What's the Angle? (But more interestingly: how did you solve it?)

So far I've filled out angles, drawn an isosceles triangle and redrawn the diagram. Thus ends my productive struggle. pic.twitter.com/Yo5gOWKCT4

— Michael Pershan (@mpershan) May 11, 2015

I saw this problem on Twitter. This high school geometry teacher wanted to know what solution paths people could come up with. (He did solve it in the end himself too!)

da Vinci Geometry students: what solution methods do you see? You've given me a new appreciation for Law of Sines, so I used that; I'm curious whether you see other paths you like. Let me know and I will add them to the online discussion! (I'll put a link to that, too, eventually, but I want to see what original stuff you come up with first.)