Question

Prove that If one pair of opposite angles of a quadrilateral is supplementary, then quadrilateral is cyclic .

Answer 1

Given : quadrilateral ABCD, in which ∠ABC + ∠ADC = 180° and ∠BAD + ∠BCD = 180°

To Prove : ABCD is a cyclic quadrilateral.

Construction : Draw a circle passing through points A, B, C but not D. But cuts AD at E. Join EC

Proof : quadrilateral ABCE is cyclic

Thus, ∠ABC + ∠AEC = 180° …..(i)

But given that

∠ABC + ∠ADC = 1800 ……(ii)

From equations (i) and (ii)

∠AEC = ∠ADC ……(iii)

But It is impossible because one is exterior and other is interior angle of ΔCED. it is possible only when D and E coincides so circle will pass through A, B, C and D 

i.e., ABCD is a cyclic quadrilateral.