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 at the center.

Let us consider a circle with center O and PA and PB are two tangents to the circle from an external point P

To Prove : Angle between two tangents is supplementary to the angle subtended by the line segments joining the points of contact at center, i.e. APB + AOB = 180°

Proof :

As AP and BP are tangents to given circle,

We have,

OA AP and OB BP

[Tangents drawn at a point on circle is perpendicular to the radius through point of contact]

So, OAP = OBP = 90°

OAP + OBP + AOB + APB = 360°

90° + 90° + AOB + APB = 360°

AOB + APB = 180°

Hence Proved

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