Answer:
Option D. [tex]<R=62\°[/tex]
Step-by-step explanation:
step 1
Find the measure of angle F
we know that
In an inscribed quadrilateral, the opposite angles are supplementary
so
[tex]<Q+<F=180\°[/tex]
substitute the value of <Q
[tex]99\°+<F=180\°[/tex]
[tex]<F=180\°-99\°=81\°[/tex]
step 2
Find the measure of arc PQ
we know that
The inscribed angle measures half that of the arc comprising
so
[tex]<F=\frac{1}{2}(arc\ PQ+arc\ RQ)[/tex]
substitute values
[tex]81\°=\frac{1}{2}(arc\ PQ+92\°)[/tex]
[tex]162\°=(arc\ PQ+92\°)[/tex]
[tex]arc\ PQ=162\°-92\°=70\°[/tex]
step 3
Find the measure of angle R
we know that
The inscribed angle measures half that of the arc comprising
so
[tex]<R=\frac{1}{2}(arc\ FP+arc\ PQ)[/tex]
substitute values
[tex]<R=\frac{1}{2}(54\°+70\°)[/tex]
[tex]<R=\frac{1}{2}(124\°)[/tex]
[tex]<R=62\°[/tex]
