Study the given figure carefully and count the number of parallelograms in it.

Concept

In a straight-line diagram, a parallelogram is determined by two pairs of parallel boundary lines. Grouping shapes by the two line directions used counts every parallelogram once and prevents double-counting.

Application

1. Horizontal and right-slanting sides: between the bottom and middle horizontal levels, four right-slanting lines give C(4, 2) = 6 parallelograms. Between bottom and top, three lines give C(3, 2) = 3; between middle and top, the same three lines give 3. This family contributes 6 + 3 + 3 = 12.

2. Horizontal and left-slanting sides: the figure is symmetric about its vertical centre, so the same span-by-span count gives 6 + 3 + 3 = 12.

3. Right-slanting and left-slanting sides: label the four right-slanting lines R1–R4 from left to right at the base, and the four left-slanting lines L1–L4 from right to left at the base. Each row below selects two boundary lines from each family whose four intersections lie on drawn segments.

   Right-slant pair	Left-slant pair
   R1–R2	L2–L3
   R1–R2	L2–L4
   R1–R2	L3–L4
   R1–R3	L2–L3
   R2–R3	L1–L2
   R2–R3	L1–L3
   R2–R3	L2–L3
   R2–R4	L1–L2
   R3–R4	L1–L2

   These are the nine complete two-slant parallelograms. Every other line-pair choice reaches beyond a drawn segment, so this family contributes 9.

Cross-check

Every drawn side is horizontal, right-slanting, or left-slanting. The three order-independent direction pairs above therefore exhaust all possible parallelograms, and no shape can belong to two different direction-pair groups.

Result

The total number of parallelograms is 12 + 12 + 9 = 33.