Video: GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 13

GCSE Mathematics Foundation Tier Pack 2 • Paper 2 • Question 13

05:50

Video Transcript

In the diagram below, 𝐴𝐵𝐶𝐷 is a quadrilateral and 𝐸𝐷𝐶 is a straight line. Part (a) i) Find the size of angle 𝑥. ii) Give a reason for your answer. Angle 𝐷𝐴𝐵 is equal to angle 𝐵𝐶𝐷. Part b) Work out the size of angle 𝐵𝐶𝐷.

Now, for part a i, what we’re trying to do is actually find the size of angle 𝑥. So in order to actually find the size of angle 𝑥, what we need to do is actually use one of our angle relationships we know. And that’s that the angles on a straight line are equal to 180 degrees. And for example, in this little diagram I’ve drawn, that means that angle 𝐴 plus angle 𝐵 would be equal to 180 degrees. And in this question, we know we can use this relationship because it tells us that 𝐸𝐷𝐶 is a straight line.

Okay, great. So let’s use this and find out the size of angle 𝑥. Well, using this relationship, we know that 𝑥 plus 103 is equal to 180. So therefore, to actually find what 𝑥 is, we’re gonna subtract 103 from 180 because we’ve actually subtracted 103 from each side of the equation. So therefore, we can say that 𝑥 is equal to 77 degrees. And we got that because we took 103 away from 180. And if you take 100 away from 180 you get 80. And take three way, you get 77.

We can do a quick check by adding 103 and 77, just to make sure we make 180. And we’re gonna do that using column addition. Well, if you add three and seven, you get 10. So you put a zero in the units column and carry a one into the tens column. Then, we have zero add seven makes seven. Add the one we carried, makes eight. So we’ve got eight in the tens column. And then a one in the hundreds column. So we get 180. Okay, great. We found the size of angle 𝑥 which is 77 degrees.

Okay, so for part ii, it says give a reason for your answer. Well, we’ve already discussed the reason for our answer. And that’s because angles on a straight line are equal to 180 degrees. Well, actually, this question asked us to give a reason for our answer. But, it’s also something that you should definitely get used to doing because it’s actually a good skill to have. Because every question where you’re asked to find different angles, you also need to give reasoning. So make sure you practice this.

Okay, so now moving on to the second part, so part b, we’ve got angle 𝐷𝐴𝐵 is equal to angle 𝐵𝐶𝐷. And part b wants us to work out the size of angle 𝐵𝐶𝐷. So I’ve actually marked those two angles on. So now for part b, what we need to know and what we need to think about is the fact that 𝐴𝐵𝐶𝐷 is actually a quadrilateral. So this means it has four sides. So what we need to use here with our quadrilateral is the fact that the sum of interior angles of a quadrilateral is equal to 360 degrees.

To help us remember this, we can actually think about a square because a square is a quadrilateral because it has four sides. And a square has four right angles, each at 90 degree. Well, if you multiply 90 degrees by four, you get 360 degrees. There are a couple of other ways that actually you could’ve worked it out. And this is also particularly useful if we have another type of polygon.

So, first of all, we have a formula. And that is that the sum of interior angles of a polygon is equal to 180 multiplied by 𝑛 minus two, where 𝑛 is the number of sides of the polygon. So in our case, that would be, the sum would be equal to 180 multiplied by four minus two, and that’s because it has four sides, which will be equal to 180 multiplied by two which will be 360 degrees.

And also, the other way that we can think about it, if we found the formula was kind of confusing, was to actually think about our shape itself. So we look at our quadrilateral and think about the least or minimum number of triangles it can be divided into. So if you start at one corner and you draw straight lines out until you’ve actually formed the least number of triangles you can. So in this case, we’ve got two triangles. So there are two triangles in our quadrilateral. So therefore, two times 180 is 360 degrees. So these are all just ways that you can think of, remembering the interior angles for a quadrilateral or other polygons.

Okay, so now let’s find angle 𝐵𝐶𝐷. So the first step, now that we know the interior angles are 360 degrees, is to actually add up the two angles we do know. So we know that the angle 𝐴𝐵𝐶 is 137 degrees. And we know the angle 𝐴𝐷𝐶 is 77 degrees cause we found that in part a. So if we actually add those together, so we have 77 add 137. This is equal to 214 degrees. So therefore, we can say that the sum of angle 𝐷𝐴𝐵 and 𝐵𝐶𝐷 is gonna be equal to 360 minus 214. Because actually, they’re the two angles left. And we know that all four angles should add up to 360. Well, this is gonna be equal to 146 degrees. Okay, great. So that’s the sum of the two angles. But how do we work out the size of angle 𝐵𝐶𝐷?

Well, the key bit of information here is the fact that angle 𝐷𝐴𝐵 is equal to angle 𝐵𝐶𝐷. So they’re in fact the same. So therefore, to actually find out what one of these angles is, all we need to do is divide 146 by two. So we’re gonna do that using the bus-stop method. So we can see, first of all, how many twos go into one. Well, this is zero. So then, we carry the one across. So now, we see how many twos go into 14, which is seven. And then we see how many twos go into six, which is three. So therefore, we say that it’s going to be 73. So therefore, we can say that angle 𝐵𝐶𝐷 is equal to 73 degrees.

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