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 1
Over Lesson 5 3 Questions 1 & 2 Questions 3 & 4 What is the relationship between the lengths of RS and ST? What is the relationship between the lengths of RT and ST? What is the relationship between the measures of A and B? What is the relationship between the measures of B and C? Over Lesson 5 3 What is the relationship between the lengths of RS and ST? A. RS > ST B. RS = ST C. RS < ST D. no relationship 2
Over Lesson 5 3 What is the relationship between the lengths of RT and ST? A. RT > ST B. RT < ST C. RT = ST D. no relationship Over Lesson 5 3 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 3
Over Lesson 5 3 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 Over Lesson 5 3 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 4
Over Lesson 5 3 In TRI, m T = 36, m R = 57, and m I = 87. List the sides in order from shortest to longest. A. RI, IT, TR B. IT, RI, TR C. TR, RI, IT D. RI, RT, IT 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. 5
You wrote paragraph, two-column, and flow proofs. Write indirect algebraic proofs. Write indirect geometric proofs. indirect reasoning indirect proof proof by contradiction 6
negating the conclusion. State the Assumption for Starting an Indirect Proof 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. 7
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. Assume temporarily 3x 4y + 1 BUT realize this means: 3x < 4y + 1 or 3x > 4y + 1 Answer: 3x 4y + 1 State the Assumption for Starting an Indirect Proof 8
A. B. C. D. A. B. C. D. BUT: Choosing both A and D would also be correct! 9
A. B. MLH PLH C. D. 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 Proof Assume temporarily x 2 Case 1 x < 2 Let x = 0 Then -2(0) + 11 < 7 11 < 7 But this is false, thus x 2 Conclusion Case 2 x = 2 Then -2(2) + 11 < 7 7 < 7 But this is false, thus x 2 Thus the temporary assumption is false so the original conclusion must be true and x > 2 10
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. A. I, II, III B. I, III, II C. II, III, I D. III, II, I 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? Given: 3x + 15 < 156, where x is the cost of each class Prove: x < 47 Proof: Assume temporarily x 47. Case 1 x > 47 Case 2 x = 47 Let x = 50 Then 3(50) + 15 < 156 165 < 156 But this is false, thus x 47 Then 3(47) + 15 < 156 156 < 156 But this is false, thus x 47 Conclusion Thus the temporary assumption is false so the original conclusion must be true and x < 47 11
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. Indirect Proofs in Number Theory Write an indirect proof to show that if x is a prime number not equal to 3, then is not an integer. Given: x is a prime number not equal to 3. Prove: is not an integer. Proof: Assume temporarily is an integer Then there exists some integer n such that = So =3 But this contradicts the definition of prime number. Therefore the temporary assumption is false and is not an integer. 12
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 Geometry Proof Write an indirect proof. Given: JKL with side lengths 5, 7, and 8 as shown. Prove: m K < m L Proof Assume temporarily m K m L Then JL JK (by angle side relationship) So 7 8 But this is false, so m K < m L 13
Which statement shows that the assumption leads to a contradiction for this indirect proof? Given: ABC with side lengths 8, 10, and 12 as shown. Prove: m C > m A 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 angleside relationships, AB BC. Substituting, 12 10. This is a false statement. 14