Question

Question 7
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.

Answer 1

PQ and PR are two tangents drawn from an external point P.


$\therefore$ PQ = PR
[ the lengths of tangents drawn from an external point to a circle are equal]
$\Rightarrow \angle PQR=\angle QRP$
[ angles opposite to equal sides are equal]
Now, in $\Delta$ PQR $\angle PQR+\angle QRP+\angle RPQ={180}^{\circ }$
[ sum of all interior angles of any triangle is ${180}^{\circ }$]
$\Rightarrow$ $\angle PQR$ + $\angle PQR$ + ${30}^{\circ }={180}^{\circ }$
$\Rightarrow 2\angle PQR={180}^{\circ }-{30}^{\circ }$
$\Rightarrow \angle PQR=\frac{{180}^{\circ }-{30}^{\circ }}{2}={75}^{\circ }$
Since SR $\parallel$ QP
$\therefore \angle SRQ=\angle RQP={75}^{\circ }$ [alternate interior angles]
Also, $\angle PQR=\angle QSR={75}^{\circ }$ [ by alternate segment theorem]
In $\Delta$ QRS $\angle Q+\angle R+\angle S={180}^{\circ }$
[ sum of all interior angles of any triangle is ${180}^{\circ }$]
$\Rightarrow \angle Q={180}^{\circ }-({75}^{\circ }+{75}^{\circ })$
$={30}^{\circ }$
$\angle RQS={30}^{\circ }$