Question

In the given figure, $\angle PQR={100}^{\circ },$ where P, Q and R are the points on a circle with centre O. What will be the value of $\angle OPR$?

• A. ${30}^{\circ }$
• B. ${20}^{\circ }$
• C. ${10}^{\circ }$
• D. ${40}^{\circ }$

Answer 1

The correct option is C ${10}^{\circ }$

Take any point A on the circumcircle of the circle outside of the arc PQR. Join AP and AR.



Consider the cyclic quadrilateral APQR so formed.
Since opposite angles are supplementary in a cyclic quadrilateral,
$\angle PAR+\angle PQR={180}^{\circ }.$

i.e., $\angle PAR+{100}^{\circ }={180}^{\circ }$
$⟹\angle PAR={80}^{\circ }$

Since $\angle POR$ and $\angle PAR$ are the angles subtended by the arc PQR at the centre of the circle and the remaining part of the circle, we have

$\angle POR=2\times \angle PAR$
$=2\times {80}^{\circ }={160}^{\circ }.$

In $△POR,$ we have
OP=OR (radius of the circle).
Since angles opposite to equal sides are equal,
$\angle OPR=\angle ORP.$
Using angle sum property,
$\angle POR+\angle OPR+\angle ORP={180}^{\circ }.$
$\Rightarrow {160}^{\circ }+\angle OPR+\angle OPR={180}^{\circ }$
$\Rightarrow 2\angle OPR={20}^{\circ }$
$\Rightarrow \angle OPR={10}^{\circ }$