#### theorem

**Theorem :** The diagonals of a parallelogram bisect each other. **proof : **

Since ABCD is a parallelogram

`therefore ` AB || DC and AD || BC

Now, AB parallel to DC and AC is the transversal intersecting them at A and C respectively.

∠BAC =∠ DCA (Alternate interior angles)

Thus, ∠ BAO = ∠ DCO ... (1)

As , AB || DC and BD intersect them at B and D respectively.

∠ ABD= ∠CDB (Alternate interior angles)

So, ∠ ABD= ∠CDO ...(2)

Now , In triangle AOB and Triangle COD , we have

∠BAO = ∠ DCO [From (1)]

AB = CD ( Opposites sides of a parallelogram )

∠ ABO =∠ CDO [From (2)]

`therefore` ∆ AOB ≅ ∆ COD (by ASA congruence criterion)

Thus , OA = OC and OB = OD (C.P.C.T)

#### Shaalaa.com | Theorem : The diagonals of a parallelogram bisect each other.

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