I have two tricky geometry problems begging for proofs. I have tried and failed to find classic geometry proofs for these.

1) A quadrangle ABCD satisfies the following relation on the angles:

2A + C = 360 degrees

and the following on the edge lengths:

BC = CD = DA

Prove that angle D is 60 degrees.

It is easy to see that given the three equal side lengths, and D=60, then it follows that 2A+C=360. If you change angle D then the angle 2A+C also changes and so can no longer be 360. However I don't know how to prove the opposite direction in a nice way without using trig.

I'm doing a bit of research into tiling patterns, and the above quadrangle makes quite a nice tiling, and it surprised me when I noticed that the requirements imposed on the tile by the tiling forced one angle to be constant. Here is a picture of the tiling pattern using this tile, with two different choices of side length AB.

Here is a second, and I think more difficult one I ran into.

2) A quadrangle ABCD satisfies the following relation on the angles:

A + C = 180 degrees

and the following on the edge lengths:

AB = DA

BC = CD + DA

Prove that D is 120 degrees.

I have no clue how to prove this classically, and even using trig to prove it seems to get unwieldy quickly. Is there a relatively simple proof of this?