Question

In the following figure, $Q$ is the centre of a circle and $PM,PN$ are tangent segments to the circle. If $\angle MPN={30}^{\circ }$, find $\angle MQN$

Answer 1

In the given figure, $Q$ is the centre of the circle.

$PM$ and $PN$ are tangents from an external common point ${}^{\prime }{P}^{\prime }$.
$\angle MPN={30}^{\circ }$, to find $\angle MQN$
In $\square PMQN$,
$\angle PMQ=\angle PNQ={90}^{\circ }$
(Radius is $\perp$ to tangent at point of contact from the centre)$
$\therefore \angle PMQ+\angle PNQ+\angle MPN+\angle MQN={360}^{\circ }$
(Sum of measures of interior angles of quadrilateral)
$\therefore {90}^{\circ }+{90}^{\circ }+{30}^{\circ }+\angle MQN={360}^{\circ }$
$\angle MQN={360}^{\circ }-({90}^{\circ }+{90}^{\circ }+{30}^{\circ })$
$\angle MQN={150}^{\circ }$.