When information is given about sides and angles of a triangle, sometimes there are no triangles possible with the given information, sometimes there is 1 triangle possible and other times 2 triangles possible.
C = 55 degrees, a = 10, c = 14
Use the law of sine to find angle a
c/Sin C = a/Sin A
14/sin 55 = 10/sin A
0.585 = Sin A
A = 36 degrees
So we know there is at least 1 triangle with angles, 55, 36, and 89 degrees.
But angle A could also be 180 - 36 = 144 since sine is positive in both the first and second quadrants. But that would be impossible because 144 + 55 (measure of angle C) = 189 which is greater than 180 degrees that all 3 triangles must add up to.
So therefore there is only 1 triangle possible.
If in the first step when solving for angle A, you get a value outside of -1 to 1, then there is no solution and no triangle exists.
If 180 - A plus measure of angle C is less than 180, then there are two triangles possible.