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Splash Screen. Five-Minute Check (over Lesson 5–3) CCSS Then/Now New Vocabulary Key Concept: How to Write an Indirect Proof Example 1: State the Assumption for Starting an Indirect Proof Example 2: Write an Indirect Algebraic Proof Example 3: Indirect Algebraic Proof

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 5–3) CCSS Then/Now New Vocabulary Key Concept: How to Write an Indirect Proof Example 1: State the Assumption for Starting an Indirect Proof Example 2: Write an Indirect Algebraic Proof Example 3: Indirect Algebraic Proof Example 4: Indirect Proofs in Number Theory Example 5: Geometry Proof Lesson Menu

  3. What is the relationship between the lengths of RS and ST? ___ ___ A.RS > ST B.RS = ST C.RS < ST D. no relationship 5-Minute Check 1

  4. What is the relationship between the lengths of RT and ST? ___ ___ A.RT > ST B.RT < ST C.RT = ST D. no relationship 5-Minute Check 2

  5. What is the relationship between the measures of A and B? A.mA > mB B.mA < mB C.mA = mB D. cannot determine relationship 5-Minute Check 3

  6. What is the relationship between the measures of B and C? A.mB > mC B.mB < mC C.mB = mC D. cannot determine relationship 5-Minute Check 4

  7. Using the Exterior Angle Inequality Theorem, which angle measure is less than m1? A. 3 B. 4 C. 6 D. all of the above 5-Minute Check 5

  8. __ __ ___ A.RI, IT, TR B.IT, RI, TR C.TR, RI, IT D.RI, RT, IT __ __ ___ __ __ __ __ ___ __ In ΔTRI, mT = 36, mR = 57, and mI = 87. List the sides in order from shortest to longest. 5-Minute Check 6

  9. Content Standards G.CO.10 Prove theorems about triangles. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 2 Reason abstractly and quantitatively. CCSS

  10. You wrote paragraph, two-column, and flow proofs. • Write indirect algebraic proofs. • Write indirect geometric proofs. Then/Now

  11. indirect reasoning • indirect proof • proof by contradiction Vocabulary

  12. Concept

  13. A. State the assumption you would make to start an indirect proof for the statement is not a perpendicular bisector. Answer: is a perpendicular bisector. State the Assumption for Starting an Indirect Proof Example 1

  14. State the Assumption for Starting an Indirect Proof B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1. Answer: 3x≠ 4y + 1 Example 1

  15. State the Assumption for Starting an Indirect Proof Example 1

  16. A. B. C. D. Example 1

  17. A. B. C. D. Example 1

  18. A. B.MLH  PLH C. D. Example 1

  19. Write an Indirect Algebraic Proof Write an indirect proof to show that if –2x + 11 < 7, then x > 2. Given: –2x + 11 < 7 Prove: x > 2 Step 1 Indirect Proof: The negation of x > 2 is x ≤ 2. So, assume that x < 2 or x = 2 is true. Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2. Example 2

  20. Write an Indirect Algebraic Proof Step 2 Make a table with several possibilities for x assuming x < 2 or x = 2. When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7. Example 2

  21. Write an Indirect Algebraic Proof Step 3In both cases, the assumption leads to a contradiction of the given information that –2x + 11 < 7. Therefore, the assumption that x ≤ 2 must be false, so the original conclusion that x > 2 must be true. Example 2

  22. Which is the correct order of steps for the following indirect proof? Given: x + 5 > 18 Prove: x > 13 I. In both cases, the assumption leads to a contradiction. Therefore, the assumption x ≤ 13 is false, so the original conclusion that x > 13 is true. II. Assume x ≤ 13. III. When x < 13, x + 5 = 18 and when x < 13, x + 5 < 18. Example 2

  23. A. I, II, III B. I, III, II C. II, III, I D. III, II, I Example 2

  24. Indirect Algebraic Proof EDUCATION Marta signed up for three classes at a community college for a little under $156. There was an administration fee of $15, and the class costs are equal. How can you show that each class cost less than $47? Let x be the cost of each class. Step 1 Given: 3x + 15 < 156Prove:x < 47 Indirect Proof:Assume that none of the classes cost less than $47. That is, x≥ 47. Example 3

  25. Indirect Algebraic Proof Step 2 If x≥ 47, then x + x + x + 15 ≥ 156 or47 + 47 + 47 + 15 ≥ 156. Step 3 This contradicts the statement that the total cost was less than $156, so the assumption that x ≥ 47 must be false. Therefore, one class must cost less than $47. Example 3

  26. SHOPPING David bought four new sweaters for a little under $135. The tax was $7, but the sweater costs varied.Can David show that at least one of the sweaters cost less than $32? A. Yes, he can show by indirect proof that assuming that every sweater costs $32 or more leads to a contradiction. B. No, assuming every sweater costs $32 or more does not lead to a contradiction. Example 3

  27. Write an indirect proof to show that if x is a prime number not equal to 3, then is not an integer. Step 1 Given: x is a prime number not equal to 3. Prove: is not an integer. Indirect Proof: Assume is an integer. This means = n for some integer n. x x x x __ __ __ __ 3 3 3 3 Indirect Proofs in Number Theory Example 4

  28. Step 2 = n Substitution of assumption x __ 3 Indirect Proofs in Number Theory x= 3n Multiplication Property Now determine whether x is a prime number. Since x≠ 3, n ≠ 1. So, x is a product of two factors, 3 and some number other than 1. Therefore, x is not a prime Example 4

  29. x x __ __ 3 3 Indirect Proofs in Number Theory Step 3 Since the assumption that is an integer leads to a contradiction of the given statement, the original conclusion that is not an integer must be true. Example 4

  30. You can express an even integer as 2k for some integer k. How can you express an odd integer? A. 2k + 1 B. 3k C.k + 1 D.k + 3 Example 4

  31. Write an indirect proof. Given:ΔJKLwith side lengths 5, 7, and 8 as shown. Prove:mK < mL Geometry Proof Example 5

  32. Step 1Assume that Step 2 By angle-side relationships, By substitution, . This inequality is a false statement. Geometry Proof Indirect Proof: Step 3 Since the assumption leads to a contradiction, the assumption must be false. Therefore, mK < mL. Example 5

  33. Which statement shows that the assumption leads to a contradiction for this indirect proof? Given:ΔABCwith side lengths 8, 10, and 12 as shown. Prove:mC > mA Example 5

  34. A. Assume mC≥mA + mB. By angle-side relationships, AB > BC + AC. Substituting, 12 ≥ 10 + 8 or 12 ≥ 18. This is a false statement. B. Assume mC ≤ mA. By angle-side relationships, AB ≤ BC. Substituting, 12 ≤ 8. This is a false statement. Example 5

  35. End of the Lesson

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