Question

Prove that the sum of opposite angles of a cylic quadrilateral is 180

Answer 1

Let ABCD be a cyclic quadrilateral with center of the circle at O.
Join BO and DO.
 
Arc DAB makes ∠2x  at center O and hence ∠x (half of angle at O) at C.
Arc DCB makes ∠2y  at  O and hence ∠y (half of angle at O) at A.

We know ,   ∠2x + 2y = 360°        =>  ∠x + ∠y = 180°
Hence,   ∠A + ∠C = x+y = 180°

So   ∠B + ∠D = 360 - ∠A + ∠C = 180°

Hence Proved