Note all the points where two or more lines meet. Label all the lines that join two points with consecutive numbers. Where three edges enclose a space there’s a triangle. Identify the triangle by writing down its sides in numerical order, for example, 3-4-9. Find all the triangles which have single numbers for their side labels and list them.
Next, where two edges join to form a longer edge, this may be the side of a larger triangle, so look for two other sides that enclose a space. You may find that the other sides are also composed of two shorter edges. For example, (1-2)-(7-9)-(11-15) could identify a triangle. List all such triangles.
Where three edges join to form a longer edge, it may be the side of an even larger triangle. Continue to look for and identify these larger triangles by checking to see if three longer edges enclose a space to form a triangle. List all occurrences. And so on, until all edges have been examined.
You can check each list of triangles by ensuring that you have included all the labels.
Finally, count the number of triangles in each list to get the total.