Question

In the given figure, tangents PQ and PR are down to a circle such that angle $RPQ={30}^0$. A chord RS is drawn parallel to the tangent PQ. Find the angle RQS.

Answer 1

Given. $\angle RPQ={30}^0$ and $PR$ and $PQ$ are tangents drawn from $P$ to the same circle.

Hence $PR=PQ$ since tangents drawn from an external point to a circle are equal in length)

$\therefore \angle PRQ=\angle PQR$ [angles opposite to equal sides are equal in a $△le$]

In $△PQR\angle RQP+\angle QRP+\angle RPQ={180}^0$ (angle sum property of a ${△}^u$)

$2\angle RQP+{30}^0={180}^0\angle RQP={150}^0\angle RQP=75$

Hence $\angle RQP=\angle QRP={75}^0\angle RQP=\angle RSQ={75}^0$ (by alternate segment theorem)

Given, $RS||PQ$ $\therefore RQP=\angle SRQ={75}^0$ (Alternate angles)

$\angle RSQ=\angle SRQ={75}^0$

$\therefore QRS$ is also an isosceles ${△}^u$. (since ride opposite to equal angles of a ${△}^u$ are equal )

$\angle RSQ+\angle SRQ+\angle RQS={180}^0$

(angle sum property of a ${△}^u$)

${75}^0+{75}^0\angle KQS={180}^0$

${150}^0+\angle RQS={180}^0$

$\therefore \angle RQS={30}^0$