In \( \triangle A B C \), if \( m \angle A \) is thirteen less than \( m \angle C \) and \( m \angle B \) is eleven less than four times \( m \angle C \), find the measure of each angle.

Let's denote the measure of angle \( C \) as \( x \). According to the problem: - The measure of angle \( A \) is \( x - 13 \). - The measure of angle \( B \) is \( 4x - 11 \). By the triangle angle sum property, we know that the sum of the angles in a triangle is 180 degrees. Therefore, we can write the equation: \[ A + B + C = 180 \] Substituting the expressions for angles \( A \), \( B \), and \( C \): \[ (x - 13) + (4x - 11) + x = 180 \] Now, combine the terms: \[ x - 13 + 4x - 11 + x = 180 \] \[ 6x - 24 = 180 \] Now, add 24 to both sides: \[ 6x = 204 \] Dividing both sides by 6 gives: \[ x = 34 \] Now we can find each angle: - \( m \angle C = x = 34^\circ \) - \( m \angle A = x - 13 = 34 - 13 = 21^\circ \) - \( m \angle B = 4x - 11 = 4(34) - 11 = 136 - 11 = 125^\circ \) Thus, the measures of the angles in triangle \( ABC \) are: \[ m \angle A = 21^\circ, \quad m \angle B = 125^\circ, \quad m \angle C = 34^\circ \]