format for a proof where one column provides statements of what is being done, and the second is to justify each statement 3. Formal Proof: Another name for a two-column proof Wednesday, November 6, 13
7. 7. Properties of Real Numbers Wednesday, November 6, 13
8. 8. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
9. 9. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
10. 10. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
11. 11. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
12. 12. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
13. 13. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
14. 14. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
15. 15. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
16. 16. Properties of Real Numbers Addition Property of Equality: If a = b, then a + c = b + c Subtraction Property of Equality: If a = b, then a − c = b − c Multiplication Property of Equality: If a = b, then a × c = b × c Division Property of Equality: If a = b and c ≠ 0, then a ÷ c = b ÷ c Reflexive Property of Equality: a=a Symmetric Property of Equality: If a = b, then b = a Transitive Property of Equality: If a = b and b = c, then a = c If a = b, then a may be replaced by b Substitution Property of Equality: in any equation/expression Distributive Property: Wednesday, November 6, 13 a(b + c) = ab + ac
17. 17. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13
18. 18. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13 Given
19. 19. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 10 − 6a − 4a − 28 = 92 Wednesday, November 6, 13 Given
20. 20. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property Wednesday, November 6, 13
21. 21. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 Wednesday, November 6, 13
22. 22. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 Substitution Property Wednesday, November 6, 13
23. 23. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property Wednesday, November 6, 13
24. 24. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property Wednesday, November 6, 13 Addition Property
25. 25. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 Wednesday, November 6, 13 Addition Property
26. 26. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 Substitution Property Wednesday, November 6, 13 Addition Property
27. 27. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 −10 −10 Substitution Property Wednesday, November 6, 13 Addition Property
28. 28. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 −10 −10 Substitution Property Division Property Wednesday, November 6, 13 Addition Property
29. 29. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 −10 −10 Substitution Property Division Property a = −11 Wednesday, November 6, 13 Addition Property
30. 30. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Given 10 − 6a − 4a − 28 = 92 Distributive Property −10a −18 = 92 +18 +18 Substitution Property −10 a = 110 −10 −10 Substitution Property Division Property a = −11 Substitution Property Wednesday, November 6, 13 Addition Property
31. 31. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Wednesday, November 6, 13
32. 32. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements Wednesday, November 6, 13 Reasons
33. 33. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. Wednesday, November 6, 13 Reasons
34. 34. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 Wednesday, November 6, 13 Reasons
35. 35. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 Wednesday, November 6, 13 Reasons Given
36. 36. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. Wednesday, November 6, 13 Reasons Given
37. 37. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t Wednesday, November 6, 13 Reasons Given
38. 38. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t Wednesday, November 6, 13 Reasons Given Subtraction Property
39. 39. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t 3. Wednesday, November 6, 13 Reasons Given Subtraction Property
40. 40. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t d−5 =t 3. 20 Wednesday, November 6, 13 Reasons Given Subtraction Property
41. 41. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t d−5 =t 3. 20 Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
42. 42. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t d−5 =t 3. 20 4. Wednesday, November 6, 13 Reasons Given Subtraction Property Division Property
43. 43. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t d−5 =t 3. 20 4. Wednesday, November 6, 13 d −5 t= 20 Reasons Given Subtraction Property Division Property
44. 44. Example 2 If the distance d an object travels is given by d = 20t +5, d −5 the time t that the object travels is given by t = . 20 Write a two-column proof to verify this conjecture. Statements 1. d = 20t +5 2. d −5 = 20t d−5 =t 3. 20 4. Wednesday, November 6, 13 d −5 t= 20 Reasons Given Subtraction Property Division Property Symmetric Property
45. 45. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Wednesday, November 6, 13
46. 46. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements Wednesday, November 6, 13 Reasons
47. 47. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. Wednesday, November 6, 13 Reasons
48. 48. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Wednesday, November 6, 13 Reasons
49. 49. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Wednesday, November 6, 13 Reasons Given
50. 50. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° 2. Wednesday, November 6, 13 Reasons Given
51. 51. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° 2. Wednesday, November 6, 13 m∠A = m∠B Reasons Given
52. 52. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° 2. Wednesday, November 6, 13 m∠A = m∠B Reasons Given Def. of ≅ angles
53. 53. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° 2. 3. Wednesday, November 6, 13 m∠A = m∠B Reasons Given Def. of ≅ angles
54. 54. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° 2. m∠A = m∠B 3. m∠A = 2m∠C Wednesday, November 6, 13 Reasons Given Def. of ≅ angles
55. 55. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property Wednesday, November 6, 13
56. 56. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. Wednesday, November 6, 13
57. 57. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. m∠A = 2(45°) Wednesday, November 6, 13
58. 58. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. m∠A = 2(45°) Substitution Property Wednesday, November 6, 13
59. 59. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. m∠A = 2(45°) Substitution Property 5. Wednesday, November 6, 13
60. 60. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. m∠A = 2(45°) Substitution Property 5. m∠A = 90° Wednesday, November 6, 13
61. 61. Example 3 If ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45°, then m∠A = 90°. Write a two-column proof to verify this conjecture. Statements 1. ∠A ≅ ∠B, m∠B = 2m∠C, and m∠C = 45° Reasons Given 2. m∠A = m∠B Def. of ≅ angles 3. m∠A = 2m∠C Transitive Property 4. m∠A = 2(45°) Substitution Property 5. m∠A = 90° Substitution Property Wednesday, November 6, 13
62. 62. Problem Set Wednesday, November 6, 13
63. 63. Problem Set p. 137 #1-27 odd, 36 “The greatest challenge to any thinker is stating the problem in a way that will allow a solution.” - Bertrand Russell Wednesday, November 6, 13