Question

In Fig. $PQR=100°,$where $P,Q$ and $R$are points on a circle with center $O.$Find $OPR.$

Answer 1

Given, $\angle PQR=100°$

Reflex$\angle POR=2\angle PQR$[The angle subtended by an arc at the center is double the arc subtended by it any point on the remaining point of the circle]

Reflex$\angle POR=2\times 100=200°$

So,$\angle POR=360°-200°$

$=160°$

In $△POR,OP=QR$ [Radii of the same circle]

So, $\angle P=\angle R$$=x$ (say)

$\therefore \angle P+\angle O+\angle R=180°$[Angle sum property of triangle]

$\Rightarrow$ $x+160+x=180$

$\Rightarrow$ $2x+160=180$

$\Rightarrow$ $2x=180-160$

$\Rightarrow$ $2x=20$

$\Rightarrow$ $x=10°$

Hence the value of $\angle OPR=10°.$