Geometry Section 5-4 1112

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Indirect Proof

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Geometry Section 5-4 1112

  1. 1. Section 5-4 Indirect Proof
  2. 2. Essential Questions How do you write indirect algebraic proofs? How do you write indirect geometric proofs?
  3. 3. Vocabulary 1. Indirect Reasoning: 2. Indirect Proof: 3. Proof by Contradiction:
  4. 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:
  5. 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 it leads to a contradiction 3. Proof by Contradiction:
  6. 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 it leads to a contradiction 3. Proof by Contradiction: Another name for indirect proof
  7. 7. Steps to Write an Indirect Proof
  8. 8. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false (opposite).
  9. 9. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false (opposite). 2. Follow a logical argument based on your false assumption to find a contradiction in your new assumption.
  10. 10. Steps to Write an Indirect Proof 1. Identify what you are trying to prove, then assume the conclusion to be false (opposite). 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.
  11. 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.
  12. 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.
  13. 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.
  14. 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.
  15. 15. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2.
  16. 16. Example 2 Write an indirect proof to show that if −2x + 11 < 7, then x > 2. Assume that x ≤ 2.
  17. 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 < 7
  18. 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 < 7
  19. 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 < 7
  20. 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 true
  21. 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 −4 + 11 < 7 7 < 7 Not true Also, −2(1) + 11 < 7
  22. 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 −4 + 11 < 7 7 < 7 Not true Also, −2(1) + 11 < 7 −2 + 11 < 7
  23. 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 −4 + 11 < 7 7 < 7 Not true Also, −2(1) + 11 < 7 −2 + 11 < 7 9 < 7
  24. 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 −4 + 11 < 7 7 < 7 Not true Also, −2(1) + 11 < 7 −2 + 11 < 7 9 < 7 Also not true
  25. 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 −4 + 11 < 7 7 < 7 Not true Also, −2(1) + 11 < 7 −2 + 11 < 7 9 < 7 Also not true Both cases provide a contradiction, so therefore x > 2 must be true.
  26. 26. Example 3 Write an indirect proof to show that if x is a prime number not equal to 3, then x/3 is not an integer.
  27. 27. Example 3 Write an indirect proof to show that if x is a prime number not equal to 3, then x/3 is not an integer. Assume x/3 is an integer.
  28. 28. Example 3 Write an indirect proof to show that if x is a prime number not equal to 3, then x/3 is not an integer. Assume x/3 is an integer. Then, x 3 = n.
  29. 29. Example 3 Write an indirect proof to show that if x is a prime number not equal to 3, then x/3 is not an integer. Assume x/3 is an integer. Then, x 3 = n. This means x = 3n.
  30. 30. Example 3 Write an indirect proof to show that if x is a prime number not equal to 3, then x/3 is not an integer. Assume x/3 is an integer. Then, x 3 = n. 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. By contradiction, x/3 is not an integer.
  31. 31. Example 4 Prove by contradiction. K J L 8 5 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L
  32. 32. Example 4 Prove by contradiction. K J L 8 5 7 Given: ∆JKL with side lengths 5, 7, and 8. Prove: m∠K < m∠L Assume m∠K ≥ m∠L.
  33. 33. Example 4 Prove by contradiction. K J L 8 5 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.
  34. 34. Example 4 Prove by contradiction. K J L 8 5 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.
  35. 35. Example 4 Prove by contradiction. K J L 8 5 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 by contradiction, m∠K < m∠L
  36. 36. Problem Set
  37. 37. Problem Set p. 354 #1-29 odd, 43, 51, 53 “It's not that I'm so smart, it's just that I stay with problems longer.” – Albert Einstein

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