Find the number of triangles in the given figure.

Concept: When a figure is built from several straight lines that cross each other, break it down into the smallest, non-overlapping triangular regions first. Every triangle in the figure is then either one of these smallest regions by itself, or a combination of two or more adjacent regions. Counting systematically means tallying every triangle in each size class — built from 1 region, then 2 regions, then 3 regions, and so on — and adding the tallies together, rather than trying to spot every triangle at a glance.

Application: Label the figure as shown, with F and G marking the two points where internal lines intersect.

Grouping every triangle by how many of the smallest regions it is made of:

• Smallest (1-region) triangles — 8 in number: ABF, BFG, BCG, CGH, GHD, GED, EFG, AFE.

• Triangles made of two regions — 5 in number: ABG, BGE, AGE, ABE, GCD.

• Triangles made of three regions — 4 in number: BCD, CDE, BED, BCE.

Adding the three groups: 8 + 5 + 4 = 17 triangles in total.

Cross-check: The figure is symmetric about the horizontal line through A and H, which swaps B with E and C with D. Under this symmetry the triangles pair up as (ABF, AFE), (ABG, AGE), (BFG, EFG), (BCG, GED), (CGH, GHD), (BCD, CDE) and (BCE, BED) — seven pairs — while ABE, GCD and BGE each map to themselves. That gives 7 × 2 + 3 = 17, confirming the count from the grouping above.