In today's video we're going to run through five relatively simple rules that you need to understand in order to do questions involving angles. The first of these rules is that the angles in a triangle always add up to 180 degrees. So if we label our angles A, B and C then we could say that A plus B plus C equals 180 degrees. The main place you'll have to use this rule. is when they give you two of the angles and ask you to work out the third.
For example in a question like this. Here they've given us two of the angles, 65 degrees and 45 degrees, and asked us to work out the missing angle x. So because we know that these three angles should add together to make 180 degrees, we can just do 65 plus 45 plus x equals 180. and then we can rearrange that to find x. So if we combine the 65 and 45, that will give us 110 plus x equals 180, and then if we subtract the 110 from each side, we'll be left with x equals 70. So we know that our missing angle was 70 degrees.
The next rule is that all of the angles around a point on one side of a straight line always add up to 180 degrees. So when we only have one angle like this, that angle will be 180 degrees, but if we had two angles, say A and B, then A plus B would equal 180 degrees. Or if we have a point with three angles, A, B and C, then those angles would all add up to 180 degrees, and so on for however many angles we had. The third rule is that the angles in a quadrilateral add up to 360 degrees.
And remember a quadrilateral is just a four-sided shape. So we can represent this rule as a plus b plus c plus d equals 360 degrees. So if we had this question here where we're trying to find out the missing angle y, we could say that 120 plus 140 plus 58 plus y equals 360. Or if we simplify it, then 318 plus y equals 360. Then we can just subtract 318 from both sides to find that y is 42 degrees.
The next rule is that the angles around a point add up to 360 degrees. And like with a straight line, it doesn't matter how many angles there are. So for this drawing, a plus b plus c plus d equals 360. But for this one here, a plus b plus c equals 360. The last rule we need to cover is for isosceles triangles. It isn't really one of the angle rules, but it comes up a lot in exams so it's worth covering.
Hopefully you remember that what makes isosceles triangles special is that two of their sides are the same length, which we show with these little dashes. And importantly, this means that these two angles at the bottom will always be the same size. To see how this is relevant, let's have a go at this question here. So in this question, they tell us that PRQ is an isosceles triangle, which we can tell for ourselves anyway, because it has these two dashes to tell us that these two sides are the same. And they're asking us to find out the missing angle x.
Now if this was a normal triangle, then we wouldn't be able to do anything because we only have one angle out of three. But because this is an isosceles triangle, we know that this angle here is also 35 degrees, and so we now have two out of the three angles. Then if we think back to our first rule, which remember said that the angles in a triangle add up to 180 degrees, then we know that 35 plus 35 plus x must equal 180. So we can simplify that to 70 plus x equals 180, and then subtract the 70 to get x equals 110 degrees. So to sum up the video so far, we've seen five rules.
Rule one was that the angle's in a triangle at 180 degrees. Rule two was that the angle's on a straight line at 180 degrees. Rule three was that the angle's in a quadrilateral. At 360 degrees, Rule 4 was that the angles in a circle add to 360 degrees, and rule 5 was that the Lysosceles triangles have two equal angles because they have two equal sides.
Before we finish though, I just want to show you an example of a trickier exam question, where you have to use multiple rules to find the missing values. Now questions like this often look really tricky because there's loads of stuff going on. The best way to approach them though is to ignore the particular letters that you're trying to find, so ignore the x and the y in this case, and just take a step back and look at what you have.
For example in this figure we have a quadrilateral on the left, and remember we know that the angles in a quadrilateral add up to 360, and we have a triangle on the right, so we know that inside this triangle the three angles will add to 180 degrees. and because it's an isosceles triangle we also know that the bottom two angles will be the same, so we could add an x down here too. Then the last thing to notice is that in the middle here at b we have a straight line that's been separated into these two angles, so we know that these two angles must add to make 180 degrees.
Now that we've identified all of the different rules that we might want to use we can start to work out some angles. and it's normally easiest to just work out whichever angles you can one by one until you end up finding the angles that you need to answer the question. So here we can start with the isosceles triangle and say that 120 plus x plus x equals 180 which we can then simplify so 120 plus 2x equals 180 then 2x equals 60 and finally x equals 30. so we know that both of these angles are 30 degrees. Now that we know this we can move on to our straight line at b where we know that this unknown angle plus 30 must equal 180, so we can subtract 30 from both sides to find that the unknown angle must be 150 degrees. And lastly now that we know three out of the four angles in the quadrilateral we can write it out as 108 plus 54 plus 150 plus y equals 360, or 312 plus y equals 360, so y equals 48 degrees.
And that's it, we've found all of the angles, so we can now fill in our answers, as x is 30 degrees and y is 48 degrees. So just remember that questions like this are quite tricky. Don't worry about trying to find the exact letters you're asked for straight away.
Just work out whatever you can and fill in all the missing angles until you get your answers. Anyway that's everything for this video. So if you want to practice questions on this topic then just click on the link in the description below or in the pinned comments and that'll take you over to our platform where you can practice a whole bunch of questions on this stuff.
Otherwise hopefully we'll see you again soon.