I spent (wasted?) several hours working this problem out years ago but saw it posted recently on Google Plus. I soon realized that either my solution predated my blog or I never actually posted my solution.
Here’s the problem and my solution is below: (ref: http://thinkzone.wlonk.com/MathFun/Triangle.htm)
Using only elementary geometry, determine angle x. Provide a step-by-step proof.
You may use only elementary geometry, such as the fact that the angles of a triangle add up to 180 degrees and the basic congruent triangle rules (side-angle-side, etc.). You may not use trigonometry, such as sines and cosines, the law of sines, the law of cosines, etc.
Spoiler Alert: Don’t read any further if you want to give this a shot yourself!