Question

Prove that if the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Answer 1

Given: A quadrilateral ABCD in which diagonals AC and BD intersect at point O such that OA = OC and OB = OD.

To prove: ABCD is a parallelogram.

Proof:

In ΔAOD and ΔBOC:

OA = OC (Given)

OD = OB (Given)

∠AOD = ∠BOC (Vertically opposite angles)

∴ ΔAOD ≅ ΔCOB (By SAS congruence criterion)

⇒ ∠OAD = ∠OCB (By c.p.c.t.)

∴ AD || BC ... (1) (If a transversal intersect two lines in such a way that a pair of alternate interior angles are equal then the two lines are parallel)

Similarly, AB || DC ... (2)

From (1) and (2), we get:

AB || DC and AD || BC

We know that a quadrilateral is a parallelogram, if both the pairs of its opposite sides are parallel.

Hence, ABCD is a parallelogram.