Indirect Proof (Proof by Contradiction)

What Is Indirect Proof?

An indirect proof (also called proof by contradiction) is a method where you assume the opposite of what you want to prove, then show that this assumption leads to a contradiction.

Steps for Writing an Indirect Proof

1. Assume the opposite: Assume that the conclusion you want to prove is false (negate the conclusion).
2. Use logical reasoning: Use definitions, theorems, and given information to reason from the assumption.
3. Reach a contradiction: Show that the assumption leads to a statement that contradicts a known fact, a given, or a theorem.
4. Conclude: Since the assumption leads to a contradiction, the original statement must be true.

Example

Prove: A Triangle cannot have two obtuse angles.

1. Assume a Triangle has two obtuse angles, say ∠A > 90° and ∠B > 90°.
2. Then ∠A + ∠B > 180°.
3. But the sum of all angles in a Triangle is 180°. So ∠A + ∠B + ∠C = 180°.
4. This means ∠C < 0°, which is impossible.
5. Contradiction! Therefore, a Triangle cannot have two obtuse angles.

When to Use Indirect Proof

• When the statement says something cannot happen or something must be true.
• When a direct approach seems difficult or impossible.
• When the conclusion involves uniqueness ("there is exactly one...").