Notice that the angles UWV and YWX are vertical angles. Then, they have the same measure.
Additionally, the angles VUW and XYW are specified to have the same measure, as well as the sides VW and WX. Then, the pair of triangles UVW and YXW have two angles in common as well as one side.
Then, the congruence theorem that can be used to identify that ΔUVW ≅ ΔYXW is: