In Fig. 7, tangents PQ and PR are drawn from an external point P to a circle with centre O, such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS.
We know that tangents from an external point are equal in length.
∴ PQ = PR
PQ = PR
∴ ∠PQR=∠PRQ (Angles opposite to equal sides are equal.)
Now in ∆PQR,
Also, radius is perpendicular to the tangent at the point of contact.
∴ ∠OQP = ∠ORP = 90°
Now, in □PQOR,
∠RPQ + ∠ORP + ∠OQP + ∠QOR = 360°
⇒ 30° + 90° + 90° + ∠QOR = 360°
⇒ ∠QOR = 360° − 210° = 150°
Since, angle subtended by an arc at any point on the circle is half the angle subtended at the centre by the same arc.
Also, ∠QSR = ∠SQT (Alternate interior angles)
∴ ∠SQT = 75°
∠SQT + ∠RQS + ∠PQR = 180°
⇒ 75° + ∠RQS + 75° = 180°
⇒ ∠RQS = 180° − 150°