To find the measure of angle \( \angle F \), we first need to solve for \( x \) using the information given about the supplementary angles \( \angle D \) and \( \angle E \).
Since \( \angle D \) and \( \angle E \) are supplementary, we can set up the equation:
\[ m\angle D + m\angle E = 180^\circ \]
Substituting the expressions we have for \( m\angle D \) and \( m\angle E \):
\[ (4x + 8) + (x + 17) = 180 \]
Now, combine the like terms:
\[ 4x + x + 8 + 17 = 180 \]
This simplifies to:
\[ 5x + 25 = 180 \]
Now, subtract 25 from both sides:
\[ 5x = 155 \]
Next, divide by 5:
\[ x = 31 \]
Now that we have \( x \), we can find \( m\angle D \):
\[ m\angle D = 4x + 8 = 4(31) + 8 = 124 + 8 = 132^\circ \]
Since \( \angle F \) is a vertical angle with \( \angle D \), they are equal. Therefore:
\[ m\angle F = m\angle D = 132^\circ \]
Thus, the measure of \( \angle F \) is:
\[ \boxed{132^\circ} \]