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In Figure, tangents PQ and PR are drawn to a circle such that \angleRPQ = 30^{\circ}. A chord RS is drawn parallel to the tangent PQ. Find the \angleRQS.

Answers (1)

In the given figure PQ and PR are two tangents drawn from an external point P. 
\therefore PQ = PR      [\mathbb{Q} lengths of tangents drawn from on external point to a circle are equal]
\Rightarrow \angle PQR=\angle QRP   [  angels opposite to equal sides are equal]
  In \bigtriangleupPQR
\angle PQR+\angle QRP+\angle RPQ= 180^{\circ}
                              [  sum of angels of a triangle is 180°]
\angle PQR+\angle PQR+\angle RPQ= 180^{\circ}
\left [ \angle PQR= \angle QRP \right ]
2\angle PQR+30^{\circ}= 180^{\circ}
\angle PQR= \frac{180^{\circ}-30^{\circ}}{2}
\angle PQR= \frac{150^{\circ}}{2}
\angle PQR= 75^{\circ}

SR||OP (Given)

\therefore  \angle SRQ= \angle RQP= 75^{\circ} [Alternate interior angles]      

  Also     \angle PQR= \angle QRS= 75^{\circ}   [Alternate segment angles]    
In \bigtriangleupQRS
\angle Q+\angle R+\angle S= 180^{\circ}
\angle Q+75^{\circ}+75^{\circ}= 180^{\circ}
\angle Q= 180^{\circ}-75^{\circ}-75^{\circ}
\angle Q= 30^{\circ}
\therefore \angle RQS= 30^{\circ}

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