Question

what is the measure of ∠prq? 102° 70° a. 102° b. 172° c. 86° d. 70°

Explanation:

Step1: Recall circle - angle property

The sum of the measures of the arcs of a circle is 360°.

Step2: Find the measure of arc $PS$

Let the measure of arc $PS$ be $x$. Then $102^{\circ}+70^{\circ}+x = 360^{\circ}$. Solving for $x$ gives $x=360^{\circ}-(102^{\circ} + 70^{\circ})=188^{\circ}$.

Step3: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. $\angle PRQ$ is an inscribed angle that intercepts arc $PS$. So $m\angle PRQ=\frac{1}{2}m\overset{\frown}{PS}$.

Step4: Calculate the measure of $\angle PRQ$

Since $m\overset{\frown}{PS} = 188^{\circ}$, then $m\angle PRQ=\frac{1}{2}\times188^{\circ}=86^{\circ}$.

Answer:

C. $86^{\circ}$