In figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find the ∠RQS.

Given : Tangents PQ and PR are drawn to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ.

To Find : ∠RQS

PQ = PR [Tangents drawn from an external point to a circle are equal]

∠PRQ = ∠PQR [Angles opposite to equal sides are equal] [1]

In △PQR

∠PRQ + ∠PQR + ∠QPR = 180°

∠PQR + ∠PQR + ∠QPR = 180° [Using 1]

2∠PQR + ∠RPQ = 180°

2∠PQR + 30 = 180

2∠PQR = 150

∠PQR = 75°

∠QRS = ∠PQR = 75° [Alternate interior angles]

∠QSR = ∠PQR = 75° [angle between tangent and the chord equals angle made by the chord in alternate segment]

Now In △RQS

∠RQS + ∠QRS + ∠QSR = 180

∠RQS + 75 + 75 = 180

∠RQS = 30°

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