Parallels and Parallelograms

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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

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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 ...

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