Find the unknown angles in triangle $\mathrm{ABC}$ for each triangle that exists. \[ A=80.8^{\circ} \quad b=9.5 \mathrm{ft} \quad a=10.4 \mathrm{ft} \] Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (Round to the nearest tenth as needed.) A. There is only one possible solution for the triangle. The measurements for the remaining angles are $\mathrm{B}=\square^{\circ}$ and $\mathrm{C}=\square^{\circ}$. B. There are two possible solutions for the triangle. The measurements for when $\mathrm{B}$ is larger are $\mathrm{B}=\square^{\circ}$ and $\mathrm{C}=\square^{\circ}$. The measurements for when $\mathrm{B}$ is smaller are $\mathrm{B}=\square^{\circ}$ and $\mathrm{C}=\square^{\circ}$. C. There are no possible solutions for the triangle.

Step 1 ：We are given one angle (A) and two sides (a and b) of a triangle. We can use the Law of Sines to find angle B. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. So, we can write: \[\frac{a}{\sin A} = \frac{b}{\sin B}\]

Step 2 ：We can solve this equation for B to find: \[B = \sin^{-1}\left(\frac{b \sin A}{a}\right)\]

Step 3 ：Once we have angle B, we can find angle C by subtracting the sum of angles A and B from 180 degrees, because the sum of the angles in a triangle is always 180 degrees: \[C = 180 - A - B\]

Step 4 ：We need to be careful, though, because the Law of Sines can sometimes give two possible solutions for a triangle. This happens when the angle we are trying to find (in this case, B) is obtuse. The sine function has the same value for an acute angle and its supplementary angle (180 degrees minus the angle), so we need to check if the supplementary angle of B is a valid angle for the triangle. If it is, then we have two possible solutions for the triangle.

Step 5 ：However, the supplementary angle of B is not a valid angle for the triangle because it results in a negative value for angle C. Therefore, there is only one possible solution for the triangle.

Step 6 ：Final Answer: \[\boxed{B = 64.4^\circ, C = 34.8^\circ}\]