We know that the four angles of a quadrilateral must sum up to 360º. So, we have:
[tex]X+Z+W+Y=360^{\circ}[/tex]
Since X and Z are congruent, and W and Y are congruent, we have:
[tex]\begin{gathered} X+X+W+W=360^{\circ} \\ \\ 2X+2W=360^{\circ} \\ \\ 2(X+W)=360^{\circ} \\ \\ X+W=180^{\circ} \end{gathered}[/tex]
Thus, X and W are co-interior angles, which means segments XY and XZ are parallel.
And since Y and W are congruent, X and Y are also co-interior angles, which means segments XW and YZ are parallel.
Therefore, WXYZ is a parallelogram.
