Question

In the given figure, O is the centre of the circle. PQ is a chord of the circle and R is any point on the circle. If $\angle PRQ=l$ and $\angle OPQ=m,$ then find $l+m$.

Answer 1

Since the angle subtended by an arc at the centre of a circle is double the angle subtended by it on any remaining point of the circle, we have
$\angle POQ=2\angle PRQ=2l$

In $△PQO,OP=OQ$ [Radii of the same circle]
Since angles opposite to equal sides are equal,
$\angle OQP=\angle OPQ=m$
Using angle sum property,
$\angle OPQ+\angle OQP+\angle POQ={180}^{\circ }$
$⟹m+m+2l={180}^{\circ }$
i.e.,$2(l+m)={180}^{\circ }$
$⟹l+m={90}^{\circ }$