All you need do is to draw one diagonal, say, PR.

 

Then AD is a midline of triangle PRS and thus parallel to and half
the length of PR. 

Similarly, BC is a midline of triangle PRQ and is 
thus ALSO parallel to and half the length of PR.  So AD and BC are both>
parallel and equal in length.  That's enough to prove ABCD is a 
parallelogram.  Now go write that up in a two-column proof.

Edwin