5.2.1 Indirect Proof and Inequalities in One Triangle (continued) Example 2 Ordering Triangle Side Lengths and Angle Measures
There is a relationship between the lengths of a side of a triangle and the measures of the opposite angle. The first theorem here states that for any two noncongruent sides of a triangle, the angle opposite of the longer of the two sides will be larger than the angle opposite of the shorter side. Therefore, if the lengths of the three sides of a triangle are known, then the measures of the three angles can be ordered from smallest to largest where the largest angle is opposite of the longest side. The second theorem relates a triangle’s angle measures to the lengths of its sides. For any two noncongruent angles in a triangle, the side opposite of the larger of the two angles will be longer than the side opposite of the smaller angle. Therefore, if a triangle’s angle measures are known, then the lengths of the three sides can be ordered from shortest to longest where the longest side is opposite of the largest angle. The triangle’s three side lengths are given in this example. To relate the angle measures, first relate the side lengths. The side lengths from shortest to longest are LN , MN , and LM , since 16.2 < 22.4 < 22.5. So, the angles from smallest to largest must be the angle opposite of LN , then the angle opposite of MN , and then the angle opposite of LM . Therefore, the angles from smallest to largest are ∠ M , ∠ L , and ∠ N . In this example, only two of the triangle’s angle measures are given. So, before the lengths of all three sides can be related, the measure of the third angle, ∠ D , must be found. By the Triangle Sum Theorem, the sum of the measures of the three angles must be 180 ° . Therefore, 68 ° + 46 ° + m ∠ D = 180 ° . Solve this equation to find m ∠ D . Once the measures of the three angles are known, use these measures to relate the lengths of the sides where the shortest side is opposite of the smallest angle and the longest side is opposite of the largest angle.
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