Given:   is a right triangle, is a right angle.

Prove:    Hypotenuse of is the longest side. That is  .

Indirect Proof:

Let us assume that the hypotenuse is not the longest side, that is  .

If  , then  . Since , . So, .

Simillarly by the same reason .

This is contradiction because we know that sum of the angle measures of a triangle is 180.

The assumption leads to the contradiction of a known fact. Therefore, the assumption that   must be false, which means hypotenuse of a triangle is the longest side.

The given theorem was proved that hypotenuse of a triangle is the longest side.