Five-Minute Check (over Lesson 5–3) Mathematical Practices 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

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

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

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

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

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

In ΔTRI, mT = 36, mR = 57, and mI = 87 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 __ ___ 5-Minute Check 6

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

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

proof by contradiction indirect reasoning indirect proof proof by contradiction Vocabulary

Concept

Answer: is a perpendicular bisector. State the Assumption for Starting an Indirect Proof A. State the assumption necessary to start an indirect proof for the statement is not a perpendicular bisector. If EF is not a perpendicular bisector is false, then EF is a perpendicular bisector must be true. Answer: is a perpendicular bisector. Example 1

Answer: BH is not congruent to LB. State the Assumption for Starting an Indirect Proof B. State the assumption necessary to start an indirect proof for the statement, If B is the midpoint of LH and LH = 26, then BH is congruent to LB. The conclusion of the conditional statement is BH is congruent to LB. The negation of the conclusion is BH is not congruent to LB. Answer: BH is not congruent to LB. Example 1

A. B. C. D. Example 1

A. B. C. D. Example 1

A. B. MLH  PLH C. D. Example 1

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

When x < 2, –2x + 11 > 7 and when x = 2, –2x + 11 = 7. 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

Write an Indirect Algebraic Proof Step 3 In 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

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 2

Let x be the cost of each class. Indirect Algebraic Proof EDUCATION Marta signed up for three classes at a community college for a little under $1226. There was a general education fee of $360 and a student life fee of $118. The class costs are equal. How can you show that each class cost less than $250? Let x be the cost of each class. Step 1 Given: 3x + 478 < 1226 Prove: x < 250 Indirect Proof: Assume that none of the classes cost less than $250. That is, x ≥ 250. Example 3

Indirect Algebraic Proof Step 2 If x ≥ 250, then x + x + x + 478 ≥ 1226 or 250 + 250 + 250 + 478 ≥ 1226. Step 3 This contradicts the statement that the total cost was less than $1226, so the assumption that x ≥ 250 must be false. Therefore, each class must cost less than $250. Example 3

SHOPPING David bought four new sweaters for a little under $135 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

Step 1 Given: x is a prime number not equal to 3. 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 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 3 Example 4

Step 2 = n Substitution of assumption x 3 Indirect Proofs in Number Theory Step 2 = n Substitution of assumption __ x 3 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

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. __ x 3 __ x 3 Example 4

You can express an even integer as 2k for some integer k 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

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

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

Given: ΔABC with side lengths 8, 10, and 12 as shown. 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 ≥ 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