All the triangles can stick together. The triangles counted is the independent triangles, triangles made up of two shapes, a triangle made up from 3 shapes, or the outline of the shape consisting of 4, 5 or 6 lines.
I seem to recall that the solution to this problem for $n$ lines is
$n \choose 3$, so for 4 lines it's 4, for 5 lines it's 10 and for 6 lines it's 20.
The idea is that
each combination of 3 lines generates exactly one triangle. They intersect in at most 3 points, which form the triangle. Actually, as long as you don't choose parallel lines, they will intersect in exactly 3 points. There are $n \choose 3$ of these combinations.
For example, here is the solution for 6 lines: