Triangle Problem

I would love for someone to prove me wrong, but you can't solve it with just those equations; I've tried it on my first attempt.
With those only those equations, the solution a = 85, b = 75, c = 55, theta = 55 is possible, but looking at the picture, clearly, c =/= theta.

EDIT: Oh, I didn't see that you used guess and check, but I'm just gonna point it out there that using substitution on those 4 equations will always get you 0 for whatever variable you decide to solve for first. The incorrect solution above was obtained from plugging the same equations into a matrix in my calculator. The correct answer of 30 is explained in my earlier post.

We could also include a rule for the angles to say that a specific one must be bigger than another or that some can't be the same, but yeah, I can't really think how you can solve it beyond just educated guessing.