Q: 8    If the non-parallel sides of a trapezium are equal, prove that it is cyclic.

Answers (1)
S seema garhwal

Given: ABCD is a trapezium.

Construction: Draw AD || BE.

Proof: In quadrilateral ABED,

                      AB || DE    (Given )

                      AD || BE    ( By construction )

Thus, ABED is a parallelogram.

     AD = BE     (Opposite sides of parallelogram )

    AD = BC      (Given )

so,  BE = BC

In \triangleEBC,

           BE = BC    (Proved above )

      Thus, \angle C = \angle 2...........1(angles opposite to equal sides )

              \angle A= \angle 1...............2(Opposite angles of the parallelogram )

From 1 and 2, we get

    \angle 1+\angle 2=180 \degree           (linear pair)

     \Rightarrow \angle A+\angle C=180 \degree

Thus, ABED is a cyclic quadrilateral.