Question

In the adjoining figure tangent PQ and PR are drawn to the circle such that angle RPQ 30 degree A chord RS is drawn parallel to the tangent PQ find angle RQS

Answer 1

PQ = PR
Since tangents drawn from an external point to a circle are equal.
And PQR is an isosceles triangle
thus, ∠RQP = ∠QRP
∠RQP + ∠QRP + ∠RPQ = 180° [Angle sum property of a triangle]
2∠RQP + 30° = 180°
2∠RQP = 150°
∠RQP = ∠QRP = 75°
∠RQP = ∠RSQ = 75° [ Angles in alternate Segment Theorem states that angle between chord and tangent is equal to the angle in the alternate segment]
RS is parallel to PQ
Therefore ∠RQP = ∠SRQ = 75° [Alternate angles]
∠RSQ = ∠SRQ = 75°
Therefore QRS is also an isosceles triangle
∠RSQ + ∠SRQ + ∠RQS = 180° [Angle sum property of a triangle]
75° + 75° + ∠RQS = 180°
150° + ∠RQS = 180°
∠RQS = 30°