(Q1) Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line – segment joining the points of contact at the centre.

OPQT

Given A circle with center 'O'

Two tangents PQ and PT from an external point P.

Let Q, T be the points of contact

R.T.P ∠P and ∠QOT are supplementary

Proof OQ ⟂PQ

[radius is perpendicular to the tangent at the point of contact] also OT ⟂PT

∠OQP + ∠OTP = °+° = °

∠OTP+∠TPQ+∠PQO+∠QOT = 360°(angle sum property)

° + ∠P+∠QOT = 360°

∠P+∠QOT = 360° - ° = ° Hence proved