Final answer:
With the given sides and angle (c=64, a=87, and C=110°), applying the Law of Sines reveals only one possible triangle since angle C is obtuse, ensuring a unique solution.
Step-by-step explanation:
To determine the number of triangles ABC possible with the given parts, c=64, a=87, and C=110°, we can use the Law of Sines. The Law of Sines states that for any triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides of the triangle. In this case, however, since we have an angle C that is greater than 90° (an obtuse angle), there can only be one unique triangle.
Let's verify this by trying to calculate angle A using the Law of Sines:
\(rac{{a}}{{\sin(A)}} = \frac{{c}}{{\sin(C)}}\)
\(rac{{87}}{{\sin(A)}} = \frac{{64}}{{\sin(110°)}}\)
This gives us:
\(\sin(A) = \frac{{87 \times \sin(110°)}}{{64}}\)
If \(\sin(A)\) calculated is greater than 1 or less than -1, then a triangle is not possible, as the sine of an angle can only have values between -1 and 1.
For angle B, because the sum of the angles in a triangle must be 180°, angle B would be calculated as 180° - C - A. However, since angle C alone is already 110°, and angle A will have some positive value, it further assures us that there will not be two different triangles fitting this criteria.
Thus, there is only one possible triangle with the given measurements.