Parallels and Parallelograms

When parallel lines are intersected by parallel transversals to form parallelograms, relationships can be drawn between the number of parallelograms, transversals, and parallel lines. The number of parallelograms follow a triangular sequence when two parallel lines are intersected by transverals. In contrast, when three parallel lines are intersected by transversals, the number of parallelograms follow a tetrahedral number sequence.

Figure 1 below shows a pair of horizontal parallel lines and four parallel transversals. Six parallelograms are formed: ,,,,, .

Figure 1. Fourth parallel transversal crossing two parallel lines

A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: ,,,,,, ,,,

Figure 2. Fifth parallel transversal crossing two parallel lines.

A sixth parallel transversal is added to the diagram as shown in Figure 3. 15 parallelograms are formed: ,,,,,, ,, ,, ,,,, .

Figure 3. Sixth transversal crossing two parallel lines

A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:,,,,,,,,,, , ,,, ,,,,, ,

Figure 4. Seventh parallel transversal crossing two parallel lines

After repeating the process of adding transversals consecutively, Figure 5 shows the conclusions that were made between the relationship between transversals and number of parallelograms formed ...