DaQuita has taught high school mathematics for six years and has a master's degree in secondary mathematics education.
Lines, Lines Everywhere
Take a look at your surroundings. Are you sitting at a desk? Are you close to a window with blinds? If you look out that window, can you see the next street or a highway? If you answered yes to any of these questions, then you are surrounded by lines, which are everywhere!
In this lesson, we are going to take a closer look at parallel lines, perpendicular lines and transverse lines. Each of these types of lines are classified as coplanar lines, meaning that they are located on the same plane, which is a flat, two-dimensional surface. Let's examine and practice with each one.
Parallel lines are defined as coplanar lines that do not intersect. They have the same slope and, just as the definition states, will never, ever meet at any point. Think about it: since slope is referred to as rise over run, having the same slope means that two lines will rise and run at the exact same rate, ensuring that they will never intersect each other. Let's take a look at real-life examples of parallel lines.
First, we have a window with blinds. Here, you can see that each blind is moving in the same direction and never touches another blind. Next, we have a parking lot. Notice that all of the lines are going in the same direction.
Perpendicular lines are coplanar lines that intersect and form a 90-degree angle. So, any time you have perpendicular lines, you will also have right angles and vice versa.
The slopes of perpendicular lines are opposite reciprocals of each other. Being opposite means that one slope will be positive and the other will be negative. Being reciprocals means that one slope will be the upside down or flipped version of the other.
Perpendicular lines are also visible in the real world. Take a look at a desk. Can you see how the top of it lays flat on all the legs? This means that the top of the desk is perpendicular to the legs and forms ninety-degree angles, which keeps things from sliding off of it.
Parallel, Perpendicular or Neither?
Now, let's practice what we've learned so far. If line g = 3x + 7 and line h = -3x - 2, are these lines parallel, perpendicular or neither?
Let's begin by looking at their slopes, which are the numbers in front of the x variables. Line g has a slope of three and line h has a slope of negative three. Their slopes are the same number, but one is positive and the other is negative. so they are not exactly the same. For this reason, we know that line g is not parallel to line h. Also, though their slopes are opposites, they are not reciprocals of each other. Therefore, we can also conclude that these two lines are not perpendicular.
For our next example, line j = 4/3x + 2 and line k = -3/4x + 5. Are these two lines parallel, perpendicular or neither?
By looking in front of the x variables, we see that line j has a slope of four-thirds, and line k has a slope of negative three-fourths. These slopes are not congruent, so the lines cannot be parallel. However, one slope is positive and the other slope is negative. Additionally, these slopes are reciprocals or flipped fractions of each other. Therefore, we can conclude that the lines are perpendicular.
A transversal is a line that intersects two or more coplanar lines at different points. For example, in this figure, line t is a transversal because it intersects both line a and line b. Transversals have an important role in geometry because they are needed to form alternate interior angles, alternate exterior angles, consecutive interior angles and corresponding angles.
In the real world, transversals are highly visible on street maps. Take a look at this one. Here, you can see that Elm Street is a transversal to Asbury Street, W. Taylor Street and Villa Avenue.
Now, let's practice identifying a transversal. Take a look at the following scenario. Which line is the transversal?
Line a does not intersect any other line. Line b intersects line c. Line c intersects line b and line d, and line d intersects line c. Therefore, since a transversal must intersect at least two lines, we can conclude that line c is the transversal.
In review, remember that all of the lines we discussed are coplanar. Parallel lines have congruent slopes, perpendicular lines have opposite reciprocal slopes, and to be a transversal, a line must intersect at least two other lines at different points.
From office furniture to highway road maps, lines are everywhere. Whether parallel, perpendicular or transverse, lines provide structure for our everyday lives.
After finishing this lesson, you should be able to:
- Define coplanar lines
- Recognize the congruent slopes in parallel lines
- Remember that opposite reciprocal slopes are perpendicular lines
- Recall that you need at least two intersections to have a transversal line
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Parallel, Perpendicular and Transverse Lines Quiz
Instructions: Choose an answer and click 'Next'. You will receive your score and answers at the end.
Which two lines are parallel?
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