In the given figure, tangents PQPQ and PRPR are drawn from an external point PP to a circle with center OO, such that RPQ=60RPQ=60. A chord RSRS is drawn parallel to the tangent PQPQ. Find the measure of RQSRQS.
R O P Q S 60°


Answer:

6060

Step by Step Explanation:
  1. Let us join OQOQ and OROR. Also, produce PQPQ and PRPR to MM and NN respectively.
    R O N M P Q S 60°
  2. 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, RPQ+ROQ=180ROQ=180RPQ=18060=120
  3. 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, RSQ=12ROQ=12×120=60 As, RS//PQ, SQM=RSQ=60 [Alternate Interior Angles]  Also, PQR=RSQ=60 [Alternate Segment Theorem] 
  4. We know that the sum of angles on a straight line is 180.

    As PM is a straight line. SQM+RQS+PQR=180 Therefore, RQS=180(SQM+PQR)=180(60+60)=60
  5. Thus, the measure of RQS is 60.

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