Complementary angles are two angles whose measures add up to 90 degrees. 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, let's simplify and solve for \( x \):
- Combine like terms:
\[ 4x - 5 + x + 20 = 90 \]
This simplifies to:
\[ 5x + 15 = 90 \]
- Next, subtract 15 from both sides:
\[ 5x = 90 - 15 \] \[ 5x = 75 \]
- Finally, divide both sides by 5:
\[ x = \frac{75}{5} \] \[ x = 15 \]
Thus, the value of \( x \) is \( \boxed{15} \).