Question

$PQRS$ is a cyclic quadrilateral. Given $\angle$QPS $={73}^{\circ }$, $\angle PQS={55}^{\circ }$ and $\angle PSR={82}^{\circ }$, calculate $\angle RQS.$

Answer 1

Here, $PQRS$ is a cyclic quadrilateral.

We know, in a cyclic quadrilateral, the sum of opposite angles are supplementary, i.e. ${180}^{\circ }$.

Here, $\angle PSR={82}^{\circ }$ and $\angle PQR$ are opposite angles.

Then, $\angle PSR+\angle PQR={180}^{\circ }$

$\Rightarrow {82}^{\circ }+\angle PQR={180}^{\circ }$

$\Rightarrow \angle PQR={180}^{\circ }-{82}^{\circ }$

$\Rightarrow \angle PQR={98}^{\circ }$

Now,

$\angle PQS+\angle RQS=\angle PQR$

$\Rightarrow$ ${55}^{\circ }+\angle RQS={98}^{\circ }$

$\Rightarrow$ $\angle RQS={98}^{\circ }-{55}^{\circ }$

$\Rightarrow \angle RQS={43}^{\circ }$.

Hence, $\angle RQS$ is equal to ${43}^{\circ }.$