Parallel Lines

Parallel lines have many interesting properties in the world of math. More importantly, they have effects that are predictable and constant. This means that we can create all kinds of useful "theorems" or rules to use when we see parallel lines. But what exactly are these rules? Why are parallel lines so special? What is the technical definition of parallel lines, and what can these lines teach us about math? Let''s find out:

What are parallel lines?

Parallel lines are coplanar. This means that they lie in the same plane. Not only that, but parallel lines also never intersect. There is only one possible way for this to happen, and this is if parallel lines are perfectly aligned with each other. In other words, parallel lines point in the same direction but do not touch. In a two-dimensional plane, this means that parallel lines have exactly the same slope. Remember that we see "slopes" when lines go up or down on a graph.

Visualizing parallel lines

Like many aspects of math and geometry, it helps if we can visualize how parallel lines work. Take a look at the following diagram:


They are also parallel because they will never touch each other. They point in the same direction and line up perfectly. They also have the same slope. Therefore, we can safely say that these lines meet the definition of parallelity.

Examples of parallel lines

But what do parallel lines actually look like when we see them in equations?

Let''s take a look at an example:

What do these lines look like on a graph? Take a look:


What is it about our equations that made these lines parallel? Easy: Both equations had a slope of 3. If the equations are different but their slopes are the same, we know that the lines can be parallel.

Parallel lines, angles, and proportionality

There are many other interesting properties associated with parallel lines. One property involves pairs of angles formed by parallel lines crossed by a transversal. Many of these angles are congruent, including the corresponding angles, alternate exterior angles, and alternate interior angles. Additionally, vertical angles are always congruent, regardless of whether the lines are parallel or not.

We also know that when we add the consecutive interior angles or supplementary angles together in this situation, we are always left with 180 degrees.

Another interesting property of parallel lines has to do with proportionality. When parallel lines cut through a triangle or across transversals, they create proportional segments.

Note that we can use many of these rules to prove that parallel lines are indeed parallel. Sometimes, it might not be clear-especially since there''s no way of tracking these parallel lines through infinite space to see if they ever intersect. By using transversals and studying the resulting angles, we can determine whether two lines are parallel.

Shapes with parallel lines

You might also be aware that many shapes have parallel lines. The obvious examples would be a square or a rectangle, which contain two pairs of parallel lines. Parallelograms also contain parallel lines (their name is a dead giveaway). Many other shapes have some degree of parallelism-even complex shapes like octagons and hexagons.

Symbols for parallel lines

Writing the word "parallel" can be a little tiresome at times, especially when we need to remember how many "ls" go where. Fortunately, there is a handy symbol that represents parallelism in the world of math: "||."

Even better, there''s a symbol for non-parallelism: "∦."

This is a lot easier than writing out "is parallel to" or "is not parallel to."

Why parallel lines are important

Sometimes, it is very important that we build things with parallel lines. Think about the building industry. What would happen if two vertical sides of a skyscraper were not parallel? The building wouldn''t last very long. Or what about train tracks? What if the two tracks collided or veered too far apart? That would be bad news for the train. Parallel lines allow us to build things with accuracy, beauty, and stability.

Topics related to the Parallel Lines

Parallel Lines

Parallel Lines and Slopes

Volume of a Cylinder

Flashcards covering the Parallel Lines

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Parallel Lines

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

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