To find the measure of angle \( \angle F \), we first need to determine the values of \( \angle D \) and \( \angle E \) based on the information given.
Since \( \angle D \) and \( \angle E \) are supplementary angles, their measures add up to \( 180^\circ \). We are given:
\[ m\angle D = 4x + 8 \] \[ m\angle E = x + 17 \]
Setting up the equation for supplementary angles:
\[ (4x + 8) + (x + 17) = 180 \]
Now, combine like terms:
\[ 4x + x + 8 + 17 = 180 \] \[ 5x + 25 = 180 \]
Next, isolate \( x \):
\[ 5x = 180 - 25 \] \[ 5x = 155 \] \[ x = \frac{155}{5} \] \[ x = 31 \]
Now we can find the measures of angles \( \angle D \) and \( \angle E \):
\[ m\angle D = 4(31) + 8 = 124 + 8 = 132° \]
\[ m\angle E = 31 + 17 = 48° \]
Since \( \angle F \) is a vertical angle with \( \angle D \), their measures are equal. Thus:
\[ m\angle F = m\angle D = 132° \]
Therefore, the measure of angle \( \angle F \) is
\[ \boxed{132^\circ} \]
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