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!
1. since it corresponds to (intersection of two parallel lines by another)
2. since it corrresponds to
3. This implies by sum of triangle angles that , and hence is equilateral, giving us
4. Also notice that since it corresponds to , which now means that .
5. Now, consider that is similar to since they share the same angle measures.
6. Also notice that is congruent to since , , and they share the side AC. (SAS)
7. Now since then , since Corresponding Parts of Congruent Triangles are Congruent (CPCTC).
8. Now (7) together with (5), means that is congruent to .
9. This means that (CPCTC).
10. By (3), this means that , so that is an isosceles triangle. So, because they must be equal and .
11. Now, using the fact that (from the fact the we know and ) we just subtract this from and we have that .