Step 1 :Given that we know the angle C and the sides b and c, we can first find the angle B using the formula: \(B = \sin^{-1}\left(\frac{b \sin C}{c}\right)\)
Step 2 :Then, we can find the angle A using the fact that the sum of the angles in a triangle is 180 degrees: \(A = 180 - B - C\)
Step 3 :We need to consider the ambiguous case of the Law of Sines, where two different triangles satisfy the given conditions. This occurs when the angle B is acute (less than 90 degrees) and the side a (opposite to angle A) is less than the height of the triangle (which is b sin B). In this case, there is another possible angle for B, which is: \(B' = 180 - B\) And the corresponding angle A' is: \(A' = 180 - B' - C\)
Step 4 :We need to calculate these angles and check if they satisfy the conditions for the ambiguous case.
Step 5 :The calculations have been done and it seems that there is no ambiguous case for this triangle, as the calculated angles A' and B' are the same as A and B, respectively.
Step 6 :Therefore, there is only one possible solution for the triangle. The measurements for the remaining angles are A = 101° 5' and B = 31° 15'.
Step 7 :Final Answer: \(\boxed{\text{The correct choice is A. There is only one possible solution for the triangle. The measurements for the remaining angles are } A=101^{\circ} 5^{\prime} \text{ and } B=31^{\circ} 15^{\prime}}\)