Got it!

I spent a week working on this.

What ended up helping?

• Checking Geogebra and finding the exact values of x and all the other angles, and realizing that this meant that the diagram was made of an equilateral triangle and an isosceles triangle. I spent a lot of time trying to prove that the triangles were either isosceles or equiliateral, to no avail.
• In a previous problem from GoGeometry I had been asked to show that an angle was 30 degrees. I was stumped, but when I looked at some solutions I saw that people used the side lengths to show that a right triangle was 30/60/90 and that (therefore) the given angle was 30 degrees. I realized that this was a possible strategy.
• I know that sometimes it helps to draw an altitude.
• I was on the look out for similar triangles, since triangle relationships (especially similarity ones) often open a problem up.

Then, essentially, it was a week of randomly trying to make each of these individual things work. I had (and I have) no understanding of why I should have known that these three strategies should be used at once. It took me a week to solve this, I think, until my continued attempts happened to let me combine these four tools at once.