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.
Transverse Lines
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 dintersects line c. Therefore, since a transversal must intersect at least two lines, we can conclude that line cis the transversal.
Lesson Summary
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.