Video: Angles in Triangles

Learn how to apply your knowledge of the fact that the sum of the interior angles in a triangle is 180° in a variety of contexts, including problems involving solving algebraic equations and problems involving ratio.

13:25

Video Transcript

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 angle.

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 nine ratio.

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 10.

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 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 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 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 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.

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