In this video, we’re going to look
at a key fact about the angles in triangles and then use it to solve some problems
in this area.
So, the key fact that we need about
the angles in triangles, is this. The sum of the interior angles in
any triangle is 180 degrees. So, by interior angles, we mean
those angles that are inside the triangle. So, the angles that are labelled
𝑎, 𝑏, and 𝑐 in the diagram here. Now, we’re going to look at one way
of proving this fact.
I’m gonna use the diagram here. And the first thing I’m going to do
is I’m gonna add in a line that is parallel to the base of this triangle. So, this line that I’ve added at
the top of the diagram, is parallel to the base. And you can see, I’ve marked that
on using the arrows. Now, this proof relies on some key
facts about angles in parallel lines. And that’s why I chose to add this
extra line in.
So, I’m going to think, first of
all, about this angle here, that I’ve marked in red. Now, if you look carefully at the
diagram, you should see that it has a particular relationship with angle 𝑎, this
angle here. They’re what’s referred to as
alternate interior angles in parallel lines. And if you recall a key fact about
those, is that alternate interior angles are equal, which means that this angle here
is the same as angle 𝑎. So, I can label it as 𝑎. So, if I write the reason down as
well, it’s because, as we already said, alternate interior angles are equal.
Now, let’s think about this angle
here. And again, if you look carefully at
the diagram, you’ll see that this angle, which I’ve marked in green, has a special
relationship with angle 𝑏. And it’s the same relationship. They are also alternate interior
angles in parallel lines. Now, given that alternate interior
angles are equal, that means I can label this little angle here, I can give it the
letter 𝑏. And that’s due to the same
reasoning as before, just in a different part of the diagram.
Right. Finally, if we think about this top
part of the diagram, so this part here. I now have angles 𝑎, 𝑐, and 𝑏,
all next to each other on a straight line. And again, if you recall a key fact
about angles on a straight line, it’s that they add up to 180 degrees. So, what this tells me, is that 𝑎
plus 𝑏 plus 𝑐, I’ll put them in that order, is equal to 180 degrees. And the reason, remember, was that
angles on a straight-line sum to 180 degrees.
So, that’s what we were trying to
show. We were trying to show that 𝑎, 𝑏
and 𝑐, which were the angles in the triangle, add to 180 degrees, by using facts
about angles in parallel lines, specifically alternate interior angles. And using the fact that angles on a
straight line sum to 180 degrees, we’ve proven this fact. Right, now, let’s look at how we
can apply this fact to answering a couple of questions.
So, we have a diagram and we’re
asked to find the measure of angle 𝐵𝐴𝐶. So, that means the angle formed
when we move from 𝐵 to 𝐴 to 𝐶, so it’s this angle here.
So, that’s the angle we’re looking
to find. But we can’t work it out
straightaway because we only currently know one of the angles in the triangle. We can, however, work out what the
other angle in the triangle is, so angle 𝐵𝐶𝐴. Because what you’ll notice is that
angle 𝐵𝐶𝐴 is on a straight line with this angle of 163 degrees.
So, we can use that fact about
angles on a straight line, to work out angle 𝐵𝐶𝐴 first. So, the measure of angle 𝐵𝐶𝐴 is
180 minus 163, and that gives us 17 degrees. And the reasoning for that, which
I’ve written at the side, is that angles on a straight line sum to 180 degrees. So, I can mark angle 𝐵𝐶𝐴 on to
the diagram. And there it is.
Now, we have enough information to
work out this angle 𝐵𝐴𝐶 that I was originally asked for because, again, we know
the angles in a triangle sum to 180 degrees. And if I know two of them, I can
work out the third. So, to find angle 𝐵𝐴𝐶, we’re
gonna do 180 minus 100 minus 17. That’s subtracting both of the
other two angles in the triangle. And so, that gives us 63 degrees
for the measure of angle 𝐵𝐴𝐶. And the reasoning, as we said,
angles in a triangle sum to 180 degrees.
So, often in questions like these,
you can’t work out the angle you’re looking for immediately. You may have to work out other
angles in the diagram first, by using facts about angles in a triangle or angles on
a straight line. And once you’ve got those, you can
then work out the angle you’re looking for.
Okay, here’s the next problem we’re
going to look at.
We’re given a diagram of a
triangle. And we’re asked to find the value
of 𝑥. And if you look at the diagram,
you’ll see that all three of the angles in this triangle are expressed in terms of
this unknown variable 𝑥.
So, thinking about how to approach
this problem, we need the key fact about angles in a triangle. And of course, it’s this, that the
angles in a triangle sum to 180 degrees. So, think about how you can use
this fact to help you answer this problem. Well, we don’t know what the values
of the angles are, in terms of their numeric measures, but we do know them in terms
of this unknown letter 𝑥.
So, we can start writing an
equation. So, if I add up all of the
different angles in this triangle. So, that will be 𝑥 plus two 𝑥
minus 10 plus 𝑥 plus 30, if I add them altogether, it has to give me 180
degrees. So, here is an equation, or the
beginning of an equation, that I can use to work out this letter 𝑥.
So, the next thing I’d want to do
is to simplify this equation. If I look at the left-hand side,
I’ve got 𝑥 plus two 𝑥 plus 𝑥, so I’ve got four 𝑥 overall. And then, I’ve got minus 10 plus
30, so overall, I’ve got plus 20. So, simplifying this equation, I
have four 𝑥 plus 20 is equal to 180. Now, I want to solve this
equation. So, the first step is gonna be to
subtract 20 from both sides. And when I do, I get four 𝑥 is
equal to 160.
Next, to work out the value of 𝑥,
I need to divide both sides of the equation by four. And when I do that, it gives me 𝑥
is equal to 40, which is therefore the answer to this question. So, this question involved using
that fact about the angle sum in a triangle, but also some algebra skills, in terms
of setting up and then solving an equation, in order to work out the value of this
unknown letter 𝑥.
Okay, the next question.
The ratio of the three angles in a
triangle is five to four to nine. Find the size of the smallest
So, with questions involving ratio,
there are lots of different approaches that you can take. I’ll demonstrate two of these
approaches, and then you could decide which of them you prefer. So, the first approach is an
algebraic approach, where we say, well, we don’t know what these angles are, but we
know they’re in the ratio of five to four to nine. Which means I could call these
angles five 𝑥, four 𝑥, and nine 𝑥, where 𝑥 is just representing some unknown
value. But this keeps the five to four to
Now, remember our key fact, that
the sum of the angles in a triangle is 180 degrees. So, I can turn this into an
equation. So, if I add plus signs between
those three terms, then it’s equal to 180. So, what I’ve done is set up an
equation involving this unknown letter 𝑥. So, now, I can simplify this
equation. Add five 𝑥 plus four 𝑥 plus nine
𝑥 becomes 18 𝑥. So, I have 18 𝑥 is equal to
180. To solve the equation then, I need
to divide both sides of the equation by 18, and this gives me that 𝑥 is equal to
Now, the question, remember, said
find the size of the smallest angle. So, the smallest angle is this four
𝑥, the one that has the least parts of this ratio. So, in order to work out the
smallest angle, I need to multiply 𝑥 by four. So, I have four 𝑥. Four times 10 is equal to
forty. And this gives me the answer to the
problem, which is that the size of the smallest angle is 40 degrees.
So, that’s one way of approaching
it, treating it like an algebra problem and setting up an equation. The other way that I’d like to
think about ratio problems, is thinking about the parts of the ratio. So, we have a ratio of five to four
to nine. If I add them together, five plus
four plus nine is 18. So, in total, there are 18 equal
parts in this ratio. Now, as we’ve said many times in
this video, the sum of the angles in a triangle is 180 degrees. So, those 18 parts together are
worth 180. I want to work out the size of the
smallest angle, so I want to know what four parts are worth.
And there are lots of different
ways I could do this. I could work out what one part is,
by dividing by 18. So, that would give me one part is
equal to 10. And then, to find four parts, I’d
have to multiply by four. And so, of course, that gives me
the same answer, as before, of forty degrees. I could, perhaps, have approached
this ratio in a slightly different way.
Instead of finding one part, I
could’ve found, perhaps, two parts. So, that would’ve meant dividing
both sides by nine, and then I’d just have doubled it to find the four parts as
40. So, whichever approach you prefer,
either an algebraic approach involving setting up an equation or thinking about
ratio in terms of equal parts and dividing down and then scaling back up to however
many parts you’re looking for.
Okay, the final question that we’re
going to look at.
It says, one angle in an isosceles
triangle is 50 degrees. What could the other angles be?
Now, there are a couple of words
that jump out at me from that question. The first is isosceles. Now, remember, an isosceles
triangle is a particular type of triangle which has two sides the same length. But also, in terms of angles, it
has two angles that are equal to each other. The other word that jumps out at me
is that word could. Because whenever I see that word,
that suggests that there’s more than one possible answer to this question. So, I need to think carefully about
why that might be the case.
So, let’s have a go at this problem
then. I’ve drawn an isosceles triangle,
and I’ve indicated two sides as the same length by including this little line on
them. Now, it tells me, one angle in the
isosceles triangle is 50 degrees. So, it could be the case that this
angle here is 50 degrees. Now, if that is the case, because
isosceles triangles have two equal angles, and because they’re the base angles where
these equal sides join the third side. That would mean that this angle
here would also have to be 50 degrees because they’d have to be equal to each
The third angle then, so this angle
here, I would work out using my fact about angles in a triangle. So, if the angles in the triangle
sum to 180 degrees, this angle here would be worked out by doing 180 minus 50 minus
50, so subtracting the other two angles. So, this would give me 80 degrees
for the third angle. So, there is one possibility for
the three angles in this triangle. They could be 50 degrees, 50
degrees, and 80 degrees.
But the question said could. And as I said, that suggests that
perhaps there is more than one answer. So, I’ll draw the diagram out
again. So, perhaps instead of the 50
degrees being one of those two equal base angles, perhaps the 50 degrees is actually
the third angle in the triangle. This angle here. And now, if I want to work out the
other two. Well, the angles in a triangle sum
to 180 degrees. If I subtract this 50 degrees, then
I’ve got 130 degrees left.
And as these two angles are both
equal, they must both be half of that. So, they’re both equal to 180
subtract 50 and then divided by two. And so, that gives me 65 degrees
for each of these angles. So, that gives me another
possibility of 50 degrees, 65 degrees, 65 degrees for the angles in this
triangle. So, that gives me my two
So, in summary then, in this video,
we’ve looked at the key fact that the sum of the interior angles in any triangle is
180 degrees. We’ve seen how to prove it using
facts about alternate interior angles in parallel lines. And then we’ve applied this fact to
answer some problems.