To avoid the abstract notion of a cut, you can assign a decimal to an arbitrary angle geometrically “just” as Kaplan assigns a decimal geometrically to an arbitrary point on a line:

Pick a point on the line.

Divide the line into 10 segments (HS geometry).

Pick the segment which contains the line and divide that into ten segments.

Continue indefinitely or until the point falls on the end of a segment.

You can do the same thing with an angle with an interesting caveat. You can’t divide an angle into 10 segments using HS geometry but you can bisect it. So let 90deg = 1 and pick an arbitrary angle. Bisect angle 1 and pick the segment which contains the angle. Then bisect that segment. Continue indefinitely or until your angle falls on the edge of a segment. You then have a binary representation of any angle which you can convert to a decimal.