To find the measure of angle AEF, we can use the fact that lines AB and CD are perpendicular at point E. Since line AB is horizontal and line CD is vertical, we know that angle AEF and angle BEF are angles that together form a straight line at point E.
The angles AEF and BEF are supplementary since they are on the same line along line AB.
Given that angle BEF is 120 degrees, we can calculate angle AEF as follows:
\[ \text{Angle AEF} + \text{Angle BEF} = 180^\circ \]
Substituting the value of angle BEF:
\[ \text{Angle AEF} + 120^\circ = 180^\circ \]
Now, solving for angle AEF:
\[ \text{Angle AEF} = 180^\circ - 120^\circ = 60^\circ \]
Thus, the measure of angle AEF is \( \boxed{60} \).