Two angles are complementary when their measures add up to \(90^\circ\). Given the measures of the angles as \((4x - 5)^\circ\) and \((x + 20)^\circ\), we can set up the equation:
\[ (4x - 5) + (x + 20) = 90 \]
Now, combine like terms:
\[ 4x - 5 + x + 20 = 90 \]
\[ 5x + 15 = 90 \]
Next, subtract 15 from both sides:
\[ 5x = 75 \]
Now, divide both sides by 5:
\[ x = 15 \]
Thus, the value of \(x\) is \( \boxed{15} \).