Geom. Two Column Proof: If the diagonals of trapezoid ABCD bisect each other, prove

[cderuwe]

I have to come up with a two column proof for the following statement:
If the diagonals of trapezoid ABCD bisect each other, prove the trapezoid is an isosceles trapezoid.

Can anyone help me with this?

I know some givens:
The bases are parallel, if the diagonals bisect each other, then the two non-parallel sides would be equal in lenght (I think, right?) I can maybe figure out that the alternate interior angles are congruent based on the fact that two parallel lines are cut by a transversal... I feel like I am just missing something with proving. Any help is greatly appreciated!

 

[stapel]

I have to come up with a two column proof for the following statement:
If the diagonals of trapezoid ABCD bisect each other, prove the trapezoid is an isosceles trapezoid.

Can anyone help me with this?

I know some givens:
The bases are parallel, if the diagonals bisect each other, then the two non-parallel sides would be equal in lenght (I think, right?) I can maybe figure out that the alternate interior angles are congruent based on the fact that two parallel lines are cut by a transversal... I feel like I am just missing something with proving. Any help is greatly appreciated!

Are you sure that you're suppose to prove this? Because it isn't true, as a quick sketch will reveal.

 

[mrarthur]

If the diagonals of a trapezium (as we call it in england) bisect each other, there would be vertically opposite angles where they bisect, and since they are bisected there are two triangles which have two sides the same length and an included angle of the same size, therefore they are congruent (SAS). Therefore, the pair of sides which are not parallel (not necessarily so far) must be the same length.
However you could then have a parallelogram, or if you want to make it into an isosceles trapezium then it will end up being the rather special kind of isosceles trapezium better known as the rectangle.