# Type I - Parallels and Parallelograms

Parallels and Parallelograms Math Portfolio

Introduction:

This investigation aims at finding a relationship between the numbers of horizontal parallel lines and the transversals. When these lines intersect they form parallelograms. The aim of this investigation is examine and determine a general statement for transversals and horizontal lines and how they affect the number of parallelograms formed within the figure. A diagram of a parallelogram and a transversal is shown below.

These lines represent two transversals; however they are supposed to intersect with a horizontal parallel line.

These lines represent two horizontal parallel lines. They are intersected with a number of transversals.

When there is one horizontal line and two transversals, this makes one parallelogram, A1, as shown below:

This diagram shows that when there are two horizontal lines and two transversals, one parallelogram is formed. This parallelogram is called A1. However, when another transversal is added to the same diagram and same pair of horizontal lines, it looks like this:

This figure shows that when another transversal is added to a pair of horizontal parallel lines, three parallelograms are formed. Although the third parallelogram is not labeled, it is clear that the total of A1  A2 makes a big parallelogram, A3. So therefore, there is A1, A2, and A3. Another transversal added to this figure is shown below:

In this figure, since a fourth transversal was added to the pair of horizontal parallel lines, this contains six different parallelograms within the diagram. Firstly you have A1, A2, and A3. However, A1  A2 form another parallelogram, which makes A4. Also, A2  A3 form another parallelogram, which creates A5. Lastly, A1  A2  A3 create a final parallelogram, A6. Therefore, there are six parallelograms formed when a fourth transversal is added to the pair of horizontal parallel lines. When five transversals are added to this figure, the figure appears as:

As a fifth transversal is added to the same figure, this contains much more parallelograms. In this diagram there are ten parallelograms. This diagram firstly contains A1, A2, A3, and A4. Also A5 is formed when A1 and A2 are combined, A1  A2. A6 is formed because A2  A3 can be combined to form another parallelogram. Also A3  A4 make another parallelogram, forming A7. When A1  A2 A3 are combined this forms another parallelogram, making A8. Then A2  A3  A4 make a parallelogram which is A9. Then lastly combining all of these together, A1  A2  A3  A4, this makes the last parallelogram in this figure, A10. The figure below represents six transversals added to the same figure.

In this figure, there are six transversals added to the pair of horizontal parallel lines. With six transversals, there are fifteen parallelograms within this diagram. The figure firstly obviously shows five parallelograms, A1, A2, A3, A4, and A5. A6 is formed when you combine A1 with A2, A1  A2. Then when you combine A1 with A2 and A3 (A1  A2  A3), A7 is formed. Also A1  A2  A3  A4 create another parallelogram, A8. Another parallelogram is formed when A1  A2  A3  A4  A5 are combined. This makes A9. When A2  A3 are combined this makes A10. Also when A2  A3  A4 are combined, this makes another parallelogram, therefore A11 is created. A2  A3 ...