Solving Indirect Proof

Introduction:

         Any factual confirmation that helps to offer the truth of something is called as proof. There are three types of proofs named direct proofs, indirect proofs, and proofs by contradiction. Indirect proof is a type of proof in which the statement to be proved is assumed false and if the assumption tends to contrariety, then the statement assumed false has been proved to be true.

 

Procedure for an Indirect proofs:

 

Step 1: State all possibilities.

Step 2: Assume the opposite of what you want to prove is true.

Step 3: Reason correctly until a contradiction of a given fact is obtained.

Step 4: State that negation assumed in Step 2 is false. Therefore, this follows that one of the other possibilities should be true.

Step 5: Therefore the remaining possibility is the desired conclusion.

 

Sample Problems for indirect proofs:

 

Solving Problem 1:

Let triangle PQR is an isosceles triangle with vertex P. Solving that the angles 1 and 2 are congruent

Proof:

  • Given the triangle is an isosceles triangle with vertex P.
  • Assume that the angles 1 and 2 are not congruent.
  • PQ is equal to QR since the sides of an isosceles triangle are congruent.
  • Since the opposite angles in a congruent side of a triangle are congruent, the angles 1 and 2 are said to be congruent.
  • The angles 1 and 2 are said to be congruent.

Solving Problem 2:

Solving the following statement using an indirect proof:
ΔABC has at most one right angle.

Proof:

  • Step1: Assume ΔABC has more than one right angle. That is, assume that angle L and angle M are both right angles.
  • If  B and C are both right angles, then b∠A = b∠B = 90 
  • b∠A + b∠B + b∠C = 180 (The sum of the measures of the angles of a triangle is 180). Substituting and solving gives b∠C =0.
  • This means that there is no ΔABC, which contradicts the given statement. So, the assumption that ∠A and ∠B are both right angles must be false.          
  • Therefore, ΔABC has at most one right angle.