Two angles are supplementary when the sum of their measures equals 180 degrees. Given the measures of angles \( \angle D \) and \( \angle E \):
\[ m\angle D = (4x + 8)^\circ \] \[ m\angle E = (x + 17)^\circ \]
We can set up the equation for supplementary angles:
\[ m\angle D + m\angle E = 180^\circ \]
Substituting in the expressions for \( m\angle D \) and \( m\angle E \):
\[ (4x + 8) + (x + 17) = 180 \]
Now, combine like terms:
\[ 4x + x + 8 + 17 = 180 \] \[ 5x + 25 = 180 \]
Next, subtract 25 from both sides:
\[ 5x = 155 \]
Now, divide by 5:
\[ x = 31 \]
Now that we have the value for \( x \), we can find the measures of \( \angle D \) and \( \angle E \):
\[ m\angle D = 4x + 8 = 4(31) + 8 = 124 + 8 = 132^\circ \] \[ m\angle E = x + 17 = 31 + 17 = 48^\circ \]
Since \( \angle F \) is a vertical angle to \( \angle D \), its measure will be the same as \( m\angle D \):
\[ m\angle F = m\angle D = 132^\circ \]
Thus, the measure of \( \angle F \) is:
\[ \boxed{132^\circ} \]