Given   A cyclic quadrilateral ABCD.

TO PROVE 

CONSTRUCTION   Join AC and BD.

PROOF  Consider side AB of quadrilateral ABCD as the chord of the circle. Clearly, are angles in the same segment determined by chord AB of the circle.

         [ since angles in the same segment are equal] ....(i)

Now, consider the side BC of quadrilateral ABCD as the chord of the circle. We find that     are angles in the same segment

      [ Angles in the same segment are equal ] ....(ii)

Adding equations (i) and (ii) , we get

      [ The sum of the angle of a triangle is ]

Hence,  

The converse of this theorem is also true as given below.

Converse Theorem: If the sum of any pair of opposite angles of a quadriletral is , then the quadrilateral is cyclic.

Given: A quadrilateral ABCD in which

Tp Prove: ABCD is a cyclic quadrilateral

Construction: Draw a circle passing through A, B and C.

Proof: Suppose the circle meets CD or CD produced at E. Join AE

Now ABCE is a cyclic Quadrilateral.

As Opposite angles of a cyclic quadrilateral are supplementary

     .........(1)    

But,     ........(2)           [Given]

From Equation (1) and (2) we get

But this is not possible is an exterior angle to the in acse (1) or is an exterior angle to the in case (2)

And we know that the exterior angle of a triangle is equal to sum of interior opposite angles

That is exterior angle is always greater than interior opposite angle

Thus our assumption is wrong

Hence ABCD is a cyclic quadrilateral.

Case (1)

Case (2)

Solution: ACDF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary

      .....................(1)

Also ABEF is a cyclic quadrilateral and In a cyclic quadrilateral opposite angles are supplementary

      ..................(2)

   ..............(3)

From Equation 1 and 3 we get

But they are corresponding angles when BE and CD are two straight lines and BC is the transversal

As they are equal, So BE || CD