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

ABCD is a quadrilateral.

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.

Hence, AD || BC.

Combining all the results we get;

AD || BC and AB || CD.

Since opposite sides are parallel to each other, the given quadrilateral is a parallelogram.

Hence in quadrilateral, if diagonal bisect each other it is a parallelogram.