1. 5.6 Inequalities in One Triangle The angles and sides of a triangle have special relationships that involve inequalities. Comparison Property of Inequality –If a = b + c and c > 0, then a > b.

2. Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles. 1 2 3

3. Theorem 5.10 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

4. Theorem 5.11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

5. Theorem 5.12 – Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

6. Using the Triangle Inequality Theorem Can a triangle have sides with the given lengths? a.3 ft., 7 ft., 8 ft. 3 + 7 > 8 10 > 8 3 + 8 > 7 11 > 7 7 + 8 > 3 15 > 3 Yes. The sum of the lengths of any two sides is greater than the length of the third side.

7. Using the Triangle Inequality Theorem Can a triangle have sides with the given lengths? b. 5 ft., 10 ft., 15 ft. 5 + 10 > 15 15 is not greater than 15. No. The sum of 5 and 10 is not greater than 15. This contradicts the Triangle Inequality Theorem.

8. Finding Possible Side Lengths You have a triangular sandbox. Two of the sides are 5 ft. and 8 ft. long. What is the range of possible lengths for the third side? x + 5 > 8 x > 3 x + 8 > 5 x > - 3 5 + 8 > x x < 13 Numbers that satisfy x > 3 and x > -3 must be greater than 3. So, the third side must be greater than 3 ft. and less than 13 ft.

9. More Practice!!!!! Homework – Textbook p. 328 – 329 #6 – 32 even.