Parallels and Parallelograms - When parallel lines are...

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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: 1 A , 2 A , 3 A , 1 A 2 A , 2 A 3 A , 1 A 3 A . Figure 1. Fourth parallel transversal crossing two parallel lines Sum of Parallelogram(s) Evidence 1 1 A , 2 A , 3 A 2 1 A 2 A , 2 A 3 A 3 1 A 3 A A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: 1 A , 2 A , 3 A , 4 A , 1 A 3 A , 2 A 4 A , 1 A 2 A , 2 A 3 A , 3 A 4 A , 1 A 4 A 1 A 2 A 3 A
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Figure 2. Fifth parallel transversal crossing two parallel lines. Sum of Parallelogram(s) Evidence 1 1 A , 2 A , 3 A , 4 A 2 1 A 2 A , 2 A 3 A , 3 A 4 A 3 1 A 3 A , 2 A 4 A 4 1 A 4 A A sixth parallel transversal is added to the diagram as shown in Figure 3. 15 parallelograms are formed: 1 A , 2 A , 3 A , 4 A , 5 A , 1 A 4 A , 1 A 5 A , 2 A 5 A , 1 A 3 A , 2 A 4 A , 3 A 5 A , 1 A 2 A , 2 A 3 A , 3 A 4 A , 4 A 5 A . 1 A 2 A 3 A 4 A 5 A Figure 3. Sixth transversal crossing two parallel lines 2 A 4 A 3 A 1 A
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Sum of Parallelogram(s) Evidence 1 1 A , 2 A , 3 A , 4 A , 5 A 2 1 A 2 A , 2 A 3 A , 3 A 4 A , 4 A 5 A . 3 1 A 3 A , 2 A 4 A , 3 A 5 A 4 1 A 4 A , 2 A 5 A 5 1 A 5 A A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed: 1 A , 2 A , 3 A , 4 A , 5 A , 6 A
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