SOLUTION: how to prove this theorem.. Theorem 7.5 If the diagonal of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Question 937428: how to prove this theorem..
Theorem 7.5 If the diagonal of a quadrilateral bisect each other, the quadrilateral is a parallelogram.
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
draw your quadrilateral ABCD so that A is top left, B is top right, C is bottom right, E is bottom left.

draw your diagonals AC and BD.

let the intersection of the diagonals be point E.

you are given that AE is congruent to EC and DE is congruent to EB.

angle AEB is congruent to angle CED by virtue of the fact that they are vertical angles and all vertical angles are congruent.

similarly, angle AED is congruent to angle CEB for the same reason.

you get triangle AEB is congruent to triangle CED and you get triangle AED is congruent to triangle CEB by the SAS postulate that says that triangle are congruent if 2 corresponding sides and the angle between them are congruent.

you then get that AB is congruent to DC and you get AD is congruent to BC because corresponding parts of congruent triangles are congruent.

you then get that quadrilateral ABCD is a parallelogram because a quadrilateral is a parallelogram if the opposite sides are congruent.

a picture of your quadrilateral is shown below:

$$$

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