Question

if the opposite angles of quadrilateral are supplementry then prove that quadrilateral is cyclic

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°

Answer 2

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