# Prove that the straight lines joining the mid points of the opposite sides of a parallelogram are parallel to the other pairs of parallel sides.

Let we have a parallelogram ABCD , and E and F are mid points of AB and CD respectively . We form our diagram , As :

Here Diagonals AC and BD and EF intersect at " O " .

We know diagonals of parallelogram bisect each other , So

AO = CO

In $\u2206$ ABC , we have

AE = BE , as we assumed E is mid point of AB

And

AO = CO , from property of parallelogram .

SO,

From conserve of mid point theorem , we get

EO | | BC , SO

EF | | BC ( As EO is part of line EF )

And

We know BC | | DA , from property of parallelogram , So

We can say

BC | | DA | | EF

So,

**Joining the mid points of the opposite sides of a parallelogram are parallel to the other pairs of parallel sides. ( Hence proved )**

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