#### theorem

**Theorem :** If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.**proof : **

In ∆ AOD and ∆ COB (Given)

OD = OB (Given)

∠AOB = ∠COD (Vertically opposite angles are equal)

Therefore , ∆AOD ≅ ∆COB (By SAS criterion of congruence)

So, ∠OAD = ∠OCB ...(1) (C.P.C.T)

Now , lines AC intersects AD and BC at A and C respectively,

such that ∠OAD = ∠OCB ...[from(1)]

`therefore` ∠OAD and ∠ OCB form a pair of alternate interior angles are equal.

Thus, AD || BC

Similarly , we can prove that AB || DC

Hence , quadrilatera; ABCD is a parallelogram.

#### Shaalaa.com | Theorem : If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

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