In order to prove that the diagonals bisect each other, we have to show that WM ≅ MY and ZM ≅ XM. To do that we can isolate △ XWM and △ ZYM in the diagram.
Examining the triangles, we see that both ∠ WMX and ∠ YMZ form a pair of vertical angles. According to the Vertical Angles Theorem, these are congruent. Let's add this information to the diagram.
Since the triangles have at least two pairs of congruent angles we can claim that they are similar by the AA (Angle-Angle) Similarity Theorem. In these triangles WM and MY are corresponding sides, as are ZM and XM.
In a parallelogram opposite sides are congruent. Since these sides are also corresponding sides in △ XWM and △ ZYM, we know that the triangles are congruent. Therefore, WM ≅ MY and ZM ≅ XM, which means they bisect each other.
