If the non-parallel sides of a trapezium are equal, prove that it is cyclic.
It is given that ABCD is a trapezium with and
We need to prove that ABCD is a cyclic quadrilateral.
Construction: Draw and .
In and , we have
(Each equal to )
(Distance between two parallel lines is constant.)
Therefore, by RHS congruence rule, we have
(Corresponding parts of congruent triangles are equal) (1)
We also have (Co-interior angles, ) (2)
From (1) and (2), we can say that
ABCD is a cyclic quadrilateral.
(If the sum of a pair of opposite angles of a quadrilateral is , the quadrilateral is cyclic.)