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Lesson Menu 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

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

Over Lesson 5–3 5-Minute Check 2 A.RT > ST B.RT < ST C.RT = ST D.no relationship What is the relationship between the lengths of RT and ST? ___

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

Over Lesson 5–3 5-Minute Check 4 A.m  B > m  C B.m  B < m  C C.m  B = m  C D.cannot determine relationship What is the relationship between the measures of  B and  C?

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

Over Lesson 5–3 5-Minute Check 6 In ΔTRI, 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 __ ___ __ ___ __ _____

CCSS 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.

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

Vocabulary indirect reasoning indirect proof proof by contradiction

Concept

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

Example 1 State the Assumption for Starting an Indirect Proof B. State the assumption you would make to start an indirect proof for the statement3x = 4y + 1. Answer: 3x ≠ 4y + 1

Example 1 State the Assumption for Starting an Indirect Proof

Example 1 A. B. C. D.

Example 1 A. B. C. D.

Example 1 A. B.  MLH   PLH C. D.

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

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

Example 2 Write an Indirect Algebraic Proof Step 3In both cases, the assumption leads to a contradiction of the given information that –2x must be true.

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

Example 3 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 1Given: 3x + 15 < 156 Prove: x < 47 Indirect Proof: Assume that none of the classes cost less than $47. That is, x ≥ 47.

Example 3 Indirect Algebraic Proof Step 2If x ≥ 47, then x + x + x + 15 ≥ 156 or ≥ 156. Step 3This 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 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. 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?

Example 4 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. __ x 3 Step 1Given: 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 3 x 3 x 3

Example 4 Indirect Proofs in Number Theory Step 2 = nSubstitution of assumption __ x 3 x = 3nMultiplication 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 Indirect Proofs in Number Theory Step 3Since 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. __ x 3 x 3

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

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

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

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

Example 5 A.Assume m  C ≥ m  A + m  B. By angle-side relationships, AB > BC + AC. Substituting, 12 ≥ 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.

End of the Lesson