In a right triangle, the longest side is the
From the assumption, it can be inferred that at least one of the legs of the right triangle is larger than the hypotenuse. Let be the leg whose length is greater than the length of the hypotenuse Therefore, by the Triangle Longer Side Theorem, the measure of the angle opposite is greater than the measure of the angle opposite
Since and is greater than the sum of the measures of these two angles is greater than According to the Interior Angles Theorem, the sum of the three interior angles of any triangle is equal to For this equation and the previously derived inequality to hold true, must be less than This conclusion contradicts the fact that the measures of all interior angles of any polygon are greater than
The contradiction resulted from the assumption that the hypotenuse of a right triangle is not the longest side. Therefore, the assumption — the negation of the original statement — must be false. That indirectly proves that the original statement is true.
Consider a conditional statement. The first step of a proof by contradiction is to assume the negation of the statement desired to be proven. Stating the negation of an if-then statement may not make sense, however. For this reason, a statement logically equivalent to the negation of the statement in if-then form can be used. Consider the statement that is formed using a conjunction. These statements are logically equivalent because both have the same truth table.
Therefore, assuming that the hypothesis is true and the conclusion is false will facilitate the process of deriving a contradiction.