Chapter 5 Lesson 5 Objective To use inequalities

Chapter 5 Lesson 5 Objective: To use inequalities involving angles and sides of triangles.

Comparison Property of Inequality If a = b + c and c > 0, then a > b. 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. and 3 2 1

Example 1: Applying the Corollary In the diagram, m 2 = m 1 by the Isosceles Triangle Theorem. Explain why m 2 > m 3. By the corollary to the Exterior Angle Theorem, m 1 > m 3. So, m 2> m 3 by substitution.

Theorem 5 -10 If two sides of a triangle are not congruent, then the larger angle lies Y opposite the longer side. If XZ > XY, then X Z

Theorem 5 -11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. B If , then BC > AC. C A

Example 2: Using Theorem 5 -11 In ∆TUV, which side is shortest? By the Triangle Angle-Sum Theorem, m T = 60. The smallest angle in ∆TUV is U. U It follows, by Theorem 5 -11, that the shortest side is. T V

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. XY + YZ > XZ YZ + ZX > YX ZX + XY > ZY

Example 3: Using the Triangle Inequality Theorem Can a triangle have sides with the given lengths? Explain. (a. ) 3 3 8 3 ft. , 7 ft. 8 ft. + 7 > 8 + 7 > 3 + 8 > 7 Yes; the sum of any two length is greater than the third length. (b. ) 3 cm, 6 cm, 10 cm 3 + 6 > 10 No; the sum of 3 and 6 is not greater than 10.

Example 4: Finding Possible Side Lengths A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths possible for the third side. Let x represent the length of the third side. By the Triangle Inequality Theorem, The third side must be longer than 2 cm and shorter than 18 cm.

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