5.6 Indirect Proof and Inequalities in Two Triangles When using indirect reasoning all possibilities are considered, then all but one are proved false. The remaining possibility must be true. Prove a statement is true by first assuming that its opposite is true. If the assumption leads to an impossibility, then you have proved that the original statement is true. To write an indirect proof. Assume that the statement is false; assume the opposite of what you want to prove is true. Use logical reasoning to reach a contradiction of an earlier statement, such as the given information or a theorem. State that the assumption you made was false. State that the original statement must be true. Examples: If a movie was shown 25 times in seven days and was not shown more than four times in any one day, then it was shown at least once a day. Assume the movie was not shown at least once a day. This means that the movie was shown 25 times in 6 days or less. This contradicts the given information that the movie was not shown more than four times a day (6 ∙ 4 = 24). Therefore, the movie was shown at least once a day.

Use an indirect proof to prove that a triangle cannot have more than one obtuse angle. Assume that a ∆ABC does have more than one obtuse angle. Then m A>90o and m B>90o, and m A + m B > 180o. We know, however, that the sum of all three angles of a triangle is 180o. This means that m C is zero, which is not possible. This means the assumption is false. Therefore, a triangle cannot have more than one obtuse angle. Theorem 5.14 Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the second, then the third side of the first is longer than the third side of the second. Theorem 5.15 Converse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second.