5.2.4 Applying Special Right Triangles Key Objectives

• Justify and apply properties of 45 ° -45 ° -90 ° triangles. • Justify and apply properties of 30 ° -60 ° -90 ° triangles. Theorems, Postulates, Corollaries, and Properties • 45 ° -45 ° -90 ° Triangle Theorem In a 45 ° -45 ° -90 ° triangle, both legs are congruent, and the length of the hypotenuse is the length of a leg times 2 . • 30 ° -60 ° -90 ° Triangle Theorem In a 30 ° -60 ° -90 ° triangle, the length of the hypotenuse is 2 times the length of the shorter leg, and the length of the longer leg is the length of the shorter leg times 3 . Example 1

The 45 ° -45 ° -90 ° Triangle Theorem describes the relationship between the lengths of the legs and the length of the hypotenuse in right triangles with angles 45 ° , 45 ° , and 90 ° . In these triangles, the two legs are the same length (congruent), and the length of the hypotenuse is the length of a leg times 2 . The 45 ° -45 ° -90 ° Triangle Theorem is used to determine the unknown length of the hypotenuse of a triangle in this example. The length of one leg is given as 11 units and the measure of its opposite angle is given as 45 ° . First, use the Triangle Sum Theorem to determine that the unknown angle in the triangle is 45 ° (180 ° − 90 ° − 45 ° = 45 ° ). It follows from the 45 ° -45 ° -90 ° Triangle Theorem that the length of the hypotenuse, x , is the length of a leg times 2, or 11 2. The 45 ° -45 ° -90 ° Triangle Theorem is used here to determine the unknown lengths of the legs of a triangle. It is given that the triangle is an isosceles right triangle. This means it is a 45 ° -45 ° -90 ° triangle. The hypotenuse is given as 9 units. By the 45 ° -45 ° -90 ° Theorem, the length of the hypotenuse is the length of a leg times 2 . Thus, = x 2 9. Solve for x , the length of a leg, to yield = x 9 2/2.

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