Use the triangle shown on the right and the given information to solve the triangle. \[ b=2, A=80^{\circ} \text {; find } a, c \text {, and } B \] \( \mathrm{a}=\square \) (Round to two decimal places as needed.) \( \mathrm{c}=\square^{\circ} \) (Round to two decimal places as needed.) \( \mathrm{B}=\square^{\circ} \) (Round to one decimal place as needed.)

To solve the triangle with the given information \( b = 2 \) and \( A = 80^\circ \), we can use the Law of Sines. The Law of Sines states: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] ### Step 1: Find angle \( B \) First, we need to find angle \( B \). We know that the sum of angles in a triangle is \( 180^\circ \): \[ A + B + C = 180^\circ \] Since we don't have \( C \) yet, we can express \( C \) in terms of \( B \): \[ C = 180^\circ - A - B = 180^\circ - 80^\circ - B = 100^\circ - B \] ### Step 2: Use the Law of Sines to find \( B \) Using the Law of Sines, we can set up the following equation: \[ \frac{a}{\sin A} = \frac{b}{\sin B} \] Substituting the known values: \[ \frac{a}{\sin(80^\circ)} = \frac{2}{\sin B} \] ### Step 3: Find \( a \) in terms of \( B \) Rearranging gives: \[ a = 2 \cdot \frac{\sin(80^\circ)}{\sin(B)} \] ### Step 4: Find angle \( C \) Using the Law of Sines again, we can also express \( c \): \[ \frac{c}{\sin C} = \frac{b}{\sin B} \] Substituting for \( C \): \[ \frac{c}{\sin(100^\circ - B)} = \frac{2}{\sin B} \] Rearranging gives: \[ c = 2 \cdot \frac{\sin(100^\circ - B)}{\sin B} \] ### Step 5: Solve for \( B \) To find \( B \), we can use the Law of Sines again. We know: \[ \frac{2}{\sin B} = \frac{2}{\sin(80^\circ)} \] This simplifies to: \[ \sin B = \sin(80^\circ) \] Since \( B \) must be less than \( 80^\circ \) (as \( A \) is already \( 80^\circ \)), we can conclude: \[ B = 80^\circ \] ### Step 6: Find angle \( C \) Now substituting \( B \) back into the equation for \( C \): \[ C = 100^\circ - B = 100^\circ - 80^\circ = 20^\circ \] ### Step 7: Find side \( a \) Now we can find \( a \): \[ \frac{a}{\sin(80^\circ)} = \frac{2}{\sin(80^\circ)} \] This gives: \[ a = 2 \] ### Step 8: Find side \( c \) Now we can find \( c \): \[ \frac{c}{\sin(20^\circ)} = \frac{2}{\sin(80^\circ)} \] Rearranging gives: \[ c = 2 \cdot \frac{\sin(20^\circ)}{\sin(80^\circ)} \] Calculating \( c \): \[ c \approx 2 \cdot \frac{0.3420}{0.9848} \approx 0.694 \] ### Final Answers Thus, the final answers are: \[ a \approx 2.00 \] \[ c \approx 0.69 \] \[ B \approx 80.0^\circ \] So, rounding to the required decimal places: - \( a = 2.00 \) - \( c = 0.69 \) - \( B = 80.0^\circ \)