One of the things most of you guys are missing, is that you can solve systems of equations, if you have as many equations as you have variables.
If I say "a + b = 7", you could look at that and say, "You can't solve it - a and b could be anything!" - and you'd be right. But if I also say "a -b = 3" you could also look at that by itself and say "You still can't solve it, a and b could be anything!". But if you take those pieces of information together, it's easy to solve, since you have 2 unknowns and 2 equations. The only answer is a=5, b=2.
As far as I know, it's not possible to use trig or geometry figure out any individual angle or length in the presented problem. But the angles and lengths are all properly constrained (build the model with drinking straws and pins, to prove it to yourself) and so by simultaneously solving a sufficient number of known equations, we can solve for everything. I presented 4 equations (if any of them are incorrect, tell me why you think so!) in 4 variables, therefore the problem is solvable.