To solve for the measure of angle \( \angle F \), we first need to find the measures of angles \( \angle D \) and \( \angle E \), which are supplementary. This means:
\[ m\angle D + m\angle E = 180° \]
Given: \[ m\angle D = (4x + 8)° \] \[ m\angle E = (x + 17)° \]
Substituting the expressions for \( \angle D \) and \( \angle E \) into the supplementary angle equation:
\[ (4x + 8) + (x + 17) = 180 \]
Combining like terms:
\[ 4x + x + 8 + 17 = 180 \] \[ 5x + 25 = 180 \]
Now, we isolate \( x \) by subtracting 25 from both sides:
\[ 5x = 155 \]
Dividing both sides by 5:
\[ x = 31 \]
Now we can find the measures of angles \( D \) and \( E \) by substituting \( x \) back into their expressions:
Calculating \( m\angle D \):
\[ m\angle D = 4(31) + 8 = 124 + 8 = 132° \]
Calculating \( m\angle E \):
\[ m\angle E = 31 + 17 = 48° \]
Now since \( \angle F \) is a vertical angle with \( \angle D \), it means \( \angle F \) has the same measure as \( \angle D \):
\[ m\angle F = m\angle D = 132° \]
Thus, the measure of \( \angle F \) is:
\[ \boxed{132°} \]