Q4 of 31 Page 218

Let us prove that for a quadrilateral circumscribed about a circle, the angles subtended by any two opposite sides at the centre are supplementary to each other.


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Let us assume ∠ OAD = ∠ OAB = a


∠ OBC = ∠ OBA = b


∠ OCD = ∠ OCB = c


∠ ODC = ∠ ODA = d


Since ABCD is a quadrilateral, so


2 (a + b + c + d) = 3600


⇒ a + b + c + d = 1800 …Equation (i)


In Δ AOB


∠ AOB = 1800- (a + b)


In Δ COD


∠ COD = 1800-(c + d)


∠ AOB + ∠ COD = 3600–(a + b + c + d)


Putting the value from Equation (i) we get


∠ AOB + ∠ COD = 3600–1800


⇒ ∠ AOB + ∠ COD = 1800


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