math geometry properties of parallelograms

Question

if a quadrilateral has two parallel sides and a pair of opposite equal angles could it be proven as a parallelogram

Answer 1

You can prove it is a parallelogram if one pair of opposite sides is congruent and parallel.

We can do that by showing that the two triangles created by the diagonal are congruent then using corresponding parts of congruent triangles are equal.

The diagonal is a common shared side of the two triangles. The diagonal also forms a transversal so that alternate interior angles in both triangles are equal. And the two opposite angles are given in each triangle.

Based on that you can say the triangles are congruent by AAS and the opposite sides are congruent and since they are already given as parallel, the quadrilateral is a parallologram.