Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

- Geometry Section 2-5 1112 by Jimbo Lamb 4850 views
- Geometry Section 2-4 1112 by Jimbo Lamb 4965 views
- Geometry Section 2-3 1112 by Jimbo Lamb 4724 views
- Geometry Section 2-2 1112 by Jimbo Lamb 4398 views
- Geometry Section 2-1 1112 by Jimbo Lamb 5635 views
- Geometry Section 2-7 1112 by Jimbo Lamb 4048 views

4,382 views

Published on

Algebraic Proof

No Downloads

Total views

4,382

On SlideShare

0

From Embeds

0

Number of Embeds

3,222

Shares

0

Downloads

14

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Section 2-6 Algebraic Proof Wednesday, November 6, 13
- 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. Vocabulary 1. Algebraic Proof: 2. Two-column Proof: 3. Formal Proof: Wednesday, November 6, 13
- 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. 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. 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. Properties of Real Numbers Wednesday, November 6, 13
- 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. 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. 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. 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. 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. 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. 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. 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. 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. Example 1 Solve the equation. Write a justification for each step. 2(5−3a)− 4(a +7) = 92 Wednesday, November 6, 13
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Problem Set Wednesday, November 6, 13
- 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

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment