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Parallels and Parallelograms

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Introduction

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.

image00.pngimage00.pngimage00.pngimage00.png

image14.pngimage03.pngimage13.pngimage05.pngimage04.png

Figure 1. Fourth parallel transversal crossing two parallel lines

Sum of Parallelogram(s)

Evidence

1

image15.png,image16.png,image22.png

2

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

3

image15.pngimage17.pngimage22.png

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

image01.pngimage02.pngimage01.pngimage01.pngimage02.png

image04.pngimage05.pngimage03.pngimage07.pngimage07.pngimage06.png

Figure 2.

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Middle

image08.pngimage08.pngimage08.pngimage08.pngimage08.pngimage08.png

image09.png

image20.pngimage21.pngimage22.pngimage18.pngimage19.png

image10.png

Figure 3. Sixth transversal crossing two parallel lines

Sum of Parallelogram(s)

Evidence

1

image15.png,image16.png,image22.png,image18.png,image19.png

2

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

3

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

4

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

5

image15.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

image11.pngimage11.pngimage11.pngimage11.pngimage11.pngimage11.pngimage11.png

image12.png

image20.pngimage21.pngimage22.pngimage18.pngimage19.pngimage23.png

image12.png

Figure 4.

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Conclusion


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

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

image33.png=image34.png

image35.png=image36.png

= image37.png

=image38.pngimage39.pngimage40.png

=image38.pngimage41.pngimage42.png

=image43.png


image34.pngimage44.png=


image45.png=

image46.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

Then:

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

Then,

image47.png* image49.png= image52.png

=image53.png

=image54.png

=3

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.

...read more.

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