Indirect Proof

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 2n 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 + . . . .+ 2n = 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, then  

Step 3:   [The sum of the measures of the angles of a triangle is 180.]

Step 4: Substitution gives

Step 5: Solving gives

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

Step 7: So, the assumption that  and are both right angles must be false.           

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

Related Terms for Indirect Proof

• Proof
• Statement

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