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.