When you count triangles it is very important to avoid non-triangles.
It can be easily demonstrated in the first stage of counting the basic shapes.
Look at this Pentagon, for instance:
The inner shape is also a Pentagon.
So, though there are 11 basic shapes
in it, but only 10 of them are triangles.
More than that:
When you stitch the inner Pentagon to
one of the triangles in the second phase
the shape is not a triangle also…
In short: beware!
This Pentagon riddle is solved in the book, of course.