Komplexify explains that, for any acute angle where two sides measure 3" on each ray, one can measure the distance between the two 3" rays and multiply by 20. This gives the angle in degrees. The error rate increases as the angle gets larger: However, even with a right triangle, at 90 degrees, the expected error rate is about 6%.
Trigonometry provides the best methods for measuring angles without a protractor; but in many cases a simple method is all that is needed. Given that the sum of all angles in a triangle equals 180 degrees, and that the sum of angle in a right angle equals 90, one can derive the measurement of almost any angle.
Let's say the angle one wants to measure is obtuse, or greater than 90 degrees. In this case all one needs to do is trace a ray along one side, and measure the acute angle created as described above; then, subtract that from 180 to get the measurement. If one has a right angle, and knows at least one other angle, he subtracts it from 90 to get the other angle.
Here's another way to measure angles: use the fist! If one makes a fist, and extends the arm straight, the area covered by the fist equals 10 degrees. One can make rough estimates of angular elevation in this way when hiking or looking at the night sky.