5.5 Use Inequalities in a Triangle Theorem 5.10: If one side of a triangle is longer than the other side, then the angle opposite the longest side A is _______ larger than the angle opposite the shorter side. B 8 5 C AB BC , so m ___ C m ___ A . 5.5 Use Inequalities in a Triangle Theorem 5.11: B If one angle of a triangle is larger than another angle, then the side opposite the larger angle 50o 30o C A is _______ longer than the side mA mC, opposite the smaller angle. so ____ BC ____ AB . 5.5 Use Inequalities in a Triangle Example 1 Write measurements in order from least to greatest Write measurements of the triangle in order from least to greatest. a. C b. 36o B 57o E 22 87o A 13 F D 12 Solution a. m ____ ____ A C mB m ____ BC AB AC ____ 5.5 Use Inequalities in a Triangle Example 1 Write measurements in order from least to greatest Write measurements of the triangle in order from least to greatest. a. C b. 36o B 57o E 22 87o A 13 F D 12 Solution b. m ____ ____ F E mD m ____ DF EF ____ DE 5.5 Use Inequalities in a Triangle Checkpoint. Write the measurements of the triangle in order from least to greatest. B 99o 1. A 34o 47o C mA mC mB BC AB AC 5.5 Use Inequalities in a Triangle Checkpoint. Write the measurements of the triangle in order from least to greatest. Q 80o 2. P 45o 55o R mP mR mQ QR PQ PR 5.5 Use Inequalities in a Triangle Theorem 5.12: Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is A greater than the length of the third side. ___ BC AC AB ___ AC ___ BC ___ AB ___ BC AB AC ___ B C 5.5 Use Inequalities in a Triangle Example 2 Find possible side lengths A triangle has one side of length 14 and another of length 10. Describe the possible lengths of the third side. Solution Let x represent the length of the third side. Draw diagrams to help visualize the small and large values of x. Then use the Triangle Inequality Theorem to write and solve inequalities. Small values of x 14 x 10 x 10 __ 14 __ x __ 4 Large values of x x 10 14 __ 14 __ x 10 __ 24 24 x , or x __ The length of the third side must be _______________________________. greater than 4 and less than 24 5.5 Use Inequalities in a Triangle Checkpoint. Complete the following exercise 3. A triangle has one side 23 meters and another of 17 meters. Describe the possible lengths of the third side. Small values of x Large values of x 23 x x 17 x 17 23 x6 17 23 17 23 x 40 x , or x 40 The length of the third side must be greater than 6 meters or less than 40 meters 5.5 Use Inequalities in a Triangle Theorem 5.13: Exterior Angle Inequality Theorem The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior angles. m1 m2 ____ m1 m3 ____ 3 2 1 5.5 Use Inequalities in a Triangle Example 3 Relate exterior and interior angles Write inequaliti es that relate m1 to mB and mC. Solution B and C are nonadjacen t interior angles to 1. B So, by the Exterior Angle Inequality Theorem, 70 _____ m1 60 m1 > 70 and ______>_____. o o 1 A o 60 o C 5.5 Use Inequalities in a Triangle Checkpoint. Complete the following exercise 4. Write inequaliti es that relate m1 to mA and mB. B 1 o 38 C 112 A o A and B are nonadjacen t interior angles to 1. m1 112 o m1 38 o 5.5 Use Inequalities in a Triangle Pg. 299, 5.5 #1-23