Geometric help please?

Question

Hi, I am doing math review for school, and have no clue how to solve a proof like this one. Thanks!

If the diagonals of a quadrilateral bisect each other, then the figure is a parallelogram.

Prove that this is so. What about the converse statement?

Answer 1

Hi, it is a little bit hard to explan without a picture, i would try my best.

Draw a quadrilateral ABCD. AC cut BD at M.  As in this case, we will have:

AM = CM  (bisect, given)

BM = DM  (bisect, given)

Proof:

Consider two step of proof:

Step 1:

In two triangles ABM and CDM we have.

AM = CM (given)

BM = DM (given)

angle AMB = angle BMC (vertical angles)

Therefore two triangles ABM and CDM are congruent.

Therefore: AB = CD  (1)

Step 2:

Same as step 1, but with two triangles ADM and CBM

Same steps of proving, we will have

Triangles ADM and CBM are congruent.

Therefore AD = BC (2)

Combine (1) and (2) we have, quadrilateral have two opposite pairs congruent, then the quadrilateral is a parallelogram.

Done!

PS. This is the one of the other ways.

Good Luck!