Vocabulary Indirect Proof So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion (the “prove”) is false. Then you show that this assumption leads to a contradiction and thus, the “prove” is true indirectly. This type of proof is also called a proof by contradiction.

Writing an Indirect Proof Assume the opposite (the negation) of the “prove” (leave the “given” alone) Use reasoning to show the assumption leads to a “contradiction” Conclude that since the assumption is false the “prove” must be true indirectly. Use an Indirect Proof when trying to prove something is “NOT”. Assume the opposite of the “prove” and look for a contradiction of one of the following: given information, a definition, a postulate, or a theorem. Helpful Hint

Example 1a Write an indirect proof that a triangle cannot have two obtuse angles.

Example 1b

Example 1c

The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles. 6.9 6.10

Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. Write the angles in order from smallest to largest.

Check It Out! Example 2a Write the angles in order from smallest to largest. Write the sides in order from shortest to longest.

A triangle is formed by three segments, but not every set of three segments can form a triangle. 6.11

FROM THE TRIANGLE INEQUALITY THEOREM SHORTCUT: To determine if 3 lengths can form a triangle you only need to check that the sum of the two shorter sides is greater than the third.   m n x

Example 3A: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 Tell whether a triangle can have sides with the given lengths. Explain. 2.3, 3.1, 4.6

Example 4: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side. The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

Lesson Quiz: Part I 1. Write the angles in order from smallest to largest. 2. Write the sides in order from shortest to longest.

Lesson Quiz: Part II 3. The lengths of two sides of a triangle are 17 cm and 12 cm. Find the range of possible lengths for the third side. 4. Tell whether a triangle can have sides with lengths 2.7, 3.5, and 9.8. Explain. 5. Ray wants to place a chair so it is 10 ft from his television set. Can the other two distances shown be 8 ft and 6 ft? Explain. 6. In an Indirect Proof you assume the opposite of the ____________.