Lesson Video: Parallel Lines and Transversals: Angle Applications Mathematics • 8th Grade

Video Transcript

In this video, we will learn how to use angle relationships formed by parallel lines and transversals to solve problems involving algebraic expressions and equations. First of all, let’s remember some properties of parallel lines and transversals. First of all, when any two lines intersect, we know that the vertically opposite angles will be congruent to one another. Opposite angles are two angles between secant lines that share a vertex, like we see here. The measure of angle 𝐴 would be equal to the measure of angle 𝐶, and the measure of angle 𝐵 would be equal to the measure of angle 𝐷.

When any two secant lines intersect, we can also say that the adjacent angles will sum to 180 degrees. In this case, the measure of angle 𝐴 plus the measure of angle 𝐷 must be equal to 180 degrees. We could also say the measure of angle 𝐷 plus the measure of angle 𝐶 must be 180 degrees. In fact, any pair of adjacent angles in this image would sum to 180 degrees.

But now we want to extend this to see what happens when two parallel lines are cut by the same transversal. Here, line two and line three are parallel, and they’re both being cut by the transversal line one. If we’re just looking at line one and line two, we’ve already shown that the vertically opposite angles are congruent to one another. But when we extend that to parallel lines cut by a transversal, we say if two or more parallel lines are cut by the same transversal, corresponding angles are congruent.

Looking at the intersection of line one and line two, we could label the angles created as position one, two, three, and four. And then, since we know that line three is parallel to line two and is also being intersected by line one, when we label them in the same direction — top left, top right, bottom right, bottom left — the corresponding angles will be congruent, which means angles in position one are congruent to one another, angles in position two are congruent to one another, and so on.

There are a few other things we should note. When two parallel lines are cut by a transversal, alternate angles are congruent. These are angles on opposite sides of the transversal. The one shown here are alternate interior angles, as they are on either side of the transversal but in between the two parallel lines. When two parallel lines are intersected by a transversal, there are two pairs of alternate interior angles. There are also two pairs of alternate exterior angles that are congruent. These are on either side of the transversal and on the outsides of the two parallel lines.

Let’s look at one final angle pair relationship. Consecutive interior angles, or sometimes called cointerior angles, are angles on the same side of the transversal and in between the two parallel lines. These angles sum to 180 degrees. It’s also worth noting that all of those properties extend to multiple parallel lines, not just two. In this image, we have three parallel lines, and for all three of the intersections, the corresponding angles will be congruent. But what if we have an additional transversal? For this other transversal, it’s of course true that the corresponding angles will be congruent. However, just because we know something about the angles of one of the transversals does not mean we can say anything about the angle relationships in the other transversal. So we have to remember to apply these properties to one transversal at a time. Let’s look at some examples.

Work out the value of 𝑥 in the figure.

First of all, let’s think about the type of lines we’re seeing in the figure. We have two parallel lines that are cut by a transversal, which makes the 61-degree angle and the 𝑥-angle alternate interior angles. And when two parallel lines are cut by a transversal, alternate interior angles are congruent. This means both of these angles are equal in measure and 𝑥 must be equal to 61.

We can look at another example with a different set of angle pairs.

Work out the value of 𝑥 in the figure.

First of all, let’s think about what we can say about these lines. We have two parallel lines cut by a transversal, and here are the two angles we’re considering. They’re on the same side of the transversal and they’re in the same position at either intersection, which means we call them corresponding angles. And when two parallel lines are cut by a transversal, corresponding angles are congruent, and therefore 𝑥 must be equal to 72.

In our next example, we’ll not only have two parallel lines, but we’ll also have two transversals.

Answer the questions for the given figure. Find the value of 𝑥 and find the value of 𝑦.

First of all, let’s think about what we know about these lines. We have two parallel lines. Let’s call them line one and line two. And then we have a transversal, line three, that intersects both line one and line two. And we have a second transversal we can call line four that also intersects line one and line two. First, let’s just consider line one, two, and three, the left side of the image. Since line three is a transversal that cuts line one and line two, the angle 𝑥 and the angle 60 degrees are consecutive interior angles, sometimes called cointerior angles. And we know that these angles must sum to 180 degrees, which means 𝑥 plus 60 must be equal to 180.

If we subtract 60 from both sides of this equation, we see that 𝑥 must be equal to 120. To solve for 𝑦, we can look at line one, two, and four. We have two parallel lines cut by a transversal, and the angle pair relationship of 𝑦 and 110 would be corresponding angles. They’re on the same side of the transversal and on the same side of their parallel lines, respectively. They’re in the same position at both intersections, and corresponding angles are congruent to one another, which means 𝑦 must be equal to 110.

Now that we have all of this information, it would be possible to find this fourth angle inside this quadrilateral. This angle is a consecutive interior angle with 110 degrees, which means 110 plus this angle must equal 180. 70 plus 110 equals 180. If we wanted to perform one final check that we’ve calculated everything correctly, we could sum the four interior angles created by these lines. We know that the four interior angles inside a quadrilateral must sum to 360 degrees. In this case, they do, and that confirms that 𝑥 equals 120 and 𝑦 equals 110.

We’re ready to look at another example.

The given figure shows a pair of parallel lines and two transversals, one of which crosses at right angles. Write an expression for 𝑑 in terms of 𝑏. Using this expression for 𝑑, find a fully simplified expression for 𝑎 in terms of 𝑏.

Let’s see what we have. We have two parallel lines. We could call them line one and line two, and then we have two transversals, which we could call line three and line four. The first thing we wanna do is write an expression for 𝑑 in terms of 𝑏. 𝑑 and 𝑏 occur at the intersection of line one and line three. They are adjacent angles on the same side of line one. This makes them supplementary angles. They must add up to 180 degrees. If we know that 𝑏 degrees plus 𝑑 degrees must equal 180 degrees and we want 𝑑 in terms of 𝑏, this means we want to try to get 𝑑 by itself and have 𝑏 on the other side of the equation. To do that, we can subtract 𝑏 degrees from both sides of the equation, which means 𝑑 degrees will be equal to 180 degrees minus 𝑏 degrees, and therefore 𝑑 will be equal to 180 minus 𝑏. This completes part one, our first expression.

Now, using this expression for 𝑑, we need to find a fully simplified expression for 𝑎 in terms of 𝑏, which means we first need to look at the relationship between 𝑎 degrees and 𝑑 degrees. When it comes to 𝑎 and 𝑑, they are in between the two parallel lines and on the same side of the transversal. They are consecutive interior angles, sometimes called cointerior angles. And when two parallel lines are crossed by a transversal, the consecutive interior angles are supplementary. They sum to 180. And that means 𝑎 degrees plus 𝑑 degrees must equal 180 degrees.

For this expression, we want to express 𝑎 in terms of 𝑏, and that means we’ll need to get 𝑎 by itself. First, we subtract 𝑑 from both sides, which gives us 𝑎 degrees equals 180 degrees minus 𝑑 degrees or 𝑎 equals 180 minus 𝑑. But remember, we want to substitute our expression we’ve already found for 𝑑. We know that 𝑑 equals 180 minus 𝑏. Remember though, we’re subtracting all of 𝑑, so we need to keep that in parentheses and then distribute that subtraction, which means subtract 180. But subtracting negative 𝑏 would be adding 𝑏, and that means 𝑎 equals 180 minus 180 plus 𝑏. 180 minus 180 equals zero, and zero plus 𝑏 equals 𝑏. A fully simplified expression for 𝑎 in terms of 𝑏 is that 𝑎 is equal to 𝑏.

In fact, if we take a closer look at 𝑎 and 𝑏, we see that the relationship between these two angles are alternate interior angles, and we know that alternate interior angles will be congruent. So 𝑑 in terms of 𝑏 would be 𝑑 equals 180 minus 𝑏, and 𝑎 in terms of 𝑏 would be 𝑎 equals 𝑏.

We’re now ready to look at our final example.

In the following figure, 𝑧 equals two 𝑥 minus 69 and 𝑤 equals two 𝑦 minus 59. Find 𝑥 and 𝑦.

First of all, let’s see what we have. We have two parallel lines. If we extend them a little further, we can call them line one and line two. And then we have two transversals, which we can call line three and line four. At first, it seems like we have a lot of variables and not as much information. But when we look at the relationship of 𝑤 degrees and 𝑧 degrees, we can say that 𝑧 degrees plus 𝑤 degrees must equal 180 degrees. In addition to that, line three cuts through line one and line two. And that means this fourth interior angle would also be equal to 𝑧 degrees since these two angles are corresponding. And this means our two transversals and our two parallel lines have given us a quadrilateral. And the interior angles of a quadrilateral must sum to 360 degrees. So we can say that 𝑧 degrees plus 𝑤 degrees plus 83 degrees plus 𝑥 must equal 360 degrees.

But we already know what 𝑧 plus 𝑤 equals. Since 𝑧 plus 𝑤 equals 180 degrees, we can say 180 plus 83 plus 𝑥 must equal 360. So 263 degrees plus 𝑥 degrees equals 360 degrees. If we subtract 263 degrees from both sides, 𝑥 degrees equals 97 degrees and 𝑥 must equal 97. And because we know that 𝑧 equals two 𝑥 minus 69, we can plug in 97 for 𝑥 and we find that 𝑧 equals 125. Putting 𝑧 equals 125 in our figure and 𝑥 equals 97, we can use this information to find 𝑤. We know that 𝑧 plus 𝑤 equals 180. We plug in 125. And when we subtract 125 from both sides, we find out 𝑤 equals 55.

But remember, our goal is to find 𝑥 and 𝑦, and that means we need to plug in what we know about 𝑤 to find 𝑦. Since 𝑤 equals two 𝑦 minus 59, we can say 55 equals two 𝑦 minus 59. Add 59 to both sides, and we get 114 equals two 𝑦. Divide both sides by two; 114 divided by two is 57. So we can say that here 𝑦 must equal 57. In this figure under these conditions, 𝑥 is 97 and 𝑦 is 57.

Before we finish, let’s review our key points. When two or more parallel lines are cut by a transversal, corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles or cointerior angles are supplementary.