Determine the remaining sides and angles of the triangle ABC. \( \mathrm{c}=6 \mathrm{mi}, \mathrm{B}=37.89^{\circ}, \mathrm{C}=34.67^{\circ} \) Find the measure of angle A . \( \mathrm{A}=\square^{\circ} \) (Type an integer or a decimal.) Find the length of side a. \( \mathrm{a}=\square \mathrm{mi} \) (Round to the nearest mile as needed.) Find the length of side b. \( \mathrm{b}=\square \mathrm{mi} \) (Round to the nearest mile as needed.)

To find the remaining sides and angles of triangle ABC, we start with angle A. Since the sum of angles in a triangle is always 180 degrees, we can calculate angle A as follows: \[ A = 180^\circ - (B + C) = 180^\circ - (37.89^\circ + 34.67^\circ) = 107.44^\circ \] Now we have: \( A = 107.44^{\circ} \) Next, we find the lengths of sides a and b using the Law of Sines, which states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] First, we calculate side b: \[ \frac{b}{\sin B} = \frac{c}{\sin C} \Rightarrow b = \frac{c \cdot \sin B}{\sin C} = \frac{6 \cdot \sin(37.89^\circ)}{\sin(34.67^\circ)} \] Calculating further: Using approximate values for the sine functions, \(\sin(37.89^\circ) \approx 0.610\) and \(\sin(34.67^\circ) \approx 0.574\): \[ b \approx \frac{6 \cdot 0.610}{0.574} \approx 6.37 \text{ mi} \quad \text{(rounding gives } b \approx 6 \text{ mi)} \] Now we calculate side a: \[ \frac{a}{\sin A} = \frac{c}{\sin C} \Rightarrow a = \frac{c \cdot \sin A}{\sin C} = \frac{6 \cdot \sin(107.44^\circ)}{\sin(34.67^\circ)} \] Using \(\sin(107.44^\circ) \approx 0.943\): \[ a \approx \frac{6 \cdot 0.943}{0.574} \approx 9.83 \text{ mi} \quad \text{(rounding gives } a \approx 10 \text{ mi)} \] In summary: \( A = 107.44^{\circ}, \quad a \approx 10 \text{ mi}, \quad b \approx 6 \text{ mi} \)