The document discusses theorems and properties related to triangles, including: 1) The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side. 2) If one side of a triangle is longer than another, the angle opposite the longer side is larger than the angle opposite the shorter side. 3) The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles.

1. Inequalities in One Triangle Objectives: - Use triangle measurement to decide which side is longest or which angle is largest. - Use the Triangle Inequality

2. Theorem If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. Longest side Largest Angle

3. Another Theorem If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. Longest side Largest Angle

4. Putting measurements in order m  G < m  H < m  J Order the segments . . . JH < JG < GH 110º 40º 30º G H J

5. Indirect Proof Given: AC > AB Prove: m  ABC > m  C A B D C 1 2 3 Draw a point D such that AB=AD. Then draw segment BD. In the isosceles ∆ ABD,  1   2. Because m  ABC = m  1+m  3, it follows that m  ABC > m  1. Substituting m  1 for m  2 produces  ABC > m  2.

6. Indirect Proof Given: AC > AB Prove: m  ABC > m  C A B D C 1 2 3 Because m  2 = m  3+m  C, it follows that m  2 > m  C. Because m  ABC > m  2 and m  2 > m  C, you can conclude that m  ABC > m  C

7. Exterior Angle Inequality The measure of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. m  1 > m  J and m  1 > m  G G H J 1

8. Triangle Inequality Theorem The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side. AB + BC > AC AC + BC > AB AB + AC > BC C A B

9. Do p. 298 1-5, 27, 28 Homework: Worksheets