Lesson video

Hello everyone and welcome to another maths lesson with me, Mr. Gratton.

Thank you so much for making the decision to join me in this lesson, where we will look at the properties of angles that lie about a point on a straight line, angles around a point that create a full turn and comparing vertically opposite angles.

Two important words that will be used throughout the lesson are supplementary, exactly two angles whose sum is 180 degrees.

And conjugate.

Again, a pair of exactly two angles, this time whose sum is 360 degrees instead.

Pause here to look through the other keywords that will be used throughout this lesson.

Before we look more thoroughly at angles in specific locations, let's look deeply into what an angle even is.

Let's find out.

An angle can be used to describe how much something is turned, such as this person here.

Before they turn at all, we can say that they have turned or rotated by an angle of zero degrees, but as they start turning on the spot, the angle that they turn will start to increase, up until a total of 78.

2 degrees, which is the amount that they rotate before they stop rotating.

A different perspective that you may be more familiar with is this top-down perspective.

Imagine that this person is standing on the vertex at the intersection of at least two lines or line segments such as on a polygon like this triangle.

An angle can describe how much you must turn to change from facing in the direction of one of the lines until they stop that rotation because they're now facing in the direction of one of the other lines that intersects at the point that the person is standing.

This person has rotated in total 10, 20, 30, 40, 50 degrees in order to face in the direction of this other line.

And Jacob says, "The size of angle A is so much bigger than the size of angle B because the lines on angle A are much longer." Is Jacob's observation correct? Well, let's have a look.

For the diagram with angle A, I start facing in the direction of that horizontal line and start rotating 16, 32, 48, 64 a total of 80 degrees in order to face in the direction of that other line.

Similarly for the diagram with angle B, I also start facing in the direction of that horizontal line and rotate 16, 32, 48, 64, 80 degrees again in order to face in the direction of that other line.

Jacob's observation is incorrect.

The length of the line has no influence on the size of an angle.

We can extend the logic that the size of a line or a line segment does not directly affect an angle to similar shapes.

If two shapes are similar, the size of their angles are invariant, meaning the angles do not change even if the side lengths of that shape increase or decrease as long as each side length has been multiplied by the same scale factor.

Okay, quick check.

Which of these statements correctly describes which angle is bigger? Pause to look through these possible options.

The answer is C.

Both angles are the same as you must turn by the same amount to go from facing along one line segment to facing along the other.

The fact that angle B has two shorter legs or line segments that create the angle, this does not mean the angle is any smaller.

What is different about the angles and the size of the legs in this check? Pause here to consider which of these statements is true for these two angle diagrams. In this case, A is correct.

Angle A is a bigger angle because you must turn more to go from facing along one line segment to facing along the other.

For example, we have a 77 degree turn for angle B, whilst in angle A, we actually have double that at a total of 154 degrees instead.

If you are facing in one direction, stay in the same spot and turn until you face in the opposite direction.

How many degrees have you turned through? If you perform a turn such that you end up facing in the exact opposite direction, we say you have turned by a total of 180 degrees.

If you're facing in one direction and turn until you face in the same direction again, how many degrees have you turned through? If you perform a turn such that you end up facing in the exact same direction, then the first time that you end up facing in this exact same direction we say you have turned by 360 degrees.

We can name or describe every single angle between zero degrees and 360 degrees.

Note that angles outside of this range do exist, but we will not be referring to them during this lesson.

So starting with any angle less than 90 degrees, they're called acute angles as we can see on screen.

However, the moment that it gets to exactly 90 degrees, we have what's called a right angle.

Any angle greater than 90 degrees, but less than 180 degrees is called an obtuse angle.

And the moment that we get to exactly 180 degrees, we get a half turn, which forms a straight line.

Any angle greater than 180 degrees, but less than that one full turn of 360 degrees is called a reflex angle as you can see on screen here.

But the moment we get to exactly 360 degrees, we would've made precisely one full turn around that fixed single point at the centre of that rotation.

But following on from this, Laura asks, "How can I tell if an angle is acute, obtuse, or reflex?" Well, I can place or visualise this template.

It looks a little bit like an upside down capital letter T.

By placing the intersection at the bottom of this template and overlapping it with the vertex of the angle, I can see that the angle is less than a right angle and so it is less than 90 degrees.

Pause here to consider which of these angles is also less than a right angle at 90 degrees.

And the only acute angle here is A.

We can also use this template to check if an angle is obtuse.

Let's have a look.

If the angle is greater than 90 degrees but doesn't quite reach the other part of the straight line segment, then it is also less than 180 degrees.

Any angle greater than the 90 degree right angle, but less than 180 degrees is obtuse.

And again, pause here to see which of these five same angles are obtuse this time.

And angles B and C are the obtuse ones from this set of five angles.

And finally, if an angle is reflex, then it is an angle greater than 180 degrees and so represents a rotation that is greater than a straight line about a point, or a rotation that goes beyond facing in the exact opposite direction and begins to rotate in such a way that it starts to face back towards its original direction.

And for the third time, pause here.

Which of these five angles are reflex angles? Angles D and E are the reflex ones.

And for this check, match the angle to its name and each name can be used more than once.

So pause here to look through all four angles and all three names.

Angle A is reflex because, well, it's almost close to a full turn at 360 degrees.

Angle B is also reflex.

It is slightly greater than a 180 degree straight line.

C is obtuse because it is less than 180 degrees, whilst D is acute.

And now pause here to give yourself time in this check to find the three pairs of angles that are of equal size.

And the three pairs are, AC, which are two obtuse angles, FD, two acute angles, and BE which are two reflex angles.

Seeing them paired up like this is a great way of showing that the length of the legs of each angle do not dictate the size of the angle itself.

And here are the practise questions.

Pause now to answer question one about the rotation of a chick.

And for question number two, which of these statements is correct and justify why you think that the statement is correct? Pause now to do question two.

And for question three, starting with the smallest angle, put these angles in order of size.

After doing so, match the angle to its description.

Pause now to do this.

And similarly for question number four, match the angle to its name and size.

Pause now to do this for all five angles.

And for question number five, match the line segment XY to these line segments, A, B, C, and D to create an acute angle, a right angle, an obtuse angle, an angle of approximately 45 degrees and an angle of approximately 170 degrees as well as a reflex angle.

Pause now to create these six angles.

And note, there may be more than one possible answer for each question.

And for the answers for question one, by how many degrees did the chick turn? By 72 degrees, which is an acute angle.

And for question two, the beetle and the spider have turned by the same angle.

Pause here to see if your explanation matches the one that's on screen.

For question three, angle C is the smallest because it is less than 90 degrees.

Then angle A is next because it is exactly 90 degrees.

Then angle D because it is between 90 and 180 degrees.

Then B is next because it is exactly 180 degrees and then E is the largest because it is greater than 180 degrees.

For question four, A is obtuse at 135 degrees.

B is reflex at 350.

C is acute at 80, D is reflex at 190 and E is acute at 10 degrees.

And for question five, an acute angle could have been made from line segments A, B, and C.

A right angle from D, obtuse angles from A, B, and C.

An angle of approximately 45 degrees from B and an angle of approximately 170 degrees from C.

A reflex angle actually could have been made from all four.

Pause now to think or discuss as to why.

Let's take our understanding of angles of all different sizes and start to apply this knowledge to groups of angles in different locations.

If you have two or more angles that meet at a single point on a straight line, then the sum of these angles is 180 degrees.

Note that this is not true if these angles meet at separate or different points at different locations on the same straight line.

Similarly, the sum of all angles that meet around a single point is 360 degrees.

Any two angles that sum to 180 degrees are called supplementary angles.

This means that two angles that meet at one point on a straight line are always supplementary.

In the same vein, any two angles that sum to 360 degrees are called conjugate angles.

This means that any two angles that meet around one single point are always conjugate with each other.

Let's show that these two angles meet at a point on a straight line.

Both angles share point A, so they do meet at the same single point.

101 plus 79 is 180.

Because these two angles share a common point and add up to 180 degrees, they certainly do lie on one straight line.

As a result, 101 degrees and 79 degrees are supplementary angles with each other.

But in this non-example, let's show that these two angles actually don't lie about a point on a straight line.

Even though both angles add up to 180 degrees, which means that 101 degrees and 79 degrees are supplementary, they do not share a common point.

101 degrees meets the line at point A while 79 degrees meets the line at point B and therefore cannot be used to determine whether the line is a straight line or not.

And in this next non-example, let's also show that these two angles actually do not lie about a point on a straight line.

Even though both angles meet at the common point of A, 101 degrees plus 81 degrees is 182 degrees, not 180 degrees.

Because these two angles do not add up to 180 degrees, this is not one straight line, rather two differently orientated straight lines that just so happen to meet at point A.

And in this next example we can see that both angles share the point A.

Furthermore, 227 degrees plus 133 degrees equals 360 degrees.

Because these two angles meet at a common point and add up to 360 degrees, then they lie on a full turn.

If two angles are conjugate because they sum to 360 degrees, then the reflex angle can be called the major angle, whilst the acute or obtuse angle that is conjugate to this major angle can be called the minor angle.

Okay, for this next check we have two different diagrams. For A, there are two angles that lie in a straight line and share a point.

And for B, we've got two angles about a point that make a full turn.

Out of this list of eight different angles, find a pair of supplementary angles and find a pair of conjugate angles.

Pause now to look through all eight options and choose the two for supplementary and the two that are conjugates.

To find a pair of supplementary angles, you need to find a pair of angles that adds up to 180 degrees.

The only pair that works from this list of eight is 145 degrees and 35 degrees.

Similarly, for conjugate angles, we need to have a pair of angles that add up to 360 degrees.

From this list we only had one pair, that was 265 degrees and 95 degrees.

When we have more than two angles that meet at a single point on a straight line, we can no longer say they're supplementary as this term and the conjugate term both refer only to pairs of angles, so two angles at a time.

However, multiple angles can still form a straight line as long as all of them still meet at the same single common point.

In this example, 75 degrees, 55 degrees and 50 degrees all meet at point A and some to 180 degrees.

And so that horizontal line is certainly one straight line.

However, in this non example, 75 plus 105 plus 130 plus 50 definitely does not equal 180 degrees.

In fact, it equals double that at 360 degrees.

This is because these angles meet at two separate points, points A and points B, rather than at one single point.

Because they meet at two separate points we apply the rule of angles on a straight line that metered a point twice.

Once for each of the two points that lie on that straight line.

Okay, let's take everything that we've learned so far and apply to something a little bit different.

If you had one straight line and a second line intersecting it, then all angles around that point of intersection will add up to 360 degrees.

This is because all four angles meet about one single point to create a full turn.

In this diagram, angles A and B lie in a straight line and share or meet at a single point.

Therefore, angles A and B add up to 180 degrees.

The same logic is true for angles B and C.

But angles A and C are called vertically opposite angles.

Both angles share a common point that is the intersection of those two straight lines.

The legs of each of these two vertically opposite angles use different parts of the two intersecting lines.

No part of either line is common to both angles except for the shared vertex point of those two angles, which is the point of intersection.

For vertically opposite angles, the size of angle A is always going to be equal to the size of angle C.

Vertically opposite angles are always equal.

Here's an example.

Let's show that these two angles are actually vertically opposite angles.

These two angles are vertically opposite because they meet at a common point that is the intersection of one straight line and a second straight line.

And the size of each angle is exactly the same at 50 degrees each.

On the other hand, in this non example, let's show that these two angles are not vertically opposite.

Whilst these two angles do meet at one shared common point, this is not one straight line, rather two separate straight line segments that meet at that common point.

Because there are not just two intersecting straight lines here, we know for certain that these two angles are not vertically opposite.

Furthermore, because these two angles are not equal in size at 60 and 50, they cannot be vertically opposite each other.

And for this check, in which of these diagrams do all the angles lie on a straight line around one single point? Pause now to investigate all four diagrams. A and D are the correct diagrams. For B 155 plus 29 does not sum to 180 degrees.

And for C, those angles meet at two separate points, A and B, not one single point.

And for this check, fill in the blank to find the size of angles B, C, and D and complete the sentence by justifying or explaining your answer for each.

Pause now for all three angles.

The size of angle B is 133 degrees because it is supplementary to the 47 degree angle.

The size of angle C is 47 degrees because it is vertically opposite the other 47 degrees and vertically opposite angles are always equal.

And for D, the size of D is 133 degrees because like angle B, it is supplementary to the 47 degree angle.

And here's the task B practise.

For question number one, which of these statements are true for the diagram that they are trying to describe? Pause now to look through all four statements for all four diagrams. For question two, we have six angles.

Two which lie about to point on a straight line and four which lie about two different points that create a full turn.

Find the most appropriate values from this list that match with each angle.

Afterwards, label each of the diagrams with conjugate or supplementary.

Pause now to do this question.

And for question three, I have six angles.

Use four of them to complete each of these four statements.

Pause now to do this.

And lastly, find the size of angles B, C, and D as well as the sum of angles B, C, and D, and choose the appropriate explanations for each.

Pause now to do this question.

And onto the answers.

For question one, A is false because 48 plus 142 is 190 degrees, not 180 degrees.

B is true, as is C.

D is incorrect because the 210 degrees plus the 150 degrees do add to 360 degrees, but they've forgotten that that right angle has a value of 90 degrees extra, making the total of that diagram 450 degrees, which is too much.

And for question two, the first diagram should have been labelled with 25 degrees and 155 degrees.

Those two angles are supplementary because they sum to 180 degrees.

For B, you should have had 171 and 189.

And for C, 267 and 93.

Both the angles at B and C are pairs of conjugate angles because each pair sums to 360 degrees.

And for question three, conjugate angles sum to 360 degrees.

For part A, that means that the minor angle is 160 degrees, and B, the major angle is 190 degrees.

Vertically opposite angles are equal and therefore the angle for C is 60 degrees, and supplementary angles sum to 180 degrees, and so the other angle is 110 degrees.

And for question four, angle B is 39 degrees because it is vertically opposite the other 39 degree angle.

Whereas C and D are 141 degrees because they are both supplementary to the 39 degree angle.

The size of angle B plus C plus D is 321 degrees because it is conjugate to the 39 degree angle.

And that's it for this lesson.

Well done on your rigorous application of angles in a lesson where we have looked at the fundamental purpose of an angle, it's the representation of how much turn or rotation there is or how much turn or rotation needs to be performed.

We've also looked at the fact that the length of the lines that enclose an angle, the angle legs, do not directly influence the size of any angle.

We've also looked at the names of different angles in different contexts as well as how supplementary and conjugate angle pairs can meet at a single point on a straight line or to create a full turn respectively.

And finally, we've also seen that vertically opposite angles are always equal in size and must meet at one single point, that is the intersection of two straight lines.

That is all for today's lesson.

Thank you so much for all the effort that you've put in.

Until next time, have a great rest of your day.

Goodbye.