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: $\angle PAO = 90°$ (radius $\perp$ tangent). $\angle PBO = 90°$
Step 2: Now $\angle PAO + \angle AOB + \angle OBP + \angle BPA = 360°$
Step 3: $\Rightarrow 90° + \angle AOB + 90° + \angle BPA = 360° \Rightarrow \angle AOB + \angle BPA = 180°$. Hence supplementary

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