In Fig., 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 ∠RSQ.

Given: ∠RPQ=30°

To find: The value of ∠RSQ.

Theorem Used:

The length of two tangents drawn from an external point are equal.

Explanation:

As P is an external point and PQ and PR are two tangents drawn from it,

From the theorem stated,

PQ = PR

∴ PQR is an isosceles triangle

⇒ ∠RQP=∠QRP
⇒ ∠RQP+∠QRP+∠RPQ=180°
⇒ 2∠RQP+30° =180°
⇒ 2∠RQP=150°
⇒ ∠RQP=∠QRP=75°

By alternate segment theorem which states that an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

∠RQP=∠RSQ=75°

As RS||PQ,

∴ ∠RQP = ∠SRQ = 75° (alternate angles)

So,

∠RSQ = ∠SRQ = 75°

SQ = RQ (sides opp. to equal angles)

∴QRS is an isosceles triangle.

In ΔQRS,

∠RSQ + ∠SRQ + ∠RQS = 180° (angle sum property)

⇒ 75° +75° + ∠RQS = 180°

⇒ 150° + ∠RQS = 180°

⇒ ∠RQS = 30°