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.

Let PA and PB are two tangents of circle with centre O. Join OA and OB.
We know that the radius is perpendicular to tangent. therefore
∠OAP = 90° and ∠OBP = 90°.
In quadrilateral OAPB, ∠OPA + ∠PBO + ∠BOA = 360°
90° + ∠APB + 90° + ∠BOA = 360° ∠APB + ∠BOA = 180°
Hence, it is proved 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.