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 $∆$ 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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