1. You will learn to use indirect reasoning to write proofs
5.5 Indirect Proof
You will learn to use indirect reasoning to write proofs

2. Vocabulary
Indirect Reasoning: all possibilities are considered and then all but one are proven false
Indirect Proof: a proof involving indirect reasoning

3. Writing an Indirect Proof
Step 1: State as a temporary assumption the opposite (negation) of what you want to prove. Step 2: Show that this temporary assumption leads to a contradiction. Step 3: Conclude that the temporary assumption must be false and that what you want to prove must be true.

4. The First Step
You should assume that the opposite of what you want to prove is TRUE.
Example: Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?
You do not have soccer practice today.
You DO have soccer practice today.
An integer n is divisible by 5.
An integer n is NOT divisible by 5.

5. Sample Problems You Try:
Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?
At least one pair of shoes you bought cost more than $25.
At least one pair of shoes you bought did NOT cost more than $25.
Triangle BOX is not acute.
Triangle BOX IS acute.

6. Identifying Contradicitons
Contradiction: You have reached a contradiction when you have two statements that cannot both be true at the same time.
To write an indirect proof, you have to be able to identify a contradiction.
Example: Which two statements contradict each other? A) B)
Triangle ABC is acute.
Triangle ABC is scalene.
Triangle ABC is equilateral.
mB  90
B is acute
B is a right angle.

7. Sample Problem
You Try: Which two statements contradict each other? A)
I. Victoria has art class from 9:00 to 10:00 on Mondays.
II. Victoria has math class from 10:30 to 11:30 on Mondays.
III. Victoria has math class from 9:00 to 10:00 on Mondays. B)
I. Triangle MNO is acute.
II. The point of concurrency for the medians and for the altitudes for Triangle MNO are at different points.
III. Triangle MNO is equilateral.

8. Lesson Summary
How is indirect reasoning used when proving something? Indirect reasoning is used by assuming that the opposite of what you want to prove is true. Then you show that this causes a contradiction.