Question

If $\angle POR=120°$, then the value of $\angle PQR=$ .

• A. 130°
• B. 60°
• C. 90°
• D. 120°

Answer 1

The correct option is D 120°

Given $\angle POR=120°.$
Consider the arc PQR. Consider a point S not on this arc, but on the circumference of the circle. Join PS and RS.

Arc PQR subtends $\angle POR$ at centre and $\angle PSR$ on the remaining part of the circle (at the point S).
We know that, the angle subtended by an arc at the centre is double the angle subtended by it on any remaining point of the circle, we have
$\angle PSR=\frac{1}{2}\angle POR.$
$⟹\angle PSR=\frac{1}{2}\times 120°=60°$
Since PQRS is a cylic quadrilateral, we have its opposite angles to be supplementary.
$⟹\angle PQR+\angle PSR=180°$
$⟹\angle PQR+60°=180°$
$⟹\angle PQR=180°-60°=120°$