Question

In the figure given, tangents $PQ$ and $PR$ are drawn from an external point $P$ to a circle with centre $O$, such that $\angle RPQ={30}^{\circ }$, A chord $RS$ is drawn parallel to the tangent $PQ$. Find $\angle RQS$.

Answer 1

Given, $RS\parallel PQ$ and $\angle RPQ={30}^o$

$PR=PQ$ (equal tangents)

$\angle PRQ=\angle PQR$ (angle opposite to equal sides)

In $△PQR$

$\angle PRQ+\angle PQR+\angle RQP={180}^{o}2\angle PQR+{30}^o={180}^o\angle PQR={75}^o$

$\angle SRQ=\angle PQR={75}^o$ (alternate angle)

Also, $\angle RQP=\angle RSQ={75}^o$ (alternate segment angles)

$\angle RSQ=\angle SRQ={75}^o$

In $△QSR$

$\angle RSQ+\angle SRQ+\angle SQR={180}^{o}75+75+\angle SQR=180\angle SQR={30}^o\angle RQS={30}^o$