# 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.

To Prove: ∠ APB + ∠ BOA = 180°

Proof:

Let us consider a circle centred at point O.

Let P be an external point from which two tangents PA and PB are drawn to the circle which touches the circle at point A and B respectively and AB is the line segment, joining point of contacts A and B together such that it subtends

AOB at centre O of the circle.

It can be observed that

OA PA (radius of circle is always perpendicular to tangent)

Therefore, OAP = 90°

Similarly, OB PB

OBP = 90°

Sum of all interior angles = 360°

OAP + APB+ PBO + BOA = 360°

90° + APB + 90° + BOA = 360°

APB + BOA = 180°

Hence, it can be observed 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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