In the given figure, tangents ^@ PQ ^@ and ^@ PR ^@ are drawn fr | Math Question and Answer

Answer:

$60^\circ$

Step by Step Explanation:

Let us join $OQ$ and $OR$ . Also, produce $PQ$ and $PR$ to $M$ and $N$ respectively.
We know that the angle between two tangents from an external point is supplementary to the angle subtended by the radii at the center.
Thus, $\begin{aligned} & \angle RPQ + \angle ROQ = 180^\circ \implies \angle ROQ = 180^\circ - \angle RPQ = 180^\circ - 60^\circ = 120^\circ \end{aligned}$
We also know that the angle subtended by an arc at the center is twice the angle subtended by the same arc on the remaining part of the circle.
So, $\angle RSQ = \dfrac { 1 } { 2 } \angle ROQ = \dfrac { 1 } { 2 } \times 120^\circ = 60^\circ$ As, $RS//PQ,$ $\implies \angle SQM = \angle RSQ = 60^\circ \text{ [Alternate Interior Angles] }$ Also, $\angle PQR = \angle RSQ = 60^\circ \text{ [Alternate Segment Theorem] }$
We know that the sum of angles on a straight line is $180^\circ$ .

As $PM$ is a straight line. $\implies \angle SQM + \angle RQS + \angle PQR = 180^\circ$ Therefore, $\angle RQS = 180^\circ - (\angle SQM + \angle PQR) = 180^\circ - (60^\circ + 60^\circ) = 60^\circ$
Thus, the measure of $\angle RQS$ is $60^\circ$ .

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