I had an inspiration. I think I may have a different solution, if you have a scientific calculator.

1) Measure the same distance out from the corner on both sides. Call that distance a.

2) now measure the distance between those two points. Call that distance b.

3) calculate (2a^2 - b^2) / 2a^2

4) push your cos^-1 button (arc-cos). The result should be the angle. I think.