1.1.3 Measuring and Constructing Angles (continued) Example 4 Finding the Measure of an Angle

When angles have the same measure, they are called congruent angles. For example, if the measure of ∠ 1 is 100° and the measure of ∠ 2 is 100°, then the angle measures are equal. m ∠ 1 = m ∠ 2 But the angles themselves are congruent . ∠ 1 ≅ ∠ 2 When a ray is an angle bisector, it divides an angle into two congruent angles. For example, if AC bisects ∠ BAD , then the two angles created by this ray are congruent. ∠ BAC ≅ ∠ CAD In this example, the fact that ∠ XZW is bisected by a ray is given. Therefore, by the definition of angle bisector, ∠ XZW is divided into two congruent angles. ∠ XZY ≅ ∠ YZW So by the definition of congruent angles, m ∠ XZY = m ∠ YZW . Algebraic expressions are given for the measures of ∠ XZY and ∠ YZW . Substitute these expressions into the equation and solve for x . m ∠ XZY = m ∠ YZW (7 x + 1)° = (5 x + 11)° Thus, x = 5. Note that the value of x is not the answer to the question. The value of x must be used to find m ∠ XZW . First, substitute 5 for x into either expression given for the small angles. Then, double that result to get m ∠ XZW .

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