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.

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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.

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Accepted Answer

Step 1: PA and PB are tangents from the external point P to the circle with centre O.
$\angle OAP = \angle OBP = 90°$
Step 2: In quadrilateral $OAPB$,
$\angle APB + \angle OAP + \angle OBP + \angle AOB = 360°$
Step 3: $\Rightarrow \angle APB + 90° + 90° + \angle AOB = 360°$
$\Rightarrow \angle APB + \angle AOB = 180°$
$	herefore \angle APB$ and $\angle AOB$ are supplementary.

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