Question

question 5 of 10 what is the measure of ∠prq? a. 86° b. 70° c. 172° d. 102°

Explanation:

Step1: Recall angle - arc relationship

The measure of an inscribed angle is half the measure of its intercepted arc.

Step2: Find the sum of arcs

The sum of the arcs of a circle is $360^{\circ}$. Let the arc $PT$ be $x$ and arc $QS$ be $y$. Given arc $PQ = 102^{\circ}$ and arc $ST=70^{\circ}$, then $x + y+102^{\circ}+70^{\circ}=360^{\circ}$, so $x + y=360^{\circ}-(102^{\circ}+70^{\circ}) = 188^{\circ}$.

Step3: Calculate $\angle PRQ$

$\angle PRQ$ is an inscribed angle that intercepts the sum of arcs $PT$ and $QS$. By the inscribed - angle theorem, $m\angle PRQ=\frac{1}{2}(m\overset{\frown}{PT}+m\overset{\frown}{QS})$. Since $m\overset{\frown}{PT}+m\overset{\frown}{QS}=188^{\circ}$, then $m\angle PRQ=\frac{188^{\circ}}{2}=86^{\circ}$.

Answer:

A. $86^{\circ}$