Question

In the following figure, $Q$ is the centre of a circle and $PM,PN$ are tangent segments to the circle. If $\angle MPN={50}^{\circ }$, find $\angle MQN$ (in ${}^{\circ }$).

Answer 1

In the given figure of the question, $Q$ is the centre of the circle, and $PM$ and $PN$ are tangents from an external common point ${}^{\prime }{P}^{\prime }$.
$\angle MPN={50}^{\circ }$
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)
$\therefore {90}^{\circ }+{90}^{\circ }+{50}^{\circ }+\angle MQN={360}^{\circ }$
$\angle MQN={360}^{\circ }-({90}^{\circ }+{90}^{\circ }+{50}^{\circ })$
$={360}^{\circ }-{230}^{\circ }={130}^{\circ }$.