A proof involving indirect reasoning.
- Often a statement and its negation are the only possibilities.

Indirect Proofs

All possibilities are considered and then all but one possibility are proved false. The remaining possibility must be true.

Indirect Reasoning

Important Vocabulary

​Steps to 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.

• The opposite of "do not have" is "do have."

• Assume temporarily that you do have soccer practice today.

B) You do not have soccer practice today.

• The opposite of "is divisible by" is "is not divisible by."

• Assume temporarily that n is not divisible by 5.

​A) An integer n is divisble by 5.

Writing the First Step of an Indirect Proof:
Suppose you want to write an indirect proof of each statement. As the first step of the proof, what would you assume?

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

Identifying Contradictions

Which two statements contradict each other?

1) FG ∥ KL
2) FG ≅ KL
3) FG ⊥ KL

• Segments can be ∥ and ≅. Statements I and II do not contradict each other.

• Segments can be ≅ and ⊥. Statements II and III do not contradict each other.

• ∥ segments do not intersect, so they cannot be ⊥. Statements I and III contradict each other.

Writing an Indirect Proof

Given: ΔABC is scalene
Prove: ∠A, ∠B, and ∠C are all different measures

Steps to Writing an Indirect Proof:
1) Temporarily assume the opposite of what you want to prove.


2) Show this assumption leads to a contradiction.


3) Conclude assumption is false and that what you want to prove is true.