Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angle at the center of the circle.
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Solution
Data : Circle withe 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.