In Exercises 19 and 20 , assume that \( \angle \boldsymbol{A} \) is supplementary to \( \angle \boldsymbol{B} \) and complementary to \( \angle \boldsymbol{C} \). Determine \( \boldsymbol{m} \angle \mathbf{A}, \boldsymbol{m} \angle \boldsymbol{B} \), and \( \boldsymbol{m} \angle \boldsymbol{C} \). \( \begin{array}{c}\text { 157 } \\ \text { 19. } m \angle A=(x+10)^{\circ}, m \angle B=(12 x+1)^{\circ}, m \angle C=(5 x+2)^{\circ} \\ \text { 20. } m \angle A=(2.5 x+17)^{\circ}, m \angle B=(21 x-25)^{\circ} . m \angle C=(8 x-11)^{\circ} \\ 33.25\end{array} 111.5 \)

To tackle this problem, we need to set up some equations based on the relationships between the angles. Since angle \(A\) is supplementary to angle \(B\), we can express that as: \[ m\angle A + m\angle B = 180^\circ \] Additionally, since angle \(A\) is complementary to angle \(C\): \[ m\angle A + m\angle C = 90^\circ \] We can substitute the expressions given for \(m\angle A\), \(m\angle B\), and \(m\angle C\) into these equations. ### For Exercise 19: 1. Substitute the expressions: \[ (x + 10) + (12x + 1) = 180 \] Simplifying gives: \[ 13x + 11 = 180 \implies 13x = 169 \implies x = 13 \] 2. Now substitute \(x = 13\) back into the angle expressions: - \(m\angle A = 13 + 10 = 23^\circ\) - \(m\angle B = 12(13) + 1 = 157^\circ\) - \(m\angle C = 5(13) + 2 = 67^\circ\) ### For Exercise 20: 1. Using the same approach, set up the equations: \[ (2.5x + 17) + (21x - 25) = 180 \] Simplifying gives: \[ 23.5x - 8 = 180 \implies 23.5x = 188 \implies x \approx 8 \] 2. Substitute \(x \approx 8\) back into the expressions: - \(m\angle A = 2.5(8) + 17 = 25^\circ\) - \(m\angle B = 21(8) - 25 = 143^\circ\) - \(m\angle C = 8(8) - 11 = 53^\circ\) This brings us to the final measures of angles. There you go, angles \(A\), \(B\), and \(C\) are now all figured out!