The proof depends heavily upon the axioms and definitions you have to use. We do not know those. So the best we can do here is to give a outline.

If you connect two non-consecutive vertices of the original quadrilateral you have a diagonal. There are two sub-triangles formed. Now the segment formed by joining the midpoints of the two adjacent sides is parallel to and one half the length of the third side, that diagonal. Do the same on the second triangle. Now you have a quadrilateral have opposite sides parallel of the same length.