Lesson video

Hello, and thank you for choosing this lesson.

My name is Dr.

Ronson, and I'm excited to be helping you with your learning today.

Let's get started.

Welcome to today's lesson from the unit of angles.

This lesson is called "Checking and Securing Understanding of Basic Angle Facts," and by the end of today's lesson, we'll be able to confidently calculate with angles at a point, angles on a line, and vertically opposite angles.

Here are some previous keywords that'll be useful during today's lesson, so you may want to pause the video if you need to remind yourselves what any of these words mean.

And then press play when you're ready to continue.

This lesson contains three learning cycles, and in the first learning cycle, we're going to be looking at angles around a point.

An angle describes how much something has turned; for example, with the image we can see here, we have a wheel.

And we consider an angle as how much the wheel turns as it rotates around that point in the centre.

An angle is often displayed as the rotation between two line segments, like you can see with the image on the right.

In other words, how much would I need to turn one of those line segments so that it's pointed in the same direction as the other line segment if I rotated it around that vertex? And angles are measured in degrees, but there are other units available as well.

A full turn is a rotation of 360 degrees, which means multiple angles around a single point sum to 360 degrees; for example, with the image on the right here, we have three angles: a, b, and c.

Those three angles would add up to 360 degrees.

Here we have Alex.

Alex says, "This wheel has 16 spokes." Each spoke is the wooden piece we can see here that goes from the centre of the wheel to its circumference.

Alex says, "It has 16 spokes, which are equally spaced.

I wonder what angle the wheel would turn if it rotated this much," what we can see here; in other words, the angle between a spoke pointing in the direction it is now and pointing in the direction that the next spoke is currently pointing.

Maybe pause the video while you think about how you could work this out.

And then press play when you're ready to continue.

Let's look at this together now.

One thing we could do is create a diagram of this situation, something that shows just the mathematical parts that we're interested in and not some of the contextual things in that photograph, such as the graphs or the people in the background, and so on.

A diagram would look a bit like this.

We could consider what is this angle here, which we've labelled now "x." Well, all of these angles around the point are all the same angle, or at least, we should assume they're the same angle based on this wheel.

Now we have 16 of these angles and because they are all around the same point, they will all add up to 360 degrees; therefore, we can make an equation a bit like this, 16x = 360, and then solve that equation by dividing both sides by 16 to get x = 22.

5; therefore, the angle is 22.

5 degrees.

Alex says, "The angle in each sector is 22.

5 degrees." Now, let's remember: a sector is part of a circle that is made up of two radii and an arc.

In the case of this wheel here, we can see the sector is part of the outer of the wheel and the two spokes that join it to the centre, and we can see that angle is 22.

5 degrees between those two spokes.

So Alex says, "I wonder what angle the wheel would turn if it rotated this much, "and here we can see that the arrow, which is showing the turn, covers five sectors of that circle.

Pause the video while you think about how you could work out what the angle of rotation is in this situation.

Then press play when you're ready to continue.

Let's take a look at it together.

If we create a similar diagram to last time, we can label each of these individual angles as 22.

5 degrees, and the angle that we are looking at is the sum of five of these angles, so we can work it out by doing 22.

5 X 5, which is 112.

5.

Here we have a diagram that shows 12 points equally spaced around a circle and a point in the centre of the circle, and what we're gonna do now is find the size of this marked angle here.

Perhaps pause the video while you think about what steps we might take to find the size of that angle there, and press play when you're ready to do it together.

Well, we could start by working out this angle here instead, these two points are next to each other as we go around the circumference of this circle, and because the points are equally spaced, we can do 360 divided by 12, and that would give 30 degrees for the angle in the centre of this sector.

That means, if we split the largest sector for the angle that we're trying to eventually find into five sectors, each of those angles will be 30 degrees, so the total angle will be 30 X 5, which is 150, so the angle we're looking for is 150 degrees.

So, let's check what we've learnt.

"Angles around a point sum to blank." What goes in that blank? Pause the video while you write it down, and press play when you're ready for an answer.

The answer is 360: "Angles around a point sum to 360 degrees." Here we have a diagram that contains three equal-size sectors.

Find the value of x.

Pause the video while you work this out, and press play when you're ready for an answer.

Well, all three angles around that point at the centre of the circle sum to 360 degrees, and they're all equal, so we can do 360/3, which is 120.

So, now the diagram shows three points equally spaced around a circle and a point at the centre, and you can see there's an angle labelled at the centre of that circle, and it's labelled "y," and you need to find the value of y.

Pause the video while you do that, and press play when you are ready for an answer.

Well, you could start by dividing 360 by 3 to get 120.

And then multiply it by 2 to get 240.

So, the fact we're working with here in this part of the lesson is that angles around a point sum to 360 degrees.

And this fact can be used to find missing angles when other angles are known; for example, here we have a diagram where we can see one angle is 148 degrees, and we want to find a value of m, which is the reflex angle at that same point.

Well, one thing we could do is create an equation out of this situation.

These two angles sum to 360 degrees, and one of them is unknown, so the equation we could create could be m + 148 = 360.

And then we could solve that equation.

We could subtract 148 from both sides, so that means m = 360 - 148, which is 212, so the angle is 212 degrees.

Here's a problem that is similar to the last one but slightly more complicated.

That's because we have four angles now around a point, and we want to find the value of n.

Well, we could do the same thing again.

We could create an equation out of what we know here.

We've got four angles that sum to 360 degrees, and one of them is unknown, so we could do n + 87 + 112 plus the 90, which is the angle on the bottom right corner of that vertex, and they equal 360.

So we could simplify this equation by adding up the angles we know, and that'll give n + 289 = 360.

And then we can solve that equation by subtracting 289 from both sides to get n = 71.

So, let's check what we've learnt.

Here you have a diagram with four angles around a point, and what you need to do is find the value of x.

Pause the video while you do that, and press play when you're ready for an answer.

The answer is one 117, and here's your working for how we can get that.

Okay, it's over to you now for task A.

This task contains two questions, and here is question one.

In this question, you have eight diagrams, and each time, you need to find the size of the marked angle.

And whenever you have a circle, assume that the points are equally spaced around the circumference of that circle.

Pause the video while you do this, and press play when you're ready for question two.

And here is question two.

You need to find the value of each unknown: a, b, c, d, e, and f.

Pause the video while you do that, and press play when you are ready for answers.

Okay, let's see how we got on.

Here are your answers to question one.

Check these against your own.

And then press play when you're ready for question two's answers.

And here are the answers to question two.

Pause, and check these against your own.

And then press play when you're ready for the next part of the lesson.

Great work so far.

Now, let's move on to the next part of today's lesson, which is looking at angles on a line and vertically opposite angles.

A half turn is a rotation of 180 degrees, and that's what we can see in this diagram here; for example, if you face directly to the left and then turn so you are facing directly to the right, the angle that you would've turned in that single rotation would be 180 degrees; therefore, if we split that up into smaller angles, it means that angles that meet about a single point on a straight line sum to 180 degrees.

Let's take a look an example of that by looking at what if we split this single angle here, of 180 degrees, into two smaller angles, like this.

We have a 30-degree angle and a 150-degree angle.

Those two angles sum to 180 degrees, and if we make the smaller of those two angles bigger, if we increased its size, the other angle will get smaller, and each time, these two numbers sum to 180 degrees.

And two angles that do sum to 180 degrees are called supplementary angles.

So, adjacent angles on a straight line sum to 180 degrees, and this fat can be used to find missing angles when other angles are known; for example, here we have AC, which is a straight-line segment, and what we need to do is find the value of x.

Well, we have one angle, which is 68 degrees, and that is adjacent to the angle labelled "x," and the two of them form a straight line, so we can create an equation.

That is x + 68 = 180.

And then we can solve that equation to get x = 112, so the angle is 112 degrees.

Here we have a similar question that is slightly more complicated.

Pause the video while you think about how we might find a value of x in this situation.

What could you do? And then press play when you're ready to continue.

Well, we could do a similar thing again, but we'll just have a slightly more complex equation.

We know that all four of those angles, which are adjacent on a straight line.

They sum to 180 degrees, so we can create this equation here.

We can simplify this equation by adding up the numbers that we know on the left-hand side of the equation to get x + 165 = 180 and then subtract 165 from both sides to get x = 15, so the angle is 15 degrees.

Now, this same fact, that adjacent angles on a straight line sum to 180 degrees, can also be used to check whether three points lie on a straight line; for example, here we have three angles: 73 degrees, a 90-degree angle, and 19 degrees.

And what we wanna do is determine whether points A, B, and C lie on a straight line.

The way we could do that is by checking if these three angles sum to 180 degrees.

If they do, then it's a straight line.

If they don't, then it's not.

If we add them together, we get 182.

That means, no, those three points, A, B, and C, do not lie on a straight line.

It's close to 180.

That's why it looks like a straight line, but it's not quite one.

It's just slightly off.

So, let's check what we've learnt.

"Angles that meet at a point on a straight line sum to blank." What goes in that blank? Pause while you write it down, and press play when you're ready for an answer.

The answer is 180: "Angles that meet at a point on a straight line sum to 180 degrees." Here we have three diagrams. In which diagram or diagrams do points A, B, and C form a straight line? Pause the video while you write your answer down, and press play when you're ready to see what the answer is.

The answer is a.

Those three angles sum to 180 degrees, but with b and c, the angles do not sum to 180 degrees, so they're not straight lines.

Here we have a diagram that has two angles: one labelled "a degrees" and the other one labelled "b degrees." True or false: a + b = 180? Is that true? Or is it false? And choose a justification from below.

Pause while you do that, and press play when you're ready for an answer.

The answer is false, and the reason why is that the angles marked a and b do not meet at a single point on the line and the short line segments are not parallel; therefore, they do not sum to 180 degrees.

Let's now take a look at vertically opposite angles.

Vertically opposite angles are a pair of opposite angles that are formed when two lines intersect at a point, and vertically opposite angles are equal; for example, here we have a diagram that shows two line segments, AC and BD, which intersect at point E.

And this diagram has C created two pairs of vertically opposite angles.

We have angle BEA and angle DEC, which are vertically opposite.

They are on opposite sides of that vertex, and therefore, they are equal, and we have angles AED and CEB, which are also vertically opposite.

They are on opposite sides of the vertex, and therefore, they are equal as well.

Now, algebra and other angle facts can be used to prove that vertically opposite angles are equal.

Now, there are multiple ways you can do it.

Let's take a look at one way together.

We could let angle AED, in this case, be equal to x degrees.

We could say that angle AED and angle BEA sum to 180 degrees, because we can see that they're adjacent on a straight line; therefore, angle BEA is equal to whatever 180 - x is in degrees.

Now, if we look at this next angle that I've highlighted, we can say that angle BEA and angle CEB sum to 180 degrees, as well, because they are adjacent on a straight line; therefore, angle CEB is equal to what we get when we do 180 - x degrees, and if we simplify that we get x.

That means angle CEB is equal to x, and so was AED, and they are vertically opposite each other; therefore, they are equal.

In the previous examples we saw, we had vertically opposite angles made by just two line segments intersecting at a point, but we can also use vertically opposite angles when there are other line segments meeting at the same point, for example, like we can see in this diagram here, so let's now find the value of x in this diagram.

One way you could do it is by adding together the four angles you know and subtracting it from 360 degrees because you know that all those angles are around the same point so they should sum to 360 degrees; however, we could do it another way, using vertically opposite angles.

We can see that AFD is a straight line, because 57 + 123 gives you 180, so we know it's a straight line.

And we can see that BFE is a straight line, because 57 + 123 on that line is also 180, so that's a straight line.

That means that angle EFA is equal to angle BFD, because they are vertically opposite each other, so they are equal; therefore, we can say that BFD is equal to the sum of those two smaller angles: BFC and CFD.

That means we can create this equation, x + 59 = 123, and solve that equation to get x = 64.

So, let's check what we've learnt.

In which diagram or diagrams show a pair of vertically opposite angles? Pause the video while you write down your choice, and press play when you're ready for an answer.

The answer is b.

In that situation, the two angles are equal to each other.

They are vertically opposite.

Here we have a diagram.

We have many angles, and you need to find the value of x.

Pause the video while you do that, and press play when you're ready for an answer.

The answer is 48.

Now, one way you could do it is by subtracting the four angles you know from 360 degrees, but the working on the screen here shows you an alternative way you could do it, using vertically opposite angles.

Okay, it's over to you again now for task B.

This task contains one question, and here it is.

You've got six figures, all of angles, and there are some unknown angles.

You need to work out the value of each unknown: a to h.

Pause the video while you do that, and press play when you're ready for some answers.

Okay, let's take a look at some answers.

a is 115.

b is 71.

c is 39.

d is 121, while e is 59.

F is 68, and g is 30, and h is 28.

Great work so far.

Now, let's move on to the third and final part of today's lesson, which is investigating joining shapes together at a vertex.

Let's start off with this problem here.

Here we have four angles, and what we want to know is whether or not these angles could fit exactly around a single point.

Pause the video while you think about how we could check whether or not that would be the case.

And then press play when you're ready to continue.

Well, let's remember the fact that angles around a point sum to 360 degrees, so if these angles do fit exactly around a point, then they would also sum to 360 degrees, so let's check if they do.

If we add them together, we get 360.

That means, yes, they would, and it could look something a bit like this.

So, here's a slightly more complicated problem now.

We have three quadrilaterals, and what we need to do is find a way to join two or more quadrilaterals together around a single point without any gaps.

Perhaps pause the video, and think about how we could go about doing that.

We could do a trial and error and just keep clipping them together until we find something that works.

Or is there a slightly more mathematical way we could work it out, without trial and error? Pause while you think about it, and press play when you're ready to continue.

Well, let's see what Aisha says.

She says, "We could look for three angles that sum to 360 degrees, using one angle from each quadrilateral"; for example, if we have 160 degrees, 110 degrees, and 90 degrees and add those together, we do get 360 degrees.

That means we could join these three quadrilaterals together at those three vertices, and it would look a bit like this.

Aisha says, "Here's one solution," and we can see that those three angles that join together at that point do fit exactly together around that point because they sum to 360 degrees.

But is this the only solution? I don't think it is.

Aisha says, "I wonder if there's another way." Can you spot another way of doing this? Pause a video while you think about it, and press play when you're ready to continue.

Here's a different solution.

If we use the 140 degrees, the 120 degrees, and 100 degrees at the centre, then those three quadrilaterals would fit exactly around that point because those three numbers sum to 360 degrees.

Aisha says, "It doesn't matter that the vertices on the outside don't connect"; for example, we can see a 90 degrees and a 30 degrees in the top left corner.

They don't connect.

That's fine.

It's the ones in the centre that we are concerned with.

So, let's check what we've learnt.

Here we have three triangles, and Jacob wants to fit these three triangles around a point without any gaps.

Would this be possible using the three vertices that are highlighted: the 150 degrees, the 60 degrees, and the 120 degrees? Pause the video while you write down either yes or no, and justify your answer, as well.

Then press play when you're ready to continue.

The answer is, no, it would not be possible to connect those three triangles together at those vertices without leaving any gaps, because 150 + 60 + 120 is 330, not 360.

So could you find, please, three vertices that Jacob could use? Pause the video while you do this, and press play when you're ready for an answer.

You could do it by using the 150 degrees, the 90 degrees, and the 120 degrees, because they add up to 360 degrees.

Okay, it's over to you now for task C.

This task contains two questions, and here they both are, and they both relate to the four triangles you can see on the screen here.

In question one, you need to find as many different ways as you can to join up two or more of the triangles together around a single point without any gaps, and for part two, you need to find as many different ways as you can to join two or more of the triangles together at a single vertex, without any gaps, to form a straight line.

So in part one, you're trying to fit triangles together around a point, and in part two, you're trying to fit triangles along a straight line.

Pause the video while you work through this, and press play when you're ready to look at some answers.

Okay, let's take a look at some answers, and let's start with question one, finding as many different ways to join the triangles together around a point without leaving any gaps.

Well, one way could be to use these angles here: 120, 150, 30, and 60.

Another way could be to use these angles instead: 20, 150, 130, and 60.

Or you could use 120, 20, 130, and 90, or you can use just three of the triangles by using 120, 150, and 90.

In question two, you need to find as many different ways as you can to join two or more triangles together at a single vertex to form a straight line.

Well, here's one way.

We could use 120, 10, 20, and 30, because they add up to 180, or you could use these angles: 40, 20, 30, and 90.

Or you could use just three of the triangles, which would be 40, 10, and 130, or you can do that by using 120, 30, and 30 or by using 20, 130, and 30 or by using 20, 130, and 30 from a different set of angles.

Or you can do it just using two triangles, for example, using 150 and 30 or 150 and 30 from a different choice of triangles.

Or you can use 120 and 60 from these triangles here.

I wonder how many you got.

If you've got any, well done.

If you've got them all, wonderful.

Fantastic work today.

Now let's summarise what we've learnt.

One thing we've learned or reminded ourselves is the fact that angles that meet around a point sum to 360 degrees, and that fact can be used in lots of different ways, like we've seen in today's lesson, same as well the next fact, that angles that make a straight line sum to 180 degrees, and also that vertically opposite angles are equal.

Well done today.

Have a nice day.