jmc

Assume the base of the triangle is on the bottom four points because a congruent triangle can be made by reflecting the base on the top four points. For a triangle with a base of length $1$, there are $3$ triangles. For a triangle with a base of length $2$, there are $3$ triangles. For length $3$, there are $2$. In total, the number of non-congruent triangles is $3+3+2=\boxed{8}$.