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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: image15.png,image16.png,image22.png,image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png, image15.pngimage17.pngimage22.png.



Figure 1. Fourth parallel transversal crossing two parallel lines

Sum of Parallelogram(s)








A fifth parallel transversal is added to the diagram as shown in Figure 2. 10 parallelograms are formed: image15.png,image16.png,image22.png,image18.png,image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png,image15.pngimage17.pngimage18.png



Figure 2.

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Figure 3. Sixth transversal crossing two parallel lines

Sum of Parallelogram(s)





image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png, image18.pngimage17.pngimage19.png.


image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image22.pngimage17.pngimage19.png





A seventh parallel transversal is added to the diagram as shown in Figure 4. Fifteen parallelograms are formed:image15.png,image16.png,image22.png,image18.png,image19.png,image23.png,image15.pngimage17.pngimage23.png,image15.pngimage17.pngimage19.png,image16.pngimage17.pngimage23.png,image15.pngimage17.pngimage18.png, image16.pngimage17.pngimage19.png, image22.pngimage17.pngimage23.png,image15.pngimage17.pngimage22.png,image16.pngimage17.pngimage18.png, image22.pngimage17.pngimage19.png,image18.pngimage17.pngimage23.png,image15.pngimage17.pngimage16.png,image16.pngimage17.pngimage22.png,image22.pngimage17.pngimage18.png, image18.pngimage17.pngimage19.png,image19.pngimage17.pngimage23.png





Figure 4.

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image30.png = image31.png= t-th triangular number

image32.png= summation of a t-th triangular number(s) from 1 to n



= image37.png







Thus this shows that if it's true for n, it's true for n + 1. Since we showed it

was true for n = 1, we now know it's also true for n = 1 + 1 = 2, and

then for n = 2 + 1 = 3, and so on, for all n >= 1.

Then, a general statement statement that satisfies both transversal and parallel lines can be drawn in that m represents horizontal parallel lines in n represents the intersected parallel transversals:

Suppose that:

image47.png=image48.png and image49.png=image50.png


image47.png* image49.png=image51.png

If m=2 in that there are 2 horizontal parallel lines and n=3 in that there are 3 parallel transversals


image47.png* image49.png= image52.png




The conclusion that 3 parallelograms formed when 2 horizontal parallel lines are intersected by 3 parallel transversals is valid. Therefore, the general statement validity is true. The limitations of the equation image47.png* image49.png=image51.png is that it only considers the number of parallelograms formed by intersecting parallel lines. However, the equation gives an accurate result of when m represents horizontal parallel lines in n represents the intersected parallel transversals.

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