12. In \( \triangle D E F \), if \( m \angle E \) is one more than three times \( m \angle F \) and \( m \angle D \) is three more than seven times \( m \angle F \), find the measure of each angle.

Let's denote \( m \angle F \) as \( x \). According to the problem, we can express the measures of the other angles in terms of \( x \): - \( m \angle E = 3x + 1 \) - \( m \angle D = 7x + 3 \) Since the sum of the angles in any triangle is \( 180^\circ \), we can write the equation: \[ x + (3x + 1) + (7x + 3) = 180 \] Simplifying this, we have: \[ x + 3x + 1 + 7x + 3 = 180 \] Combine like terms: \[ 11x + 4 = 180 \] Now, subtract 4 from both sides: \[ 11x = 176 \] Next, divide by 11: \[ x = 16 \] Now that we have \( m \angle F = x = 16^\circ \), we can find the other angles: \[ m \angle E = 3(16) + 1 = 48 + 1 = 49^\circ \] \[ m \angle D = 7(16) + 3 = 112 + 3 = 115^\circ \] Thus, the measures of the angles are: - \( m \angle D = 115^\circ \) - \( m \angle E = 49^\circ \) - \( m \angle F = 16^\circ \) In conclusion, the measures of the angles in triangle \( D E F \) are \( 115^\circ \), \( 49^\circ \), and \( 16^\circ \).