In Fig. 10.88, APB is a tangent to a circle with centre O at point P. If ∠QPB = 500, then the measure of ∠POQ is
Given:
∠QPB =50°
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, ∆OPB is right-angled at ∠OPB (i.e., ∠OPB = 90°).
∠OPQ = ∠OPB – ∠QPB
⇒ ∠OPQ = 90° – 50° = 40°
And,
∠OPQ = ∠OQP [∵ OP = OQ (radius of circle)]
Now by property 2,
∠OPQ + ∠OQP + ∠POQ = 180°
⇒ 40° + 40° + ∠POQ = 180°
⇒ 80° + ∠POQ = 180°
⇒ ∠POQ = 180° - 80°
⇒ ∠POQ = 100°
Hence, ⇒ ∠POQ = 100°