Objectives Write indirect proofs. Apply inequalities in one triangle.

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Presentation transcript:

Objectives Write indirect proofs. Apply inequalities in one triangle.

Vocabulary indirect proof

So far you have written proofs using direct reasoning 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 ___________you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Helpful Hint

Example 1 Write an indirect proof that a triangle cannot have two right angles. Step 1 Identify the conjecture to be proven. Given: Prove: Step 2 Assume the opposite of the conclusion.

Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. Step 4 Conclude that the original conjecture is true.

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

Example 2a Write the angles in order from smallest to largest.

Example 2b 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.

A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

Example 3a Tell whether a triangle can have sides with the given lengths. Explain. 8, 13, 21

Example 3b Tell whether a triangle can have sides with the given lengths. Explain. 6.2, 7, 9

Example 3c Tell whether a triangle can have sides with the given lengths. Explain. t – 2, 4t, t2 + 1, when t = 4

Example 4 The lengths of two sides of a triangle are 22 inches and 17 inches. Find the range of possible lengths for the third side.

Example 5: Travel Application The figure shows the approximate distances between cities in California. What is the range of distances from San Francisco to Oakland?