#### In Figure, tangents PQ and PR are drawn to a circle such that $\angle$RPQ = $30^{\circ}$. A chord RS is drawn parallel to the tangent PQ. Find the $\angle$RQS.

Solution
In the given figure PQ and PR are two tangents drawn from an external point P.
$\therefore$ PQ = PR      [$\mathbb{Q}$ lengths of tangents drawn from on external point to a circle are equal]
$\Rightarrow \angle PQR=\angle QRP$   [  angels opposite to equal sides are equal]
In $\bigtriangleup$PQR
$\angle PQR+\angle QRP+\angle RPQ= 180^{\circ}$
[  sum of angels of a triangle is 180°]
$\angle PQR+\angle PQR+\angle RPQ= 180^{\circ}$
$\left [ \angle PQR= \angle QRP \right ]$
$2\angle PQR+30^{\circ}= 180^{\circ}$
$\angle PQR= \frac{180^{\circ}-30^{\circ}}{2}$
$\angle PQR= \frac{150^{\circ}}{2}$
$\angle PQR= 75^{\circ}$

SR||OP (Given)

$\therefore$  $\angle SRQ= \angle RQP= 75^{\circ}$ [Alternate interior angles]

Also     $\angle PQR= \angle QRS= 75^{\circ}$   [Alternate segment angles]
In $\bigtriangleup$QRS
$\angle Q+\angle R+\angle S= 180^{\circ}$
$\angle Q+75^{\circ}+75^{\circ}= 180^{\circ}$
$\angle Q= 180^{\circ}-75^{\circ}-75^{\circ}$
$\angle Q= 30^{\circ}$
$\therefore \angle RQS= 30^{\circ}$