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In figure, tangents PQ and PR are drawn ...

In figure, tangents PQ and PR are drawn to a circle such that `angleRPQ=30^(@)`. A chord RS is drawn parallel to the tangent PQ. Find the `angleRQS`.

Text Solution

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Since, PQ and PR are the tangents to a circle from an external point P.
`:." "PQ=PR`
`implies" "anglePRQ=anglePQR" "`(angles opposite to equal sides are equal)
In `trianglePQR`
`anglePQR+anglePRQ+angleQPR=180^(@)`
`implies" "anglePQR+anglePQR+30^(@)=180^(@)`
`implies" "2anglePQR=150^(@)" "implies" "angle75^(@)`
Again`" "SR || QP`
`:." "SRQ=RQP=75^(@)`
Thus,`" "PQR=QSR=75^(@)" "`(by alternate segment theorem)
In `triangleQRS,`
`angleQRS+angleRSQ+angleSQR=180^(@)`
`" "angleSQR=180^(@)-75^(@)-75^(@)`
`implies" "angleRQS=30^(@)`
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