Lesson 5-3 Indirect Proof

Standardized Test Practice: Transparency 5-3 5-Minute Check on Lesson 5-2 Determine the relationship between the lengths of the given sides. 1. RS, ST 2. RT, ST Determine the relationship between the measures of the given angles. 3. A, B 4. B, C Refer to the figure. 5. Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m1. 6. Which angle has the greatest measure? Standardized Test Practice: A 1 B 2 C 3 D 4

Standardized Test Practice: Transparency 5-3 5-Minute Check on Lesson 5-2 Determine the relationship between the lengths of the given sides. 1. RS, ST RS < ST 2. RT, ST RT > ST Determine the relationship between the measures of the given angles. 3. A, B mA < mB 4. B, C mB < mC Refer to the figure. 5. Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m1. 3, 4, 5, 6 6. Which angle has the greatest measure? Standardized Test Practice: A 1 B 2 C 3 D 4

Objectives Use indirect proof with algebra Use indirect proof with geometry

Vocabulary Indirect reasoning – showing something to be false so that the opposite must be true Indirect proof – proving the opposite of what you assume is true Proof by contradiction – proving the assumption contradicts some fact, definition or theorem

Key Concept Step 1: Assume that the conclusion is false, so then the opposite is true. Step 2: Show that this assumption leads to a contradiction of the hypothesis, or some other fact, such as a definition, postulate, theorem or corollary Step 3: Point out that because the false conclusion leads to an incorrect statement, the original conclusion must be true (the opposite of what we assumed in step 1)

Algebraic Example Martha signed up for 3 classes at Wytheville Community College for a little under $156. There was an administrative fee of $15, but the class costs varied. How can you show that at least one class cost less than $47? Given: Martha spent less than $156 Prove: At least one class cost (x) less than $47 Step 1: Assume x  $47 Step 2: Then $47 + $47 + $47 + $15  $156 Step 3: This contradicts what Martha paid, so the assumption must be false. Therefore one class must cost less than $47!

Geometric Example Given: JKL with side lengths as shown Prove: mK < mL 8 5 7 L J Step 1: Assume mK  mL Step 2: By angle-side relationships, JL  JK Step 3: This contradicts the given side lengths, so the assumption must be false Therefore, mK < mL !

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 you would make to start an indirect proof for the statement Answer: State the assumption you would make to start an indirect proof for the statement m1 is less than or equal to m2. If m1  m2 is false, then m1 > m2. Answer: m1 > m2

State the assumption you would make to start an indirect proof for the statement If B is the midpoint of and then is congruent to The conclusion of the conditional statement is is congruent to The negation of the conclusion is is not congruent to Answer: is not congruent to

State the assumption you would make to start an indirect proof of each statement. a. is not an altitude. b. Answer: is an altitude. Answer: c. mABC is greater than or equal to mXYZ. Answer: mABC < mXYZ d. If is an angle bisector of MLP, then MLH is congruent to PLH. Answer: MLH is not congruent to PLH.

Write an indirect proof. 1 Given: ----------- = 20 2y + 4 Prove: y  -2 Indirect Proof: Step 1 Assume that . Step 2 Substitute –2 for y in the equation Substitution

Multiply. Add. This is a contradiction because the denominator cannot be 0. Step 3 The assumption leads to a contradiction. Therefore, the assumption that must be false, which means that must be true.

Write an indirect proof. Given: ABC with side lengths 8, 10, and 12 as shown. Prove: mC > mA 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C > mA.

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. How can you show that at least one of the sweaters cost less than $32? Answer: Given: David spent less than $135. Prove: At least one of the sweaters x cost less than $32. That is,

Step 3. The assumption leads to a contradiction of a. known fact Step 3 The assumption leads to a contradiction of a known fact. Therefore, the assumption that must be false. Thus, at least one of the sweaters cost less than $32. Step 1 Assume that none of the sweaters cost less than $32. Indirect Proof: Step 2 then the minimum total amount David spent is However, this is a contradiction since David spent less than $135.

Summary & Homework Summary: Homework: In an indirect proof, the conclusion is assumed to be false and a contradiction is reached Homework: pg 258-9: 4-6, 13, 14 Proofs: 11, 22