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5.3 Inequalities in One Triangle Then: You found the relationship between the angle measures of a triangle. Now: 1. Recognize and apply properties of inequalities to the measure of the angles of a triangle. 2. Recognize and apply properties of inequalities to the relationship between the angles and sides of a triangle. Angle Inequality: *Properties of inequalities can be used with measures of angles and segments. See Key Concept p. 344. Theorem 5.8 Exterior Angle Inequality: The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles. Angle Inequality Exterior Angle Inequality Theorem: and m1 m A m1 m B Example 1: Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a. Measures are less than m 1 b. Measures are greater than m 1 b. Measures are less than m 7 Example 1: d. Measures are greater than m 2 e. Measures are greater than m 5 f. Measures are less than m 4 Inequalities in One Triangle B A E C ABC F D DEF Largest angle Longest side Smallest angle Shortest side How are angle and side measures related? Angle-Side Relationships in Triangles Theorem 5.9: If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. CB AB, so mA mC Angle-Side Relationships in Triangles Theorem 5.10: If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle. mB mC, so AC AB http://www.wyzant.com/Help/Images/inequality10.gif Example 2: List the angles and sides in order from smallest to largest. a. b. Example 3: List the angles and sides in order from smallest to largest. Example 4: a. HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie? Example 4: b. List the sides of ABC in order from smallest to largest if the angles of ABC have the indicated measures. mA = 5x + 2 m B = 6x -10 m C = x + 20 Example 4: c. Order the sides from shortest to longest. D A 55 50 65 B 70 C 5.3 Assignment p. 348-351 #8-11, 14-22 evens, 23, 24, 26-40 evens, 41, 44, 48, 51, 52