1. Honors Geometry Lesson 5.5 Inequalities in Two Triangles

2. What You Should Learn Why You Should Learn It Goal 1: How to read and write an indirect proof Goal 2: How to use the Hinge Theorem and its converse You can use indirect reasoning to solve many real-life problems. For example, you can eliminate incorrect responses on a multiple- choice test to find the correct response

3. Sherlock Holmes Sherlock Holmes observed that “…when you have eliminated the impossible, whatever remains, however improbable, must be the truth…”

4. Indirect Proof (or proof by contradiction) With an indirect proof, you argue that, of all possible cases, all but one is impossible. This allows you to conclude that the remaining case must be true A B C Jeff is not in Homeroom A or Homeroom B

5. Indirect Reasoning If everyone in a class takes a test and one paper is not signed, the teacher can use indirect reasoning to determine the owner of the unsigned test.

6. Example 1 on page 246 gives a good example of an indirect proof

7. Using the Hinge Theorem Notice that and, but E is larger than B. It appears that the side opposite the 122° angle is longer than the side opposite the 85° angle. E 122° DF B A C 85°

8. Theorem 5.10 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 included angle of the second, then the third side of the first is longer than the third side of the second

9. Theorem 5.11 Converse of 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 E 12 DF B A C 8

10. Comparing Distances You and a friend are flying separate planes. You leave the airport and fly 120 miles due north. You then change direction and fly N 30° E for 70 miles. Your friend leaves the airport and flies 120 miles due south. She then changes direction and flies S 40° W for 70 miles. Each of you has flown 190 miles, but which is farther from the airport?

11. Comparing Distances You and a friend are flying separate planes. You leave the airport and fly 120 miles due north. You then change direction and fly N 30° E for 70 miles. Your friend leaves the airport and flies 120 miles due south. She then changes direction and flies S 40° W for 70 miles. Each of you has flown 190 miles, but which is farther from the airport? By applying the Hinge Theorem you can conclude that PR is longer than PT. Thus you are farther from the airport