Prove the theorem: If in a quadrilateral two sides are equal and parallel, then this quadrilateral is a parallelogram.

Let a quadrilateral ABCD, AB and CD be parallel in it.
Let us prove that side BC is parallel to side AD.
Let us draw two diagonals in the quadrilateral, intersecting at point O and dividing in half.
So AO = OС, BO = OD.
Then the angles AOB and COD, COB and AOD are vertical, then they are respectively equal.
This means that triangles AOB and COD are equal, since the sides are equal and the angles between these sides are equal.
And, therefore, triangles AOD and COB are equal.
This means that the BC side is parallel to the AD side.
It turns out that this quadrilateral is a parallelogram.