Question

What is an indirect proof?

Answer 1

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 m∠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 gives m∠N = 0.
Step 6: This means that there is no ΔLMN, which contradicts the given statement.
Step 7: So, the assumption that ∠L and ∠M are both right angles must be false.
Step 8: Therefore, ΔLMN has at most one right angle