Video Transcript
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, that’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 120 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 120, I can
work out angle 𝐴 by subtracting it from 180. So I have a measure of angle 𝐴,
one hundred and eighty degrees minus 120 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 120 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 to each other, they are congruent, which means angle 𝐶 must also be 120
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 180 degrees.
So we could work out the measure of
angle 𝐴 by subtracting the measure of angle 𝐶 from 180. So again, that gives us 60 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 tried 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 180 degrees. So what this means then is we can
write down our equation. If we sum these two angles
together, we should get 180. So we have this equation here, four
𝑥 minus 10 plus two 𝑥 plus 10 is equal to 180.
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 10 plus 10, which
cancel each other out. So this just leaves us with six 𝑥
is equal to 180. 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
30, 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 360, 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 360 minus 85 minus 140 minus 50. And so that gives me 85 degrees for
this angle. And as we said the reasoning behind
that is that angles in a quadrilateral sum to 360 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 85 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 85 degrees because, as we said,
they are alternate interior angles. So I can label that 85 degrees on
my diagram as well. Now just as an aside at this point,
the fact that there’s another 85 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 180 degrees. So I can work out the measure of
this angle 𝐶𝐷𝐸 by doing 180 minus 50 minus 85. And if I do that, it gives me 45
degrees for the measure of this angle, with the reasoning, remember, being that the
sum of the angles on a straight line is 180 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.
