To find the measure of angle ∠F, we first need to determine the measures of angles ∠D and ∠E. Since angles ∠D and ∠E are supplementary, their measures add up to 180 degrees. This gives us the equation:
\[ m∠D + m∠E = 180° \]
Substituting the expressions for m∠D and m∠E:
\[ (4x + 8) + (x + 17) = 180 \]
Now, combine like terms:
\[ 4x + x + 8 + 17 = 180 \] \[ 5x + 25 = 180 \]
Next, subtract 25 from both sides:
\[ 5x = 180 - 25 \] \[ 5x = 155 \]
Now, divide by 5:
\[ x = \frac{155}{5} = 31 \]
Now that we have x, we can find the measures of angles ∠D and ∠E:
- Calculate \( m∠D \):
\[ m∠D = 4x + 8 = 4(31) + 8 = 124 + 8 = 132° \]
- Calculate \( m∠E \):
\[ m∠E = x + 17 = 31 + 17 = 48° \]
Now we can find the measure of angle ∠F. Since ∠F is a vertical angle with ∠D, it has the same measure as ∠D:
\[ m∠F = m∠D = 132° \]
Therefore, the measure of angle ∠F is:
\[ \boxed{132°} \]