To find the measure of angle \( \angle F \), we first need to determine the measures of angles \( \angle D \) and \( \angle E \).
Since angles \( D \) and \( E \) are supplementary, we know that:
\[ m\angle D + m\angle E = 180° \]
Given:
\[ m\angle D = (4x + 8)° \quad \text{and} \quad m\angle E = (x + 17)° \]
Substituting the expressions into the supplementary angles equation:
\[ (4x + 8) + (x + 17) = 180 \]
Combining like terms:
\[ 5x + 25 = 180 \]
Now, subtract 25 from both sides:
\[ 5x = 155 \]
Dividing by 5:
\[ x = 31 \]
Now we can find the measure of \( \angle D \):
\[ m\angle D = 4(31) + 8 = 124 + 8 = 132° \]
Since \( \angle F \) is a vertical angle to \( \angle D \), the measure of \( \angle F \) is the same as that of \( \angle D\):
\[ m\angle F = m\angle D = 132° \]
Thus, the measure of \( \angle F \) is:
\[ \boxed{132°} \]