I know less than you lol. I am researching as I post this.
There are actually multiple ways to prove it, do you HAVE to use your theorem (1)?
Prove that both pairs of opposite sides are parallel.
Prove that both pairs of opposite sides are congruent.
Prove that one pair of opposite sides is both congruent and parallel.
Prove that the diagonals bisect each other.
Also, are you sure Diagonals can't be perpendicular? Because perpendicular diagonals form squares, and squares are parrellelograms.