This problem starts with a very simple picture but I want to show my first attempt down a non-productive line of thought.
Two thoughts came to me immediately. Angle EAC is almost part of an equilateral triangle if we reflect the whole triangle. Secondly ECA is half of EAC and it could be divided into two halves so we end up with many 15 degree angles. I pursued the second thought first because it created some attractive symmetry.
So I went back to my first thought. Reflect the whole triangle AEC and create a giant equilateral triangle out of it.
Then I started angle tracing at which point I realized I didn't need the full equilateral triangle the bottom 30-60-90 was interesting by itself.
This one fell out after trying 2 ideas. With some other problems I go down many more avenues. My general guiding principle is to try out reflections and angle division additions first around the figure always with an aim at more symmetry or new congruences.