In Fig. 10.95, PQ and PR are tangents drawn from P to a circle with centre O. If ∠OPQ = 35°, then
Given:
∠OPQ = 35°
Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal.
Property 2: The tangent at a point on a circle is at right angles to the radius obtained by joining center and the point of tangency.
Property 3: The sum of all angles of a triangle = 180°.
By property 1,
QP = QR (tangent from Q)
By property 2, ∆OPQ is right-angled at ∠OQP (i.e., ∠OQP = 90°) and ∆ORP is right-angled at ∠ORP (i.e., ∠ORP = 90°).
∴ OQ ⊥ QP
OR⊥ RP
Now,
∠OQP = ∠ORP = 90° [Property 1]
QP = QR [Property 2]
OP = OP [Given]
∴ ∆OPQ ≅ ∆OPR By SAS
Hence, ∠OPQ = ∠OPR = 35° By CPCTC
i.e. ∠a = 35°
By property 3,
∠OQP + ∠OPQ + ∠QOP = 180°
⇒ 90° + 35° + ∠QOP = 180°
⇒ 125° + ∠QOP = 180°
⇒ ∠QOP = 180° - 125°
⇒ ∠QOP = 55°
i.e. ∠b = 55°
Hence, ∠a = 35° and ∠b = 55°