**Definition of Indirect Proof **

- Indirect proof is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true.

**Examples of Indirect Proof**

- Sum of 2
*n*even numbers is even, where*n*> 0. Prove the statement using an indirect proof.

The first step of an indirect proof is to assume that 'Sum of even integers is odd.'

That is, 2 + 4 + 6 + 8 + . . . .+ 2*n*= an odd number

⇒2(1 + 2 + 3 + 4 + . . . +*n*) = an odd number

⇒2 × = an odd number

⇒*n*(*n*+ 1) = an odd number, a contradiction, because*n*(*n*+ 1) is always an even number. Thus, the statement is proved using an indirect proof.

**Solved Example on Indirect Proof **

Prove the following statement using an indirect proof:

ΔLMN has at most one right angle.

Step 1:Assume ΔLMN has more than one right angle. That is, assume that angle L and angle M are both right angles.

Step 2:If M and N are both right angles, thenm∠L=m∠M= 90

Step 3:m∠L+m∠M+m∠N= 180 [The sum of the measures of the angles of a triangle is 180.]

Step 4:Substitution gives 90 + 90 +m∠N =180.

Step 5:Solving givesm∠N =0.

Step 6:This means that there is no ΔLMN, which contradicts the given statement.

Step 7:So, the assumption that∠Land ∠Mare both right angles must be false.

Step 8:Therefore, ΔLMN has at most one right angle.

**Related Terms for Indirect Proof**

- Proof
- Statement