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 one hundred and eighty 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 one hundred and eighty degrees. So what this tells me, is that 𝑎 plus 𝑏 plus 𝑐, I put them in that order, is equal to one hundred and eighty degrees. And the reason, remember, was that angles on a straight line sum to one hundred and eighty 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 a hundred and eighty 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 a hundred and eighty 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 notice is that angle 𝐵𝐶𝐴 is on a straight line with this angle of one hundred and sixty-three 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 a hundred and eighty minus one hundred and sixty-three, and that gives us seventeen degrees. And the reasoning for that, which I’ve written at the side, is that angles on a straight line sum to one hundred and eighty degrees. So I can mark angle 𝐵𝐶𝐴 onto 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 one hundred and eighty degrees. And if I know two of them, I can work out the third.
So to find angle 𝐵𝐴𝐶, we’re gonna do one hundred and eighty minus a hundred minus seventeen, that’s subtracting both of the other two angles in the triangle. And so that gives us sixty-three degrees for the measure of angle 𝐵𝐴𝐶. And the reasoning, as we said, angles in a triangle sum to one hundred and eighty 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 one hundred and eighty 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’ll be 𝑥 plus two 𝑥 minus ten plus 𝑥 plus thirty, if I add them altogether, it has to give me one hundred and eighty 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 ten plus thirty, so overall I’ve got plus twenty. So simplifying this equation, I have four 𝑥 plus twenty is equal to a hundred and eighty. Now I want to solve this equation. So the first step is gonna be to subtract twenty from both sides. And when I do, I get four 𝑥 is equal to a hundred and sixty. 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 forty, 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 angle.
So the question’s 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 them 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 nine ratio.
Now remember our key fact, that the sum of the angles in a triangle is a hundred and eighty degrees. So I can turn this into an equation. So if I add plus signs between those three terms, then it’s equal to one hundred and eighty. So what I’ve done is set up an equation involving this unknown letter 𝑥. So now I can simplify this equation. Our five 𝑥 plus four 𝑥 plus nine 𝑥 becomes eighteen 𝑥. So I have eighteen 𝑥 is equal to a hundred and eighty. To solve the equation then, I need to divide both sides of the equation by eighteen, and this gives me that 𝑥 is equal to ten. 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 ten is equal to forty, and this gives me the answer to the problem, which is that the size of the smallest angle is forty 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 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 eighteen. So in total, there are eighteen equal parts in this ratio. Now as we said many times in this video, the sum of the angles in a triangle is one hundred and eighty degrees. So those eighteen parts together are worth a hundred and eighty. 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 eighteen. So that would give me, one part is equal to ten. 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 forty.
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 fifty degrees. What could the other angles be?
Now there are a couple of words that jump out to 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 to me is that word “could”. Because whenever I see that word, that’d suggest 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 fifty degrees. So it could be the case that this angle here is fifty 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 fifty degrees because they’d have to be equal to each other. 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 a hundred and eighty degrees, this angle here would be worked out by doing a hundred and eighty minus fifty minus fifty, so subtracting the other two angles. So this would give me eighty degrees for the third angle. So there is one possibility for the three angles in this triangle. They could be fifty degrees, fifty degrees, and eighty degrees.
But the question said “could”, and as I said, that suggest that perhaps there is more than one answer. So I’ll draw the diagram out again. So perhaps instead of the fifty degrees being one of those two equal base angles, perhaps the fifty 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 a hundred and eighty degrees. If I subtract this fifty degrees, then I’ve got a hundred and thirty degrees left. And as these two angles are both equal, they must both be half of that. So they’re both equal to a hundred and eighty subtract fifty and then divide it by two. And so that gives me sixty-five degrees for each of these angles. So that gives me another possibility of fifty degrees, sixty-five degrees, sixty-five degrees for the angles in this triangle. So that gives me my two possibilities.
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 one hundred and eighty 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.