Question

In figure, $\angle$PQR$={100}^o$, where P, Q and R are points on a circle with centre O. Find $\angle$OPR.

Answer 1

If $\angle PQR={100}^o,$ then $\angle OPR=?$
$\angle PQR={100}^o$
$\therefore \angle POR=?$
$\angle POQ=2\times \angle PQR=2\times {100}^o$ {Property of the angle in the circle at center}
$\angle POR={200}^o$
$\therefore \angle POR+$ Reflex angle $\angle POR={360}^o$
$\therefore \angle POR=360-200$
$\therefore \angle POR={160}^o$

In $△POR,OP=OQ$ radii
$\therefore \angle OPR=\angle ORP$
$\therefore \angle OPR+\angle ORP+\angle POR={180}^o$
$\angle OPR+\angle OPR+160=180$
$2\angle OPR+160=180$
$2\angle OPR=180-160$
$2\angle OPR=20$
$\angle OPR=\frac{20}{2}$
$\therefore \angle OPR={10}^o$