4.1.2 Angle Relationships in Triangles (continued)
The figure given here is a four-sided polygon that is divided into three triangles: △ ABE , △ BDE , and △ BCD . The measures of four angles are given. First, find m ∠ CBD . Notice that ∠ CBD is an angle in △ BCD and the measures of the other two angles in △ BCD are given, m ∠ C = 93° and m ∠ BDC = 42°. Use the Triangle Sum Theorem (sum of a triangle’s three angles is 180°) to write an equation and solve the equation to find m ∠ CBD . Now, find m ∠ EAB . Notice that ∠ EAB is an angle in △ ABE . However, the measure of only one other angle in △ ABE is given, m ∠ AEB = 48°. So, m ∠ ABE must be found before the Triangle Sum Theorem can be used. Since ∠ ABE is one of three angles that form a straight angle and the measures of the other two angles are known, use the definition of a straight angle (a straight angle measures 180°) to write an equation. m ∠ ABE + m ∠ EBD + m ∠ CBD = 180° Now substitute the known measures of ∠ EBD and ∠ CBD into the equation and solve to find m ∠ ABE . Once m ∠ ABE is found, the Triangle Sum Theorem can be used with △ ABE to find m ∠ EAB .
Example 2 Finding Angle Measures in Right Angles
The first corollary states that in a right triangle, the two acute angles must be complementary (their sum is 90°). This corollary follows from the Triangle Sum Theorem. By the Triangle Sum Theorem, the sum of the angle measures of a triangle is always 180°. In a right triangle, one of those three angles is 90°. So, the sum of the other two angles must be 180° − 90°, or 90°, which means those two angles are complementary. The second corollary states that each angle measures 60° in an equiangular triangle. Again, by the Triangle Sum Theorem, the sum of the angle measures of a triangle is always 180°. In an equiangular triangle all three angles are congruent. Therefore, the measure of each angle is 180°/3, or 60°.
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