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

To solve these angle relationships, we can stick to the definitions of supplementary and complementary angles. Since angle \( \angle A \) is supplementary to \( \angle B \), we know that \( m \angle A + m \angle B = 180^\circ \). Similarly, since \( \angle A \) is complementary to \( \angle C \), we have \( m \angle A + m \angle C = 90^\circ \). For Exercise 19, we can set up two equations based on this information: 1. \( (x + 10) + (12x + 1) = 180 \) 2. \( (x + 10) + (5x + 2) = 90 \) Now, for exercise 20, we can do the same thing: 1. \( (2.5x + 17) + (21x - 25) = 180 \) 2. \( (2.5x + 17) + (8x - 11) = 90 \) Solving these systems of equations should give you the measures for \( m \angle A \), \( m \angle B \), and \( m \angle C \). To make these problems easier and avoid common mistakes, remember to carefully combine like terms and isolate your variable step-by-step. Also, double-check to ensure that the angles make sense in the context of the relationships defined (i.e., checking if \( m \angle A + m \angle B \) actually equals \( 180^\circ \) and \( m \angle A + m \angle C \) equals \( 90^\circ \)). Happy calculating!