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High School Geometry: Help and Review13 chapters | 162 lessons

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Lesson Transcript

Instructor:
*Kimberlee Davison*

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

In this lesson, you will learn about parallel lines and how you can identify and label them. You'll also find out how they are similar to things found in everyday life.

**Parallel lines** are two lines that are always the same distance apart and never touch. In order for two lines to be parallel, they must be drawn in the same **plane**, a perfectly flat surface like a wall or sheet of paper.

Parallel lines are useful in understanding the relationships between paths of objects and sides of various shapes. For example, squares, rectangles, and parallelograms have sides across from each other that are parallel.

In formulas, parallel lines are indicated with a pair of vertical pipes between the line names, like this:

*AB* || *CD*

Each line has many parallels. Any line that has the same slope as the original will never intersect with it. Lines that would never cross, even if extended forever, are parallel.

Sometimes it is helpful to think of parallel lines as a set of railroad tracks. The two rails of the track are created for the wheels on each side of the train to travel along. Because the wheels of the train are always the same distance apart, they do not get closer, not even when they turn.

The tracks have to be the same distance apart everywhere. Since they are made by humans, railroad tracks aren't quite parallel, but in order to work properly, they have to be awfully close.

The other difference between railroad tracks and perfectly parallel lines is that tracks are built over hills and valleys. The ground they cover is not perfectly flat. When mathematicians imagine parallel lines, on the other hand, they draw them on a perfectly flat surface.

When two parallel lines are graphed, they must always be at the same angle, which means they'll always have the same **slope**, or steepness.

Here's a graph of two parallel lines:

Even though these two lines don't start in the same place, you can see that they are equally steep. They decrease or slope downward at the same rate. That ensures that they're parallel.

If you look at the equations of those two lines, you may notice something: they are exactly the same except for the numbers on the right hand side, the '6' and '12.' We can use some simple algebra and rewrite the two equations in **slope-intercept** form (*y* = *mx* + *b*), a form of a line that is the most familiar to people.

The blue line's equation will then be:

*y * = -3/2*x * + 6

While the red line will have an equation of:

*y* = -3/2*x* + 3

Again, you might notice that the two equations are exactly the same except for one thing: the number to the right of *x*. That number is called the **constant** and it tells us how high or low the line sits on the graph. When the line is in slope-intercept form, that number also tells us the **y-intercept** which is exactly where the line hits the y-axis, the vertical line marking zero on the graph. In this graph, the two lines are exactly the same except that one of them is above the other.

When lines are in slope-intercept form, the number by which *x* is multiplied (that's the number the number attached to *x* in the question) indicates the line slope. You can see that both lines in our graph have the same slope, -3/2. Parallel lines will always have the same slope. In fact, showing that the slopes of two lines are equal is one way to prove that they are parallel.

In geometry, lines are considered to go one forever. Sections of lines with beginning and end points aren't called lines but **line segments**. Even though line segments don't go on forever, they can still be parallel.

However, unlike lines, it's possible for two line segments to never meet and still not be perfectly parallel. Line segments are *not parallel* if they are closer to each other at one end than another because they would eventually meet if they were extended further.

If the two segments are straight, all you need to do is measure the distances between the segments at both ends. If those points are the same distance from one another, those two segments are probably parallel.

**Parallel lines** are lines that are always the same distance apart. Because they are always the same distance from one another, parallel lines will never intersect. You can find parallel lines in many places including the sides of geometric shapes, like squares and rectangles, and railroad tracks in your neighborhood.

When plotted on a graph, parallel lines will have the same **slope**. This is easily visible when the equations for parallel lines are listed in **slope-intercept form**. If the slope variable is the same, the lines are parallel.

Here are a few important things to remember:

- First, every line can have
**multiple parallels**. A line will be parallel with any line which has the same slope, so many lines can be parallel at once. - Second,
**line segments**might not be parallel even if they don't meet. The way to know for sure is to measure the distances at the ends. If they're the same distance from one another, then the segments have the same slope and would be parallel if they extended further.

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High School Geometry: Help and Review13 chapters | 162 lessons

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