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°