Supplementary angles are angles that add up to 180 degrees. Given that
\( m∠D = (4x + 8)^\circ \)
\( m∠E = (x + 17)^\circ \)
Since ∠D and ∠E are supplementary, we can set up the equation:
\[ m∠D + m∠E = 180^\circ \]
Substituting the expressions for the angles:
\[ (4x + 8) + (x + 17) = 180 \]
Combining like terms:
\[ 5x + 25 = 180 \]
Now, subtract 25 from both sides:
\[ 5x = 155 \]
Next, divide by 5:
\[ x = 31 \]
Now we can find the measures of angles D and E. First, calculate \( m∠D \):
\[ m∠D = 4(31) + 8 = 124 + 8 = 132^\circ \]
Next, calculate \( m∠E \):
\[ m∠E = 31 + 17 = 48^\circ \]
Since \(\angle F\) is a vertical angle to \(\angle D\), it has the same measure as \(\angle D\):
\[ m∠F = m∠D = 132^\circ \]
Thus, the measure of \(\angle F\) is:
\[ \boxed{132^\circ} \]
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