In Fig. 10.85, if PR is tangent to the circle at P and Q is the centre of the circle, then ∠POQ =
Given:
∠RPQ = 60°
Property 1: 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 2: Sum of all angles of a triangle = 180°.
By property 1, ∆OPR is right-angled at ∠OPR (i.e., ∠OPR = 90°).
OP = OQ [∵ radius of circle]
∴ ∠OPQ = ∠OQP = 30°
Now by property 2,
∠OPQ + ∠OQP + ∠POQ = 180°
⇒ 30° + 30° + ∠POQ = 180°
⇒ 60° + ∠POQ = 180°
⇒ ∠POQ = 180° - 60°
⇒ ∠POQ = 120°
Hence, ⇒ ∠POQ = 120°