Study Guide

Parallel and Perpendicular Lines - Parallel Lines

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Parallel Lines

Parallel Lines is Blondie's third album. Who's that? Seriously? Blondie was breaking hearts of glass way before Gaga got her blonde on.

The album cover is rockin' some stripes. Maybe that's where they got the title. Aside from the obvious influence that parallel lines have on everyone and everything, we mean.

Parallel lines are coplanar lines that never intersect. If pictures are overkill, you can use the symbol || to get the job done. So in the image above, we could say, "Line DE and line PQ are parallel," or we could just write DE || PQ. Of course, drawing the lines out is always an option too.

What about the other types of lines out there? Lines that lie on the same plane and do intersect are called intersecting lines. Really creative, we know. We'll talk about intersecting lines a little later, so don't get your panties in a bunch.

Lines that lie on different planes and never intersect are called skew lines. They aren't perfectly in sync like parallel lines but they don't clash like intersecting lines either. They kind of just go about their own business separately, never meeting or even acknowledging one another.

To show that lines are parallel in drawings, it's common to draw an arrowhead or two on the parallel lines. Just like tick marks are used to indicate congruent segments, these arrowheads clarify which line is parallel to which other line.

Sample Problem

Which pairs of lines are parallel?

Well, we know that intersecting lines can't be parallel, so that means that lines d and e can't be parallel to a, b, or c. From the image, we can tell that d and e both have one arrowhead each, so they must be parallel to each other. Of a, b, and c, only a and c have double arrowheads, so that means they are parallel. Therefore a || c and d || e, but b isn't parallel to any of them.

  • Equations of Parallel Lines

    Remember when we learned about the equation of a line back in algebra? You know, all that y = mx + b business? Well, if we think about parallel lines in terms of linear functions, we know there's got to be something weird going on. After all, two lines that never intersect—ever? That's some fishy business.

    The truth is that it's not weird at all. Looking at two parallel lines, we can see that they'll never crash into one another because if one of the lines moves one unit up and one unit to the right, the other does the same. If it wants to go up three units and two to the left, a line parallel to it will do the exact same. What we're getting at is that two parallel lines will always have the exact same slope. In fact, that's what makes them parallel.

    Sample Problem

    Two lines have equations of y = 6x + 3 and y = 3x + 3. Are these lines parallel?

    For two lines to be parallel, their slopes must be the same. In the case of linear equations, the slope of the line is the coefficient before x, otherwise known as m in y = mx + b. Here, the slope of the first line is 6 and the slope of the second line is 3. Since they're not equal, the lines aren't parallel. (In fact, they intersect at the y-intercept. Plug in x = 0 for both equations and you'll see what we mean.)

    If we want to find a line parallel to another line, all we need is the slope of the original line. That takes care of the m part in y = mx + b. After that, all we need to do is plug in a point on that line (either given or chosen) and solve for b.

    Be careful though! We don't want to plug in a point that's on the original line because parallel lines don't intersect. If we do that, we'll end up with the equation for the original line, not a new one that's parallel to it.

    It's like when songwriters try to rhyme a word with the same exact word. That's not rhyming; that's repetition. We're looking at you, Carly Rae Jepsen.

    Sample Problem

    What is a line that's parallel to y = ¼x – 3 and passes through the point (4, -3)?

    Our first step is to start out with our new line's equation in slope-intercept form, y = mx + b. We know that parallel lines have the same slope, so we can substitute ¼ for m to get y = ¼x + b. To find b, we can plug in the point (4, -3) for x and y.

    y = ¼x + b
    -3 = ¼(4) + b
    -3 = 1 + b
    b = -4

    Our final equation is y = ¼x – 4, which is parallel to y = ¼x – 3.

    We won't harp too much more on the algebra of it all, but it's important to understand that it all ties together. Whether you like it or not, math is all related. And one way or another, it's gonna getcha.

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