In this video, we are going to look at parallel lines and transversals. Or more specifically, the angles that are created and the relationships between them.
Now parallel lines, remember, are lines that will never meet, no matter how far they are extended. They’re always exactly the same distance from each other. Now a transversal is a line that intersects two lines at two different points. So on these parallel line diagrams that I have here, then the green and the orange lines that I’ve added, those are examples of transversals cause they’re crossing those parallel lines at two places.
Now this transversal creates eight angles. So that’s these eight angles here. And what we’re interested in, are the relationships that exist between particular pairs of angles. So there are four different types of angles that we’re going to look at. We’re going to name them, and then we’re going to look at the relationship that exists between them.
The first type of angles we’re looking at are called corresponding angles. Now the way to recognise these, is that they’re angles that are essentially in the same place, at the two points where the transversal crosses the parallel lines. So, for example, these two angles marked in red. They are both above the parallel lines and to the right of the transversal. Equally, these angles marked in green. They’re in the same position at both places, where the transversal crosses the parallel lines. So they’re another pair of corresponding angles. I could also have the two angles marked in blue, or the two angles marked in orange. So there are, in fact, four pairs of corresponding angles within the diagram. Now the key fact about these, is that corresponding angles are equal to each other. So both of those red angles are the same as each other, both of the blue angles are the same as each other, and so on. So corresponding angles are equal, or congruent would be another way of saying that.
The second type of angle we’re going to look at, are what’s called alternate interior angles. Now alternate interior angles are ones such as these ones here. Alternate means they’re on opposite sides of the transversal, and interior means they are inside the parallel lines. So another example of those would be the pair that I’ve marked in blue here. Now the key fact about those, is that alternate interior angles are also equal, or congruent, to each other. So that is the second type of angles. The third type of angle we’re going to look at are called consecutive interior angles.
So an example of those would be these two angles here. Consecutive means they’re on the same side of the transversal, it’s sort of next to each other. And interior again means they’re inside the parallel lines. Again, another pair of consecutive interior angles will be those that I’ve marked in blue here. So the key fact about these, they’re not equal to each other. In fact, you can see that one’s an acute angle and one’s an obtuse angle, so they certainly aren’t the same. But they are instead supplementary, which means the sum of these two angles is one hundred and eighty degrees. So that is the third type of angle. Consecutive interior angles are supplementary to each other. The final type of angle that we’re going to look at here are what’s referred to as alternate exterior angles.
So perhaps you can deduce from the name, alternate meaning they’re on opposite sides of the transversal, and exterior meaning they’re outside the parallel lines. So this angle and this angle here, would be alternate exterior angles. Alternatively again, the pair that I’ve marked in blue would also be another example. Now the key fact about these, is that they are also an example of angles that are equal, or congruent, to each other. So that is the fourth type of angles that we’re looking at.
Now you need to remember the specific names for the different pairs of angles. And you also need to remember whether they are congruent, or whether they’re supplementary. If you’re ever struggling to decide whether two angles are congruent or supplementary, have a look at the type of angles they are. If they are both, for example, acute angles, then they’re going to be congruent. Same thing, if they’re both obtuse angles. Whereas if one’s obtuse and one’s acute, well they can’t possibly be the same. Therefore, they’re going to be supplementary angles, in this context, if their angle’s created by a transversal in parallel lines.
So let’s look at answering our first question on this. We have a diagram and we’re asked to find the measure of angle 𝐴, which is marked on the diagram here. We’re given the angle of a hundred and twenty degrees further up.
Now there are often lots of different ways that you could answer this question. So I’ll do two different methods, so we can see some of the different routes that you could take. Now looking at the angles that are marked, the two angles there, they don’t actually fall into any of the categories that we looked at previously, which means I’m gonna need two stages of working out, or more perhaps, rather than just one cause I can’t immediately just refer to them as a particular type of angle.
Right. So method one, I’m gonna think about this angle here, first of all, which I’m gonna give the letter 𝐵 to. Now if you look at the diagram and if you recall the different names that we had for different pairs of angles before, you’ll see that this angle 𝐵 is corresponding to the angle of a hundred and twenty degrees cause they’re both in the same position but at the two different places where the transversal cuts the parallel lines. So if you recall from before, if two angles are corresponding, then they are congruent to each other, which means the measure of angle 𝐵 must be a hundred and twenty degrees. So the first stage of my working out is to write down the measure of angle 𝐵, and a reason for why that is the case.
Now in order to find angle 𝐴, we actually don’t need facts about angles and parallel lines. We just needed more basic fact about angles on a straight line, which is that they add to a hundred and eighty degrees, or a supplementary. So if I know angle 𝐵 is a hundred and twenty, I can work out angle 𝐴 by subtracting it from a hundred and eighty. So I have a measure of angle 𝐴, one hundred and eighty degrees minus a hundred and twenty degrees, which is sixty degrees. And my reasoning for that is that, the sum of the angles on a straight line.
So that is one approach that we could take to working out this angle. Another approach might be to work out this angle here first, so this angle that I’m going to call angle 𝐶. Now angle 𝐶 is in a special position relative to that a hundred and twenty degrees. And again, it’s not a fact about angles and parallel lines. It’s a general angle’s fact, which is that angle 𝐶 is vertically opposite that angle. And when two angles are vertically opposite each other, they are congruent, which means angle 𝐶 must also be a hundred and twenty degrees.
So that can be the first stage of my working out. Now if I look at angle 𝐶 and angle 𝐴, I’ll see that they are a specific type of angle that we named in the previous slide. They are inside the parallel lines and on the same side of the transversal, so they are consecutive interior angles. Now if you recall the key fact about consecutive interior angles, was that they are supplementary, i.e., they add up to one hundred and eighty degrees. So we could work out the measure of angle 𝐴 by subtracting the measure of angle 𝐶 from one hundred and eighty. So again, that gives us sixty degrees for angle 𝐴. And the reason, remember we said, was that they are consecutive interior angles.
So you see that the calculations involved are identical with both of these two methods, but the reasoning is different depending which angles we try to find out first. And there are other ways I could’ve done it as well. There are lots of different methods that you could use to answer a question like this.
Okay. In the next question, we’re given a diagram, again, with parallel lines and a transversal and we’re asked to find the value of 𝑥. And looking at the diagram, we can see that 𝑥 is used to describe the size of two of these angles here.
So we don’t know what the value of 𝑥 is. Often questions like this involve a setting up and solving an equation. And that’s exactly what we’re going to do here. Firstly, we need to identify what type of angles we’ve got. So looking carefully at the diagram, the two angles we’ve got are inside the parallel lines and on the same side of the transversal. Therefore, they are consecutive interior angles. Remember, the key fact about those was that they are supplementary to each other, i.e., they add up to a hundred and eighty degrees. So what this means then, is we can write down our equation. If we sum these two angles together, we should get one hundred and eighty. So we have this equation here, four 𝑥 minus ten plus two 𝑥 plus ten is equal to one hundred and eighty.
Now we just need to solve this equation. So looking at the left-hand side, we have four 𝑥 plus two 𝑥, which is six 𝑥 and we have minus ten plus ten, which cancel each other out. So this just leaves us with six 𝑥 is equal to a hundred and eighty. The final step in solving this equation, is we need to divide both sides by six. And so this gives us 𝑥 is equal to thirty, which is our answer to this problem.
So this question just involve looking carefully at the diagram, identifying the type of angles that we’ve been given using the fact we knew about them, which was that they were supplementary, and then setting up and solving an equation in order to work out this value 𝑥.
Okay. This is the final question that we’re going to look at. We’re given a diagram and we’re asked to calculate the measure of angle 𝐶𝐷𝐸. So looking at the diagram, that’s this angle created when we move from 𝐶 to 𝐷 to 𝐸, so it’s this angle here that I’ve marked in green.
So you may want to have a look at the diagram yourself and plan the approach that you might take. It’s not immediately obvious how we’re going to calculate angle 𝐶𝐷𝐸. So what I would do, is have a look and see, are there any other angles that I can calculate straightaway? And what you’ll notice is that 𝐴𝐵𝐶𝐷 is a quadrilateral and I know three of the angles. I also know that angles in a quadrilateral sum to three hundred and sixty, which means I can work out this final angle, angle 𝐵𝐴𝐷, using that fact. So that’s how I’m gonna begin. The measure of this angle 𝐵𝐴𝐷 is three hundred and sixty minus eighty-five minus a hundred and forty minus fifty. And so that gives me eighty-five degrees for this angle. And as we said the reasoning behind that, is that angles in a quadrilateral sum to three hundred and sixty degrees.
Now let’s see how that helps at working out this angle 𝐶𝐷𝐸. Well I’ve got a pair of parallel lines in the diagram. I can see that because of the arrows on them. And it may help just to extend both of these lines a little bit. Now having done that, you can perhaps see what’s going on a little bit more easily. And perhaps you can see that there is a relationship between this angle here and the eighty-five degrees that we’ve just worked out, the one marked in blue. If you look carefully at the diagram, you should spot that they are in fact alternate interior angles. And therefore, they must be congruent to each other. You may find it helpful to perhaps tilt your head. Or, if you’re using a tablet or something, tilt the screen so that you can see that more easily. So what this tells me then, is that the measure of this angle 𝐴𝐷𝐹, now 𝐹 is the point where I have extended this parallel line to, the measure of that angle is also eighty-five degrees because, as we said, they are alternate interior angles. So I can label that eighty-five degrees on my diagram as well. Now just as an aside at this point, the fact that there’s another eighty-five degrees in this quadrilateral, that is coincidental. It wouldn’t always be the case that that would be true.
Now finally, I want to work out this angle 𝐶𝐷𝐸. And what you can see in this part of the diagram here, is that these three angles, the blue angle, the fifty degrees, and the green angle are on a straight line together. And therefore, the sum of those three angles must be one hundred and eighty degrees. So I can work out the measure of this angle 𝐶𝐷𝐸 by doing one hundred and eighty minus fifty minus eighty-five. And if I do that, it gives me forty-five degrees for the measure of this angle, with the reasoning, remember, being that the sum of the angles on a straight line is one hundred and eighty degrees.
So in this question, we couldn’t immediately work out the required angle. We had to work out some other angles within the diagram first, then use our properties of angles in parallel lines in order to work out the angle we’re originally asked for.
So to summarize then, we’ve seen four different types of angles in parallel lines, that we need to be able to recognize and name. And we’ve seen how to apply properties about these types of angles in order to work out missing angles in different figures.