PQ is a tangent drawn from a point P to a circle with centre O and QOR is a diameter of the circle such that ∠POR = 120° , then ∠OPQ is

Given:

∠POR = 120°

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 straight line = 180°.

Property 3: Sum of all angles of a triangle = 180°.

By property 1,

∠PQO = 90°

By property 2,

∠POQ + ∠POR = 180°

⇒ ∠POQ + 120° = 180°

⇒ ∠POQ = 180° - 120°

⇒ ∠POQ = 60°

Now by property 3 in ∆OPQ,

∠POQ + ∠PQO + ∠OPQ = 180°

⇒ ∠OPQ = 180° - ∠POQ + ∠PQO

⇒ ∠OPQ = 180° - (60° + 90°)

⇒ ∠OPQ = 180° - 150°

⇒ ∠OPQ = 30°

Hence, ∠OPQ = 30°