How can I measure a ## degree angle in blender?

I’m not quite sure what you mean.

If you want to rotate something by a specific angle, press R (then maybe an axis (X, Y, Z, XX, YY, ZZ) then just type the number. Tab to use the next axis if you are doing more than one.

If you are trying to find the angle at the intersection of two edges, In Edit Mode in Editing Context>Mesh Tools 1 Panel, there is a group of buttons for measurement, one of them is Edge Angle.