Count the number of parallelograms in the given figure.

Concept: A parallelogram is a four-sided region whose two pairs of opposite sides are parallel. When a figure is built from families of parallel straight lines, every parallelogram is bounded by two parallel lines taken from one direction and two parallel lines taken from a second direction. So the parallelograms are counted systematically by taking the line-directions two at a time - this guarantees none is missed and none is counted twice.

Application: The segments in this star figure run in three directions: a first set of slanted lines, a second set of slanted lines, and the horizontal lines. Counting the lines in each direction gives 3 lines in the first slanted direction, 3 lines in the second slanted direction, and 2 horizontal lines. Now count the parallelograms for each pair of directions:

1. Both pairs of sides slanted: there are 3 ways to pick 2 of the 3 lines in the first slanted direction and 3 ways to pick 2 of the 3 lines in the second slanted direction, giving 3 x 3 = 9 parallelograms.

2. One pair of sides from the first slanted direction and one pair horizontal: 3 ways to pick 2 of the 3 slanted lines and 1 way to pick both horizontal lines, giving 3 x 1 = 3 parallelograms.

3. One pair of sides from the second slanted direction and one pair horizontal: again 3 x 1 = 3 parallelograms.

Cross-check: Every parallelogram uses exactly two of the three line-directions, and the three cases above cover each such pair of directions exactly once, so no parallelogram is missed and none is double-counted. Adding them: 9 + 3 + 3 = 15.