Indirect Proof

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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


Video Examples: Introduction to Indirect Proof


Example 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 × Indirect Proof= 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

Ques: Prove the following statement using an indirect proof: △LMN has at most one right angle.

Solution:

    Step 1: Assume △LMN 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 △LMN, which contradicts the given statement.
    Step 7: So, the assumption that ∠L and ∠M are both right angles must be false.
    Step 8: Therefore, △LMN has at most one right angle.

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