1. Identify the Sets of Lines

The figure is composed of three types of lines:

• Horizontal-ish lines ($H$): There are 5 lines (the bottom base, 3 internal lines, and the top base).
• Vertical-ish lines ($V$): There are 5 lines (the left side, 3 internal lines, and the right side).
• Diagonal line ($D$): A single line connecting the bottom-left vertex to the top-right vertex.

2. Understand Triangle Formation

For a triangle to be formed, we need three non-parallel lines that do not all intersect at the same point.

• Since all lines in set $H$ are parallel to each other, no two lines from $H$ can form a triangle.
• Similarly, no two lines from set $V$ can form a triangle.
• Therefore, every triangle in this figure must include the diagonal line ($D$) as one of its sides.
• To form a triangle with the diagonal, we must choose one line from set $H$ and one line from set $V$ that intersect the diagonal at two different points.

3. Mathematical Calculation

Each intersection of a horizontal line and a vertical line (a "grid point") represents a potential triangle vertex when paired with the diagonal. There are:

$ 5 \text{ (horizontal lines)} \times 5 \text{ (vertical lines)} = 25 \text{ intersection points} $

However, if a horizontal line, a vertical line, and the diagonal line all meet at the same point, they do not form a triangle (the area is zero). In a $4 \times 4$ grid (5 lines by 5 lines) where the diagonal connects opposite corners, the diagonal passes exactly through the grid points where the indices are equal (e.g., $(0,0), (1,1), (2,2), (3,3), (4,4)$).

• Total intersections: 25
• Points where the diagonal meets a grid intersection: 5

The number of triangles is the total number of intersections minus the number of points where the lines are concurrent:

$ \text{Total Triangles} = 25 - 5 = 20 $