To find the measure of angle \( \angle F \), we first need to determine the measures of angles \( \angle D \) and \( \angle E \). Since \( \angle D \) and \( \angle E \) are supplementary angles, their measures add up to \( 180^\circ \).
Given: \[ m\angle D = (4x + 8)^\circ \] \[ m\angle E = (x + 17)^\circ \]
Setting up the equation for supplementary angles: \[ m\angle D + m\angle E = 180^\circ \] Substituting the expressions for \( m\angle D \) and \( m\angle E \): \[ (4x + 8) + (x + 17) = 180 \] Combining like terms: \[ 5x + 25 = 180 \] Now, subtract \( 25 \) from both sides: \[ 5x = 155 \] Dividing both sides by \( 5 \): \[ x = 31 \]
Now we can find the measures of \( \angle D \) and \( \angle E \): \[ m\angle D = 4(31) + 8 = 124^\circ \] \[ m\angle E = 31 + 17 = 48^\circ \]
Since vertical angles are equal, \( \angle F \), which is vertical to \( \angle D \), will have the same measure as \( \angle D \): \[ m\angle F = m\angle D = 124^\circ \]
Thus, the measure of \( \angle F \) is: \[ \boxed{124^\circ} \]
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