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 segments joining the points of contact to the center

Answer 1

Given : PA and PB are the tangent drawn from a point P to a circle wirth center O .

Also , the line segments OA and Ob are drawn.

To prove : $\angle$ APB + $\angle$ AOB = ${180}^{\circ }$

Proof : We know that the tangents to a circle is perpendicular to the radius through the points of contact .

$\therefore$ , PA $\perp$ OA $\Rightarrow$ OAP = ${90}^{\circ }$ and

PB $\perp$ OB $\Rightarrow$ $\angle$ OBP = ${90}^{\circ }$

Therefore , $\angle$ OAP + $\angle$ OBP = ${90}^{\circ }$

hence , $\angle$ APB + $\angle$AOB = ${180}^{\circ }$

[Sum of the all the angles of a quadrilateral is ${360}^{\circ }$]