Question

how to prove that the sum of opposite side of a cyclic quadrilateral is 180 degree

Answer 1

Let ABCD is cyclic quadrilateral. To prove : ∠A + ∠C = 180°  and ∠B + ∠D = 180°. Construction : join OB and OD. 
Proof : ∠BOD = 2 ∠BAD     
∠BAD = 1/2∠ BOD
Similarly ∠BCD = 1/2 ∠DOB
 ∠BAD + ∠BCD = 1/2∠BOD + 1/2∠DOB                 
  =1/2(∠ BOD + ∠DOB)     
              = (1/2)X360° = 180°
 Similarly ∠B + ∠D = 180°

hence proved