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

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

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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 the parallelogram )

$AD = BC$ (Given)

So, $BE = BC$

In $\triangle EBC$,

$BE = BC$ (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
$\angle 1+\angle 2=180^{\circ}$
$\Rightarrow \angle A+\angle C=180^{\circ}$
Thus, $ABED$ is a cyclic quadrilateral.

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