Find the two pairs of vertically opposite angles. These are the angles that are diagonal to each other at the vertex of the angles -- the point where the two lines intersect. Label one angle pair "A" and "B" and the other angle pair "C" and "D."
Place the protractor along one of the straight lines so the zero line of the protractor aligns with the right side of the line.
Notice the line on the protractor where the intersecting line falls and note the number. This is the measurement of the angle. For example, the angle measurement may be 47 degrees.
Subtract the measurement of the first angle from 180 to find the measurement of the second angle on the same line. Using the same example, 180 - 47 = 133. The second angle is 133 degrees.
Find the vertically opposite angles to the angles you already measured. The vertically opposite angle to the first angle you measured will also be 47 degrees. The angle vertically opposite to the second angle you figured will also be 133 degrees.
Check all the angle measurements with the protractor, to ensure accuracy.