Video: Pack 3 • Paper 3 • Question 15

Pack 3 • Paper 3 • Question 15

04:15

Video Transcript

Here is an equilateral triangle and a regular pentagon. What is the size of the angle marked 𝑎? Show all your working.

Well, the first thing we can do is actually find out what the angles are, so the interior angles of the equilateral triangle. And the way we do that is by actually dividing 180 by three. And that’s because each of the angles in an equilateral triangle are the same. And what I’ve done here is I’ve actually shown the reason why, so make sure at every stage you actually show why and the reason why is because it’s the interior angle of an equilateral triangle.

So now the next thing we want to do is actually find out what the interior angle of our regular pentagon is. And the reason we want to do this is because if we find this, then this plus 60 degrees plus 𝑎 is gonna be equal to 360, because their angles are on a point and therefore allows us to solve the problem and find 𝑎.

There’re actually a couple of methods we can actually use to find the interior angle of the regular pentagon. So we’re just gonna show a couple here. The first one is using this formula, which is that the interior angle is equal to 180 multiplied by 𝑛 minus two over 𝑛, where 𝑛 is actually the number of sides of the regular polygon. And the key here is the fact that we have a regular polygon, and in this case it’s a regular pentagon. And that means that we know we can use this formula because it will only work if we do have a regular polygon.

So therefore, if we actually substitute our values in, we’re gonna have the interior angle of the pentagon, which is 𝐼 𝑝 — I’ve wrote it here — is equal to 180 multiplied by five minus two over five. And we use five because that’s the number of sides. Well, this is gonna give us 180 multiplied by three over five, which will give us 540 over five, which will give us an interior angle of 108 degrees, because if we do 540 divided by five, we get 108.

So as I said, we’re actually gonna check by using another method. And in this other method, we’re gonna have to use another formula and a relationship. And that formula is that the exterior angle of a polygon is equal to 360 divided by 𝑛, where 𝑛 again is the number of sides. And then the relationship we’re gonna use is that an interior angle plus an exterior angle is equal to 180 degrees.

So first of all, what we’re gonna do is actually work out the exterior angle of our pentagon. And we can do this using 360 divided by five, cause five is the number of sides. I’ve actually marked onto the diagram where an external angle would be. And if we divide 360 by five, we get 72 degrees. So we can see that, actually, yes, our exterior angle is gonna be equal to 72 degrees.

So now what we’re gonna use is actually our other relationship we know, which is that an interior angle plus an exterior angle is equal to 180 degrees. So we’ve got our 𝐼 𝑝, which is our interior angle of the pentagon, plus 72, because that’s our exterior angle, is equal to 180. So therefore, the interior angle of the pentagon is gonna be equal to 180 minus 72, because we’ve actually subtracted 72 from each side. And therefore, what we get is an answer of 108 degrees.

And great, yet this is actually the same interior angle that we got using the other method. So we know, yes, we definitely know that the interior angle of our pentagon is 108 degrees. So now what we’re gonna do is actually work out our angle 𝑎. And as we said earlier, what we need to do that is actually use the fact that angles at a point are equal to 360 degrees. So therefore, 60 plus 108 plus 𝑎 is equal to 360. So then we have 168 plus 𝑎 is equal to 360. So then if we actually subtract 168 from each side, we get 𝑎 is equal to 360 minus 168.

So therefore, we can say that our final answer is that 𝑎 is equal to 192 degrees. And I’ve just drawn that again on the diagram. And we’ve used at this point the relationship that angles at a point add up to 360 degrees.

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