Question

prove that if the sum of any pair of opposite angles of a quadrilateral is 180 then it is cyclic

Answer 1

Given : 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°