# Prove that if diagonal of quadrilateral bisect each other then it is parallelogram

Consider the below quadrilateral ABCD in which diagonals bisect each other.

Given:
AC & BD are diagonals bisecting each other.
Hence, AO = OC and BO = OD.

To Prove:
Prove that the given quadrilateral is a parallelogram.

Proof:
Consider triangle AOB and COD;

AO = OC (given)
∠AOB = ∠COD ( vertically opposite angle)
BO = OD (given)

By SAS congruency condition, both triangles AOB & COD are congruent.

Since both triangles are congruent, we can write;
∠BAO = ∠DCO

Both the angles are also alternate interior angles formed by line AB & CD intersected by transversal AC.

Since both the angles are equal this means that the line AB & CD are parallel to each other.
Hence, AB || CD.

Now consider the triangle AOD and COB.

OA = OC (given)
∠AOD = ∠COB ( vertically opposite angle)
OD = OB

By SAS congruency, both triangles AOD & COB are congruent.

Since both triangles are congruent, we can write ∠ODA = ∠OBC.

These are also alternate interior angles formed by line AD & BC intersected by transversal BD.

Since both angles are equal, it means the line AD & BC are parallel.