Diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that ABCD is a llgm.. It's urgent ASAP!

Chetna Khanna , Meritnation Expert added an answer, on 26/12/11

Given, a quadrilateral ABCD with AB = DC and DO = OB.

**To prove:** ABCD a parallelogram

**Construction:** Draw OX DC.

In Δ BCD,

O is the mid point of BD and OX DC.

∴By converse of mid point theorem,

X is the mid point of BC and OX = DC.

As AB = DC

∴ OX = AB.

Now, in Δ ABC,

X is the mid point of BC and OX = AB.

∴ By converse of mid point theorem,

O is the mid point of AC and OX AB

As OX AB and OX DC

⇒ AB DC

Now, AB DC and AB = DC

∴ ABCD is a parallelogram.

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