Determine the remaining sides and angles of the triangle \( A B C \). \( A=26.61^{\circ} \quad B=34.51^{\circ} \quad C=14.46 \mathrm{~m} \) \( C=\square^{\circ} \)

To determine the remaining sides and angles of triangle \( ABC \), we can use the following steps: 1. **Find the remaining angle \( A \)**: The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can find angle \( A \) using the formula: \[ A = 180^\circ - B - C \] Substituting the known values: \[ A = 180^\circ - 34.51^\circ - 26.61^\circ \] \[ A = 180^\circ - 61.12^\circ = 118.88^\circ \] 2. **Find the remaining sides \( a \) and \( b \)**: We can use the Law of Sines to find the lengths of sides \( a \) and \( b \): \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] We know \( C = 14.46 \, \text{m} \) and we have calculated \( A \) and \( B \). First, we can find side \( a \): \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] Rearranging gives: \[ a = c \cdot \frac{\sin A}{\sin C} \] Substituting the known values: \[ a = 14.46 \cdot \frac{\sin(118.88^\circ)}{\sin(14.46^\circ)} \] Now, calculate \( \sin(118.88^\circ) \) and \( \sin(14.46^\circ) \): \[ \sin(118.88^\circ) \approx 0.8746 \] \[ \sin(14.46^\circ) \approx 0.2497 \] Now substituting these values: \[ a = 14.46 \cdot \frac{0.8746}{0.2497} \approx 14.46 \cdot 3.500 \approx 50.6 \, \text{m} \] Next, we can find side \( b \): \[ \frac{b}{\sin B} = \frac{c}{\sin C} \] Rearranging gives: \[ b = c \cdot \frac{\sin B}{\sin C} \] Substituting the known values: \[ b = 14.46 \cdot \frac{\sin(34.51^\circ)}{\sin(14.46^\circ)} \] Now calculate \( \sin(34.51^\circ) \): \[ \sin(34.51^\circ) \approx 0.5660 \] Now substituting these values: \[ b = 14.46 \cdot \frac{0.5660}{0.2497} \approx 14.46 \cdot 2.270 \approx 32.8 \, \text{m} \] 3. **Summary of the triangle**: - Angle \( A \approx 118.88^\circ \) - Angle \( B = 34.51^\circ \) - Angle \( C = 14.46^\circ \) - Side \( a \approx 50.6 \, \text{m} \) - Side \( b \approx 32.8 \, \text{m} \) - Side \( c = 14.46 \, \text{m} \) Thus, the remaining angle \( A \) is approximately \( 118.88^\circ \), side \( a \) is approximately \( 50.6 \, \text{m} \), and side \( b \) is approximately \( 32.8 \, \text{m} \).