5. 1 Indirect Proof An indirect proof is useful when a direct proof is difficult to apply. Let’s take a B A D C Given: Prove: Proof: Either or Let’s assume E F
From the given information we can prove the triangles congruent by ASA, which makes A B D C But wait! This is not possible, since the given information states that E F 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 Given: Prove: R Q
Given: P Prove: S R Q Step 1: Proof: Either or Step 2: Assume . Then we can say . Step 3: Since , we know that , thus, triangle PSR is congruent to triangle QSR by ASA (Since ). Step 4: This means by CPCTC, but this contradicts the given information that therefore the assumption must be false, leaving the only other possibility:
Hang in there, one last example… C B A Given: Prove: O O