How are angles measured in light experiments?

From my experience the only time I've had to do this is when testing angles of incidence and refraction. Usually the angles of incidence are predetermined, say 20, 40 and 60 degree's. A sheet of plain white paper is placed down. A glass block is set in the middle and drawn around. Make sure you draw in your normals. Use a protractor to draw the lines which are are your angles of incidence. Then place the glass block down on the outline and use a line box with a thin slit to make the light as direct as possible. You will notice the light shining through the other side of the glass block, mark these for each incidence angle. Remove your glass block and draw proper lines for the light that was shining out of the block.
From here you should be able to draw lines through your glass block outline which will connect the rays of incidence and refraction. Using your protractor you will be able to find the angles of refraction accurate to 1/2 degree i think.
Other light experiments such as finding the critical angle which is a limiting angle which if passed will cause total internal reflection to occur. You can find the angle of incidence which causes this to happen in which ever medium you are testing by using the formula:
refractive index = 1/sin of critical angle
e.g using a diamond of refractive index 2.1 you would have:
2.1 = 1/sin C
SinC = 1/2.1
C = Sin^-1 (1/2.1)
C = 28.4 degrees (roughly)
I hope this helped, however being more specific to which light experiment you want could help!