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Method

Indirect Proof

An indirect proof, or proof by contradiction, is a proof method that uses indirect reasoning. Here, all possibilities are considered, and then all except one are proven false. Typically, in an indirect proof, there are two possibilities — a statement and its negation. The negation of the statement is eliminated by contradiction. The following statement will be analyzed using an indirect proof.

In a right triangle, the longest side is the

There are three steps to follow to prove this statement using an indirect proof.

1

Identify the statement to be proven and temporarily assume that its negation is true.

The aim is to prove that the hypotenuse is the longest side of a right triangle. For this case, consider where is a right angle.

Assume temporarily that the negation of the statement is true.

In a right triangle, the longest side is not its

2

Use logical reasoning until a contradiction is derived.

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

3

State that the original statement must be true because the contradiction shows that the assumption is false.

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.

Extra

What to assume when proving a conditional statement using an indirect proof?

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.

T T T F
T F F T
F T T F
F F T T

Therefore, assuming that the hypothesis is true and the conclusion is false will facilitate the process of deriving a contradiction.


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