A student needed to draw a circle with a 2" diameter, then draw the following angles: 100°, 120º, and 140º. She had her compass but didn't have her protractor.
First she drew the circle, then she drew 2 perpendicular diameters. Since a circle encompasses 360º, each quadrant comprising 90º. We drew the 120º angle first using an entire 90º quadrant plus 1/3 of the adjacent quadrant, erasing the unneeded line, which leaves 60º in that second quadrant.
Then we found the circumference of the circle (C=πD, or 3.14x2"=6.28"). Next we found 1/4 of the circumference (6.28"/4=1.57"). We wanted to be able find the arc length in 10º increments, so we divided the arc of one quadrant by 9 (1.57"/9=0.174"). We converted this into 1/16ths of an inch by multiplying by 16 (0.174"x16=2.79 sixteenths of an inch).
Getting back to our angles, we measured the 100º angle next by taking our remaining 60º and adding 40º of arc length to it (4x2.79=11.16 sixteenths). Note: It may be helpful to mark sixteenths of an inch along the arc and reposition the ruler after each. The final 140º angle arises naturally.
Hope this helps if you (or a student) needs to construct angles without a protractor. :)