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.

Information given

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.