Question

A circle touches the sides of the quadrilateral PQRS at W,X,Y,Z respectively. show that the angle subtended at the center by a pair of opposite sides are supplementary

Answer 1

Question:

A circle touches the sides of the quadrilateral PQRS at W,X,Y,Z respectively. show that the angle subtended at the center by a pair of opposite sides are supplementary

Answer:

Given:

The circle center O touches the side PQ, QR, RS, SP. Of a quadrilateral touches the point W, X, Y, Z

To prove:

∠POQ+∠ROS=180⁰ and

∠POS+∠QOR=180⁰

Construction:

Join, OA, OB, OC and OD

Proof:

since two tangents drawn from the external point to the circle subtend equal angle at the centre

Therefore,∠1=∠2,∠3=∠4,∠5=∠6 and ∠7=∠8

Now,∠1+∠2+∠3+∠4+∠5+∠6+∠7+∠8=360⁰

➜2+∠6+∠7)=360⁰

➜2(∠1+∠4+∠5+∠8)=360⁰

➜(∠2+∠3)+(∠6+∠7)=180⁰ And,

➜(∠1+∠4)+(∠5+∠8)=180⁰

[Therefore,∠2+∠3=∠AOB,

∠6+∠7=∠COD,

∠1+∠8=∠AOD and,

∠4+∠5=∠BOC]

➜∠AOB+COD=180⁰

➜∠AOD+∠BOC=180⁰

[HOPE THIS HELPS YOU.../]