Geometry Section 5-4 1112
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Geometry Section 5-4 1112

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Indirect Proof

Indirect Proof

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Geometry Section 5-4 1112 Presentation Transcript

  • 1. Section 5-4 Indirect ProofTuesday, March 13, 2012
  • 2. Essential Questions How do you write indirect algebraic proofs? How do you write indirect geometric proofs?Tuesday, March 13, 2012
  • 3. Vocabulary 1. Indirect Reasoning: 2. Indirect Proof: 3. Proof by Contradiction:Tuesday, March 13, 2012
  • 4. Vocabulary 1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction 2. Indirect Proof: 3. Proof by Contradiction:Tuesday, March 13, 2012
  • 5. Vocabulary 1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction 2. Indirect Proof: Prove a statement to be true by assuming that which you are trying to prove to be false, then show that is leads to a contradiction 3. Proof by Contradiction:Tuesday, March 13, 2012
  • 6. Vocabulary 1. Indirect Reasoning: Assuming that the conclusion of a conditional is false, then showing this assumption leads to a contradiction 2. Indirect Proof: Prove a statement to be true by assuming that which you are trying to prove to be false, then show that is leads to a contradiction 3. Proof by Contradiction: Another name for indirect proofTuesday, March 13, 2012
  • 7. Steps to Write an Indirect ProofTuesday, March 13, 2012
  • 8. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false.Tuesday, March 13, 2012
  • 9. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false. 2. Follow a logical argument based on your false assumption to find a contradiction in your new assumption.Tuesday, March 13, 2012
  • 10. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false. 2. Follow a logical argument based on your false assumption to find a contradiction in your new assumption. 3. Identify the contradiction, thus proving that the original conclusion must be true.Tuesday, March 13, 2012
  • 11. Example 1 State the assumption necessary to start an indirect proof for each statement. a. EF is not a perpendicular bisector. b. 3x = 4y + 1 c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.Tuesday, March 13, 2012
  • 12. Example 1 State the assumption necessary to start an indirect proof for each statement. a. EF is not a perpendicular bisector. EF is a perpendicular bisector. b. 3x = 4y + 1 c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.Tuesday, March 13, 2012
  • 13. Example 1 State the assumption necessary to start an indirect proof for each statement. a. EF is not a perpendicular bisector. EF is a perpendicular bisector. b. 3x = 4y + 1 3x ≠ 4y + 1 c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB.Tuesday, March 13, 2012
  • 14. Example 1 State the assumption necessary to start an indirect proof for each statement. a. EF is not a perpendicular bisector. EF is a perpendicular bisector. b. 3x = 4y + 1 3x ≠ 4y + 1 c. If B is the midpoint of LH and LH = 26, then BH is congruent to LB. BH is not congruent to LB.Tuesday, March 13, 2012
  • 15. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2.Tuesday, March 13, 2012
  • 16. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2.Tuesday, March 13, 2012
  • 17. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7Tuesday, March 13, 2012
  • 18. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 −4 + 11 < 7Tuesday, March 13, 2012
  • 19. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 −4 + 11 < 7 7<7Tuesday, March 13, 2012
  • 20. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 −4 + 11 < 7 7<7 Not trueTuesday, March 13, 2012
  • 21. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 Also, −2(1) + 11 < 7 −4 + 11 < 7 7<7 Not trueTuesday, March 13, 2012
  • 22. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 Also, −2(1) + 11 < 7 −4 + 11 < 7 −2 + 11 < 7 7<7 Not trueTuesday, March 13, 2012
  • 23. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 Also, −2(1) + 11 < 7 −4 + 11 < 7 −2 + 11 < 7 7<7 9<7 Not trueTuesday, March 13, 2012
  • 24. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 Also, −2(1) + 11 < 7 −4 + 11 < 7 −2 + 11 < 7 7<7 9<7 Not true Also not trueTuesday, March 13, 2012
  • 25. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2. Then −2(2) + 11 < 7 Also, −2(1) + 11 < 7 −4 + 11 < 7 −2 + 11 < 7 7<7 9<7 Not true Also not true Both cases provide a contradiction, so therefore x > 2 must be true.Tuesday, March 13, 2012
  • 26. Example 3 Write an indirect proof to show that if x is a prime number, then x/3 is not an integer.Tuesday, March 13, 2012
  • 27. Example 3 Write an indirect proof to show that if x is a prime number, then x/3 is not an integer. Assume x/3 is an integer.Tuesday, March 13, 2012
  • 28. Example 3 Write an indirect proof to show that if x is a prime number, then x/3 is not an integer. Assume x/3 is an integer. x Then, = n. 3Tuesday, March 13, 2012
  • 29. Example 3 Write an indirect proof to show that if x is a prime number, then x/3 is not an integer. Assume x/3 is an integer. x Then, = n. 3 This means x = 3n.Tuesday, March 13, 2012
  • 30. Example 3 Write an indirect proof to show that if x is a prime number, then x/3 is not an integer. Assume x/3 is an integer. x Then, = n. 3 This means x = 3n. This cannot be true, because if x is prime, then 3 cannot be a factor of x as shown in the equation x = 3n. Therefore, x/3 is not an integer.Tuesday, March 13, 2012
  • 31. K Example 4 8 5 Prove by contradiction. J L 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠LTuesday, March 13, 2012
  • 32. K Example 4 8 5 Prove by contradiction. J L 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L Assume m∠K ≥ m∠L.Tuesday, March 13, 2012
  • 33. K Example 4 8 5 Prove by contradiction. J L 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L Assume m∠K ≥ m∠L. Then, by angle-side relationships, JL ≥ JK.Tuesday, March 13, 2012
  • 34. K Example 4 8 5 Prove by contradiction. J L 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L Assume m∠K ≥ m∠L. Then, by angle-side relationships, JL ≥ JK. This mean that 7 ≥ 8, which is not true.Tuesday, March 13, 2012
  • 35. K Example 4 8 5 Prove by contradiction. J L 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L Assume m∠K ≥ m∠L. Then, by angle-side relationships, JL ≥ JK. This mean that 7 ≥ 8, which is not true. So the assumption m∠K ≥ m∠L is false, so therefore m∠K < m∠LTuesday, March 13, 2012
  • 36. Check Your Understanding Explore problems #1-10 on page 354Tuesday, March 13, 2012
  • 37. Problem SetTuesday, March 13, 2012
  • 38. Problem Set p. 355 #11-29 odd, 43, 51, 53 “Its not that Im so smart, its just that I stay with problems longer.” – Albert EinsteinTuesday, March 13, 2012