Indirect Proofs

Warm Up Classify each triangle by its angle measures. 1. 2. 3. Simplify 4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2. Which value is greater? acute right 12 85; 144; c2

Indirect Proofs Up to this point, we have been proving a statement true by direct proofs. Sometimes direct proofs are difficult and we can instead prove a statement indirectly, which is very common in everyday logical thinking.

An Indirect Proof Example You say that my dog, Rex, dug a hole in your yard on July 15th. I will prove that Rex did not dig a hole in your yard. Let’s temporarily assume that Rex did dig a hole in your yard on July 15th. Then he would have been in your yard on July 15th. But this contradicts the fact that Rex was in the kennel from July 14th to July 17th. I have bills that show this is true. Thus, our assumption is false, therefore Rex did not dig a hole in your backyard.

Things to think about… The negation of = is ≠, and vice versa. The key to an indirect proof is to know how to start it and to reason through it logically.

Example: Write the first step of an indirect proof. a) If AB = BC, then ABC is not scalene. b) If then . c) If then XY // CD

Indirect Proofs When trying to prove a statement is true, it may be beneficial to ask yourself, "What if this statement was not true?" and examine what happens.  This is the premise of the Indirect Proof or Proof by Contradiction.

Rules: List the possibilities for the conclusion. Assume negation of the desired conclusion is correct. Write a chain of reasons until you reach an impossibility. This will be a contradiction of either: the given information or a theorem definition or known fact. State the remaining possibility as the desired conclusion.

How to Write an Indirect Proof

Given: n2 > 6n Prove: n ≠ 4 Assume temporarily that n = 4. Then…n2 = 42 = 16 and 6n = 6(4) = 24. Since 16 < 24, then n2 < 6n. But this contradicts… the given that n2 > 6n. Thus our assumption is false, . . n ≠ 4.

Given: ≠ Prove: ≠ . Assume temporarily that = . b 1 2 a 3 Assume temporarily that = . Then…a//b because Corr. Angles Congruent Imply // Lines. Since a//b, then = because // lines imply that alt. int. angles are congruent. But this contradicts… the given that ≠ Thus our assumption is false, . . ≠ .

Given: Scalene Triangle REN Prove: ≠ Assume temporarily = . Then…EN=RE by the Converse to the Iso. Triangle Theorem. Thus, REN would be an isosceles triangle. But this contradicts… the given that REN is a scalene triangle. Thus our assumption is false, . . ≠ N

Either RS bisects PRQ or RS does not bisect PRQ Either RS bisects PRQ or RS does not bisect PRQ. Assume RS bisects PRQ. Then we can say that PRS  QRS. Since RS PQ, we know that PSR  QSR. Thus, ΔPSR  ΔQSR by ASA (SR  SR) PR  QR by CPCTC. But this is impossible because it contradicts the given fact that QR  PR. The assumption is false. RS does not bisect PRQ. T

Given:<H ≠ <K Prove: JH ≠ JK ~ ~ Either JH is  to JK or it’s not. Assume JH is  to JK, then ΔHJK is isosceles because of congruent segments. Then  H is  to  K. Since  H isn’t congruent to  K, then JH isn’t congruent to JK. J H K