Sherpa

Polygons are multi-sided shapes with different properties. Shapes have symmetrical properties and some can tessellate. Parallel lines are lines which are always the same distance apart and never meet. Arrowheads show lines are parallel. When a pair of parallel lines is cut with another line known as an intersecting transversal, it creates pairs of angles with special properties.

Okay, today we're going to be building up on the basics video of angles that we did previously and looking at some more advanced properties of angles. Angles on power lines, angles around polygons, and so on. So we're going to start off looking at probably the easiest property of angles, the most obvious one, visually, vertically opposite angles, then going to angles on parallel lines. It's important to know the terms of these and then on to the properties of polygons.

So the first one that you need to know about is vertically opposite angles. If you take any two lines, like these two straight blue lines that cross across the central point here, a vertex in the middle, then the opposite angles are the same, they are equal. So let's say this one B is around about 150. This 1D will also be 150. And from that we can actually work out what C and A are, because if we do 150 plus 150, we get 300.

We know that all the way around the point we've got 360. So C and A here have to add up to 60. And so we know that each of those two would be 30 degrees. That's the gist of vertically opposite angles. It is important when you cite reasons, doing questions with these vertically opposite angles, that you say vertically opposite angles.

You used to be able to say just opposite angles. That's no longer the case. So for these two B and D, if you were to be shown that B is 150 and asked what is D? You would say D is 150 degrees. Because B and D are vertically opposite angles.

This even applies, strange as it sounds, when they are horizontally opposite, there's no such word. We say they are vertically opposite either way, whether it's vertical or horizontal, they are called vertically opposite angles. If that word vertically is not used, then you don't get the marks. They are vertically opposite angles.

Moving on to alternate angles, first we're going to be looking at unparallel lines. So these horizontal lines, RNS are parallel. Usually that would be shown with some little arrows. There not going to be strictly necessary. Right now you get the idea they are parallel lines and parallel lines.

When you have a diagonal slash line T here that goes through, it sets up some angular relationships, which it's important to know. Here we have angle C and angle F on this left hand diagram. And the right hand one, we have angle D and angle E. Now these two are both called alternate angles. And again, it's important to remember this name.

You must use it when giving a reason. So if C was, let's say, 50 degrees, and the question was what is angle F? The answer would be angle F is 50 degrees. Give a reason for your answer, because angle C and F are alternate angles. And you'd have to write that down.

This is the key term to use alternate. Notice how you can tell if angles are alternate. Here you can see that these two CNF, they form a sort of Z shape. Here where these two angles are nestled in the corners of the Z. That's a key identifier of an alternative angle, this Z shape.

These two also form a Z shape. It's a sort of stretched out Z that's running but it's still a sort of Z shape there. And that's the key indicator of alternate angles. You get ZS either way. Notice that for alternate angles they are both inside the two parallel lines.

Notice that while one angle is below the parallel line the other one is above the other parallel line. Same thing with this one alternate angle. E is above the parallel line and D is underneath. And notice that when you look at the diagonal slash one is on the left, one is on the right. And again one is on the left, one is on the right.

So alternate angles when it comes to the parallel lines and the diagonals, one is one way, one is the other way. One is above the parallel, one is below the parallel, one is on the left of their diagonal, one's on the right of the diagonal. But they are always both inside the parallel lines.

Looking at corresponding angles, these are very different to alternate angles. All the properties we discussed about alternate angles are now flipped. So first off, this doesn't make a Z. This corresponding angles. What they do, and notice these can be difficult.

Sometimes they form an F shape. It can be upside down or back to front, but it's always an F shape. So if you draw a line with your finger touching all of the lines where the angles meet it will form an F shape. Again underneath. Here we draw along the line any line that touches these angles.

And we've got ourselves another F shape. Yes, it's upside down and back to front, but it is an F. And again, here we go through any lines that are touching the angles. And there's our F shape there. And lastly, here's our F shape, this one.

So that's the key for corresponding angles. You end up with that F shape. Notice the properties. Unlike with alternate angles where one angle was above the parallel and one was below with corresponding angles they are both on the same side of the parallels. So C and D here are both beneath the parallel lines.

E and A here are both above the parallel lines. B and F are both above here and D and H are both below. Notice going back again, c and G are both on the left of the diagonal slash a and E are both on the left. B and F here are both on the right and D and H are both on the right. With alternate angles, we saw that they were both inside the parallel lines with corresponding angles.

One is on the inside, one is on the outside. Again, one on the inside, one on the outside. F is on the inside. B is on the outside. And here, D is on the inside of the parallel lines and H is on the outside.

So, very different to alternate angles. Alternate angles make a z. Corresponding angles make an F with alternate angles. If one angle is on the bottom, the other is on the top. If one is on the left of the diagonal, the other is on the right.

With corresponding angles, they are both on the same side of the parallels and both on the same side of the diagonal. So that's another way that you might choose to remember them.

Quick numerical example here. If C was to be 60 degrees and someone was to say what is G? Your answer would be G is 60 degrees because C and G are corresponding angles. And again, you must use that word corresponding. You must remember it.

Otherwise, you do not get the marks.

Moving on to angles in a polygon. Right now we are looking at exterior angles of a polygon. A polygon is just any sided shape and there are two angles that we need to be looking at. One is an external or exterior angle and one is an internal or interior angle. So looking at this exterior or external angle here, I've marked them all in the red.

And external or exterior angle is set up by continuing one of the lengths of the sides of the polygon. Like here, we've continued this red line up north, continuing that and then form the angle between that line and the polygon. Again, we continue this line along here and we form the angle between this line and the side of the polygon. These are our exterior angles. And there's a formula for exterior angles.

So for any exterior angle, if we notice, I'll prove the property here for you. If we start up north and we go anticlockwise here we end up on this line to roundabout northwest and we end up on a bearing around about northwest. We started from north. Now we're northwest. We continue now we're around about southwest.

Then we continue going clockwise. Around about southeast, clockwise again, we are starting anti clockwise. It was clockwise back to the beginning then.

So looking at the exterior angles of a polygon now the exterior angle is when we continue from one side of the polygon. We continue that and then form the angle between the extended line of the polygon to the next side. Again, we take the side of the polygon, we go out and we extend it. And then we form the angle between that extended side of the polygon and the next actual side of the polygon. And then we form all these exterior angles around until we get back to the beginning.

So the property of exterior angles is that they all add up to 360 deg. So since this is a regular pentagon and I want to know what one of these exterior angles is, I add them all up and then I divide 360 degrees because these all add up to 360. I divide 360 degrees by five because it's five angles and I will get the external angle. But I'm going to prove that to you. Now that might not be obvious.

So here's what we do. We take, let's say this direction here, straight up north in red. We take this first angle and we start moving clockwise around the point.

So let's go around the point here. We go clockwise here until we hit this line. And now we're on a new bearing around about northwest. We continue here going clockwise until we're around about southwest. We continue moving around.

We're around about southeast. We continue clockwise. Now we're a bit above east. Remember, there's five angles. We're not going at 90 deg here.

And then we go back continuing. And now we're back to this angle and we're straight up north. What have we done overall? By using these five angles, we have gone a total of 360 degree. We have gone a total of 360 degrees all the way around by going through all five of these angles.

So that's why for an exterior angle, you do 360 degrees divided by the number of sides, which in this case is five. So that's 72 degrees. So we get this exterior angle here of 72 degrees. If we wanted to know the internal angle of a regular pentagon, we have to make sure all these sides are the same. And this works for any polygon.

Any polygon. The exterior angle is 360 divided by the number of sides. Right here, it's 360 degrees divided by the number of sides. Gives you the exterior angle. To find the internal angle of a regular polygon, then you just do 180 degrees, take away the exterior angle, which in this case is 108 degrees.

So we know that the interior angle, one interior angle of a regular pentagon is 108. And you could do this for any polygon as long as it's regular.

So what we're going to move on to now is calculating the interior angles of a polygon. And this doesn't have to be a regular polygon. It can be any polygon and it will still work out. So I'm going to show you the formula here and how it works. Now, if we take this triangle, we know any triangle adds up to 180 degrees.

Okay, great. If we take any point on any polygon, in this case, this polygon is an irregular hexagon, 12345, six sides. We start from any point and we draw lines to all the other vertices and we form a series of triangles. How many did we get to this? Well, we got 1234.

So for N, the number of sides being six, we got the triangles being four. Let's do another one. Here we have, take my word for it, count them up ten sides. So we have the number of sides n equal to ten. How many triangles do we form when we split this shape up?

123-4567? Eight. So the number of triangles was eight. If you keep doing this, you'll notice a pattern. That the number of triangles you form inside a polygon is two less than the number of sides.

So that's n subtract two for the number of triangles. How many degrees were there in a triangle? 180 deg. So if you want to get all the degrees inside any polygon, all you do is you add up all the degrees, all the angles of each of these triangles. So the final formula for the total number of angles in a polygon is N takeaway two, that's the number of sides.

Take away two times 180 and I'll give you all of the interior angles, not just one of them, but all of them. Now, in the previous slide, we saw that the internal angle of a regular pentagon was 108 degrees. If I times that by five, then I get 540 degrees. So that should be what we should calculate using this formula. Do we?

Let's see. So it's a pentagon, so N equals five, number of sides is five. Take away two, and so n takeaway two is three, times 180. What's three times 180? Well, it is 540.

So yes, it does indeed work. This formula of N takeaway two times 180 to find the interior angles in any polygon. It doesn't have to be regular.

Okay, let's have a look finally at a couple of exam questions here. You will almost always, if not always, be asked to give a reason for stating your angle properties. Just like this first question says, work out the side of the angle x and give a reason for your answer. Well, angle x here doesn't really share any properties. There is a special property for this called cointerior angles, but it's not necessary to learn that one you can stick to the others alternate angles and corresponding angles which we've shown.

Work out the size of the angle x. Well, sometimes it's useful to add your own angles. So I'm going to add another angle here. Let's just call this a and can you see a relationship between A and 111?

Sometimes it helps to rotate the page. They are both on the same side of the parallel lines and they're both on the same side of the diagonal slash. And you can see that they do form an F shape here. So it turns out that A and 111 are corresponding. So A is equal to 111 degrees because it corresponds with 111 corresponds with that angle 111 degrees.

And because A and x lie on a straight line.

So we can see that x must be equal to 69 because 111 plus 69 is 180. So we would say here that X is equal to 69 because angles on a straight line add up to 180 degrees. And that's a very common way to state that reason. But angles on a straight line at 180 degrees that's enough to get your marks there. This final question that I'm going to be leaving with you is a lot more difficult.

I will give you a hint. These two ticks mean that these two lines are the same side, same size. And that sets up a triangular property that you can use. But it's a much more difficult and involved question this one. So give it a best shot.

Put your answers into the comments section. I'm looking forward to hearing what they are. My name is Adam, Tutor for Sherpa. I will give you the answers if you've would like to know.

You can also discuss it amongst yourselves and any other Sherpa tutor will be able to give you that answer. Thank you and have a lovely day.