5.1 Indirect Proof Let’s take a Given: Prove: Proof: Either or An indirect proof is useful when a direct proof is difficult to apply. Let’s take a A D B E C F Given: Prove: Proof: Either or Let’s assume

Therefore, our assumption was false and we can now say , From the given information we can prove the triangles congruent by ASA, which makes A D B E C F But wait! This is not possible, since the given information states that Therefore, our assumption was false and we can now say , because this is the only other possibility.  

Indirect Proof Procedure… 1. List the possibilities for the conclusion. 2. Assume that the negation of the desired conclusion is correct. 3. Write a chain of reasons until you reach an impossibility. This will be a contradiction of either: a. Given information or b. A theorem, definition, or other known fact. 4. State the remaining possibility as the desired conclusion.

Remember to start by looking at the conclusion! P S R Given: Q Prove:

Given: Prove: Proof: Either or Assume . Then we can say . Q Step 1: Proof: Either or Assume . Then we can say . Since , we know that , thus, triangle PSR is congruent to triangle QSR by ASA (Since ). This means by CPCTC, but this contradicts the given information that therefore the assumption must be false, leaving the only other possibility: Step 2: Step 3: Step 4:

Hang in there, one last example… C B A Given: O O Prove: