We have to find all choices of positive 
integers x, y, and z such that

x + y + z = 15

x + y + z = 15
x + y > z
x + z > y
y + z > x

From
x + y + z = 15
    x + y = 15 - z

and since x + y > z,
         15 - z > z
             15 > 2z
            7.5 > z

From
x + y + z = 15
    x + z = 15 - y

and since x + z > y,
         15 - y > y
             15 > 2y
            7.5 > y

From
x + y + z = 15
    y + z = 15 - x

and since y + z > x,
         15 - x > x
             15 > 2x
            7.5 > x

So all the sides are 7 or less

And since to make sure they are non-congruent, 

we'll require that x < y < z, so

x = 1 y = 7 z = 7
x = 2 y = 6 z = 7
x = 3 y = 5 z = 7
x = 3 y = 6 z = 6
x = 4 y = 4 z = 7
x = 4 y = 5 z = 6
x = 5 y = 5 z = 5

So there are 7.

Edwin