11.2.1 Inscribed Angles (continued)
The angle measures of a quadrilateral inscribed in a circle are determined in this example. The measures of three of the angles are given as algebraic expressions in one variable, x . Since ∠ P and ∠ R are opposite angles and therefore supplementary, the sum of their measures is 180 ° . Substitute the expressions for the measures of the angles into the equation and solve for x . The solution yields x = 9. Substitute this value of x back into the expressions for the angles to find their measures. The solutions yield m ∠ P = 80 ° , m ∠ R = 100 ° , and m ∠ Q = 120 ° . The sum of the angles in a quadrilateral is 360 ° , so m ∠ S = 360 ° − (80 ° + 100 ° + 120 ° ) = 60 ° . Or, since ∠ S is opposite ∠ Q , m ∠ S = 180 ° − 120 ° = 60 ° .
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