Question

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Answer 1

Data: Circle with centre ‘O’ is circumscribed in a quadrilateral ABCD. 

To Prove: ∠AOD + ∠BOC = 180° 

Sides of the quadrilateral AB, BC, CD and DA touches at the points P, Q, R and S respectively. 

OA. OB, OC, OD and OP, OQ, OR, OS are joined. 

OA bisects ∠POS. 

∠1 =∠2 

∠3 = ∠4 

∠5 = ∠6 

∠7 = ∠8 

2(∠1 + ∠4 + ∠5 + ∠8) = 360° 

(∠1 + ∠8) (∠4 + ∠5) = 180° 

∴ ∠AOD = ∠BOD = 180°

∴ Opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.