Question

Two tangents PA and PB are drawn from an external point P to a circle with centre O, such that ∠ APB = x and ∠ AOB = y prove than opposite angles are supplementary.

Answer 1

It is given that Two tangents and are drawn from an external point to a circle with centre O, such that ∠APB and ∠AOB = y 

We know that the tangent at a point to a circle is perpendicular to radius through that point.

Therefore OA ⟂ AP and OB ⟂ BP 

So ∠OAB = 90° 

⇒ ∠OBP = 90° 

⇒ ∠OAP + ∠OBP = 90° + 90°

⇒ ∠OAP + ∠OBP = 180° …….. (i) 

In a quadrilateral AOBP the sum of all angles = 360° 

So ∠OAP + ∠OBP + ∠APB + ∠AOB = 360° 

⇒ 180° + ∠APB + ∠AOB = 360°  [from (i)] 

⇒ ∠APB + ∠AOB = 360° − 180° 

⇒ ∠APB + ∠AOB = 180° 

Hence, the sum of opposite angles are supplementary.

Answer 2

Since, AO BP is a quadrilateral

or, opp.  angle are supplementary.