Let's analyze the given information:
1. LCDE and LEDF are supplementary: \( m\angle CDE + m\angle LEDF = 180^\circ \).
2. Angles that form a linear pair are supplementary: \( x + (3x + 20) = 180^\circ \).
Now, let's solve for x:
a. Combine like terms: \( x + 3x + 20 = 180 \).
b. Combine the x terms: \( 4x + 20 = 180 \).
c. Subtract 20 from both sides: \( 4x = 160 \).
d. Divide by 4: \( x = 40 \).
So, the value of x is 40. Now, substitute this value back into the expression for angle \( (3x + 20) \):
\[ (3 \times 40 + 20)^\circ = 120 + 20 = 140^\circ \].
Therefore, the measure of angle \( (3x + 20) \) is \( 140^\circ \), and the correct answer is option (e).