In this chapter we will prove that “any quadrilateral with equal opposite sides is actually a parallelogram.

**Given:**

Consider quadrilateral ABCD in which opposite sides are equal.

AB = CD and AD = BC

**To Prove:**

Prove that the quadrilateral is parallelogram.

i.e. AB || CD and AD || CB

**Proof:**

Consider triangle ABC and CDA;

AB = CD ( given )

AC = CA ( common side )

AD = CB ( given )

By **SSS congruency**, both the triangles are congruent.

i.e. \mathtt{\triangle ABC\ \cong \triangle CDA}

Since both triangles are congruent, we can write;

∠BAC = ∠DCA and ∠BCA = ∠DAC

**Since ∠BAC = ∠DCA;**

This is possible when line AB || CD and intersected by AC as transversal.

Also **as ∠BCA = ∠DAC;**

This is possible when line AD || BC and intersected by AC as transversal.

Since both the opposite sides are parallel to each other, this means that the given figure is of parallelogram.

Hence Proved.