I’m sure there is a much more elegant solution, but given my technology interests, I thought this would be a cool way to incorporate CAS. Based on Dave’s restrictions, the following angle measures apply.
There are many ways to write equations from this setup, several of which are identical forms of the same information. One way to keep from writing dependent equations is to use combinations. Using only the smallest triangles, you get
From the top two triangles, one relationship is A+Y=180. Because the large triangle is isosceles, the bottom two triangles give X+V=W. There’s no convenient way to combine the top and bottom triangle.
Combining all three triangles, one relationship is X+W+Z=180.
That’s a system of 6 equations in 6 variables which Wolfram Alpha solves to give A=45 degrees.
I’d love to see a non-algebraic approach.