How can an angle be measured without a protractor?

Answer 1

Raise a perpendicular from one arm to the other.

This creates a right angled triangle. Measure two of the sides of the triangle and use the appropriate trigonometric ratio and a computer/calculator/slide rule/tables to convert the ratio to an angle.

eg measuring the side adjacent to the angle and the length of the side opposite the angle, which is the perpendicular raised, dividing the length of the opposite side by the adjacent side gives a value which can be looked up in arc tan tables (etc) to get the angle.