This is a problem that has tortured me for a while.  It was first given to me by Asli Yaman (a PhD candidate at the University of Southampton) at a topology conference in Montreal a while ago.  A number of grad students and a few professors spent many nights in various bars trying to solve this.

Without measuring, find the indicated angle.

You are to prove this using the knowledge of a triangle known to Euclid or any high school geometry text.

Note: Finding this angle completely determines the figure up to similarity and finding all but one of the unmarked angles is an equivalent problem.

So put away those rulers and protractors and Good luck!