Question

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 pls pls its urgent

Answer 1

in right Δ OAP and right Δ OBP, we havePA = PB [Tangents to circle from an external point P]OA = OB [Radii of the same circle]OP = OP [Common]∴ By SSS congruency,Δ OAP ≅ Δ OBP∴ Their corresponding parts are equal.∴ ∠OAA = ∠OPBAnd ∠AOP = ∠BOP⇒ ∠APB = 2 ∠OPA and ∠AOB = 2 ∠AOPBut ∠AOP = 90° − ∠OPA⇒ 2 ∠AOP = 180° − 2 ∠OPA⇒ ∠AOB = 180° − ∠APB⇒ ∠AOB + ∠APB = 180°.