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Question

In the given figure, tangents PQ and PR are down to a circle such that angle RPQ=300. A chord RS is drawn parallel to the tangent PQ. Find the angle RQS.
1266656_6df471650e594d03b4bf754badbcc730.PNG

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Solution

Given. RPQ=300 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)
PRQ=PQR [angles opposite to equal sides are equal in a le]
In PQRRQP+QRP+RPQ=1800 (angle sum property of a u)
2RQP+300=1800RQP=1500RQP=75
Hence RQP=QRP=750RQP=RSQ=750 (by alternate segment theorem)
Given, RS||PQ RQP=SRQ=750 (Alternate angles)
RSQ=SRQ=750
QRS is also an isosceles u. (since ride opposite to equal angles of a u are equal )
RSQ+SRQ+RQS=1800
(angle sum property of a u)
750+750KQS=1800
1500+RQS=1800
RQS=300

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