Lesson video

Hello, my name is Dr.

Rowlandson and I'll be helping you with your learning during today's lesson.

Let's get started.

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

This lesson is called checking and securing understanding of properties of shapes.

And by the end of today's lesson, we'll be able to identify a shape from its properties and state the properties of a given shape.

Here are some previous keywords that will be useful during today's lesson, so you may want to pause the video if you want to remind yourself what any of these words mean, and then press play when you're ready to continue.

This lesson is broken into three learning cycles, and we're going to start with identifying types of triangles and quadrilaterals.

Triangles are three-sided polygons and there are different types of triangles.

There are equilateral triangles, isosceles triangles, and scalene triangles.

The type of triangles can be identified by the number of equal sides it has.

An equilateral triangle has three equal sides.

Isosceles triangles have two equal sides, and scalene triangles have no equal sides, or the type of triangle can be identified by the number of equal angles it has.

Equilateral triangles have three equal angles isosceles triangles have two equal angles while scalene triangles have no equal angles.

For isosceles and scalene triangles, they can contain right angles and there are examples of that at the bottom of the screen.

However, equilateral triangles cannot contain right angles.

Triangles can be presented in different orientations, so rather than always having the base of a triangle horizontal at the bottom of the triangle, it can be presented in different ways.

However, the orientations do not affect what type of triangle it is.

It's down to the number of equal sides and the number of equal angles.

So let's check what we've learned, fill in the blanks for the facts about types of triangles.

Pause the video while you do that and press play when you're ready to see the answers.

Let's take a look at the answers.

Equilateral triangles have three angles of equal size.

Isosceles triangles have two angles of equal size and scalene triangles have no angles of equal size.

So what type of triangle is this? Pause the video while you choose from either A equilateral triangle, B, isosceles triangle or C scalene triangle and then press play when you're ready for the answer.

The answer is B, isosceles triangle.

We can see that because it has two equal angles and the third angle is different from the other two.

Which angles are equal to other angles in this triangle here? Pause the video while you write down your answer and then press play when you're ready to see what the answer is.

The angles which are equal to each other here are A and C.

We can see that by the sides which make up those angles.

For angle B that is made from two equal sides, whereas angles A and C are different to that.

They are made from one of the equal sides.

And the third side, which is not equal to the others.

Quadrilaterals are four sided polygons and we can see some examples of quadrilaterals on the screen.

There are different types of quadrilaterals.

There are squares, rhombi or a rhombus, trapezium or trapezius for pleural, rectangles, parallelograms and kites.

Let's take a look now at parallel sides in quadrilaterals.

These four types of quadrilaterals all have two pairs of parallel sides.

They are a square, a rhombus, a rectangle, and a parallelogram, and we can see which sides are parallel to other sides based on the feathers on the sides, the arrows, which you can see.

What is the difference between these two groups of quadrilaterals? In one group we have a square and a rectangle, and the other group we have a rhombus and a parallelogram.

What is the difference between those groups? Pause the video while you think about this and press play when you're ready to continue.

The difference is that all of the angles in squares and rectangles are right angles, but when it comes to rhombi and parallelograms, they do not necessarily contain right angles.

Another thing to bear in mind is that angles that are opposite each other in these quadrilaterals are equal and in squares and rectangles, all of the angles are equal.

They're all 90 degrees.

So let's split these four quadrilaterals into two different groups.

In one group we have a square and a rhombus, and in the other group we have a rectangle and a parallelogram.

What is the difference between these two groups of quadrilaterals? Pause the video while you think about this and press play when you're ready to continue.

Well, all of the sides in squares and rhombi are of equal length.

We can see that now with the hash marks are on the sides, whereas with rectangles and parallelograms, they have two pairs of sides that are equal to each other, and we can see that because one pair of sides has one hash mark on and the other pair of sides have two hash marks.

Let's now think about a trapezium.

Trapezia contain exactly one pair of parallel sides.

So what about the number of equal sides and the number of equal angles in a trapezium? Well, a trapezium can contain no sides of equal length, two sides of equal length or three sides of equal length like we can see in these examples here.

And when it comes to angles, trapezium can contain no pairs of equal angles, one pair of equal angles or two pairs of equal angles.

But all trapezia contain two pairs of co-interior angles between parallel lines which are supplementary.

In other words, they sum to 180 degrees.

Let's now think about kites.

Kites contain two pairs of equal sides.

The equal sides in kites are adjacent to each other.

With the example on the left, we can see that the top two sides are equal to each other and the bottom two sides are.

With the example on the right, we can see the leftmost two sides, they're equal to each other and the two sides on the right, they're equal to each other.

And what about angles? Kites have one pair of equal angles, and these are the angles which are between sides of different lengths.

Quadrilaterals then can be identified by the well defined properties based on how many equal sides they have, how many equal angles they have, whether it has right angles or not, whether it has parallel sides or not.

Jun says, "Do I really need to know the properties of each shape? Won't I just recognise what type of quadrilateral it is by how it looks?" Well, classifying shapes based on appearance can be unreliable because some may not look like the typical examples.

For example, here we have two quadrilaterals that look a bit different to the usual quadrilaterals we see when we are studying examples of each type.

So classifying these can be difficult.

If we know the properties though we could do this more reliably.

Jun says, "The additional information means that I can classify them based on the defining properties." One of these is a trapezium and the other one is a kite.

Can you see which way around they go? The one on the left that's a trapezium because it has one pair of parallel sides, whereas the shape on the right is a kite because it has two pairs of equal sides and those equal sides are adjacent to each other.

And we can see as well that the angle that is on the left and the angle that is opposite on the right, they are equal, but the other two angles are not equal.

Classifying shapes based on appearance can also be unreliable because the difference between types of shapes can sometimes be very subtle.

For example, take a look at these two shapes here.

Jun says, "These look like the same shape." They both look like they could be parallelograms. But when we look at the properties of these shapes and look at the angles and the number of parallel sides, hmm, what do we think now? One of these shapes is a parallelogram, while the other shape is a trapezium, can you spot which way around they go? The shape on the left is a parallelogram because it has two pairs of parallel sides and two pairs of equal sides, and we can see that the opposite angles are equal to each other, whereas the shape on the right, that's a trapezium because it has only one pair of parallel sides and we can't see whether any of the sides are equal to other sides.

And also we can see that the diagonally opposite angles are not equal to each other.

Here we have a quadrilateral.

Is it possible to identify whether this shape contains parallel sides based on the information given? And you'll notice that there are no feathers on any of the sides.

None of those arrows will indicate parallel sides, but it has given us the angles.

Can you determine whether this side has parallel sides based on the angles? Pause the video while you think about that and press play when you're ready to continue.

Yes, you can and that's because co-interior angles sum to 180 degrees between parallel lines.

And we can see here that we have these co-interior angles which sum to 180 degrees.

So the top side and bottom side must be parallel.

And also we can see that these two co-interior angles sum to 180 degrees, which means the side on the left and the side on the right must also be parallel.

So yes, we can determine that it has parallel sides based on the angles.

So let's check what we've learned, sort the quadrilaterals into the correct columns of the table.

And for this activity, assume that the kite is not a square or a rhombus.

Pause the video while you do this and press play when you're ready to see the answers.

Let's take a look at some answers.

The kite has no parallel sides.

The trapezium has one pair of parallel sides and the other shapes have two pairs of parallel sides.

That is a parallelogram, a rectangle, a rhombus, and a square.

So what type of quadrilateral could the shape be? Pause the video while you choose from A, kite, B parallelogram, C, rhombus and D trapezium and then press play when you're ready for an answer.

The answer is D trapezium.

We can see that because when we look at the co-interior angles on the top with a bottom, we can see they sum to 180 degrees 107 plus 73 and 71 plus 109.

They both make 180 degrees, which means the top side and bottom side are parallel.

But if we do that going from left to right with those pairs of co-interior angles, they do not sum to 180 degrees.

107 plus 71 is not 180.

73 plus 109 is not 180.

So the left side and right side are not parallel.

Okay, it's over to you now for task A, this task contains three questions and here is question one.

You have four quadrilaterals and four names of types of quadrilaterals.

You need to match the words to the quadrilaterals based on the information you're given.

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

Here is question two.

We have three shapes that are partially covered and we have a table with different types of shapes in.

For each shape tick the cells in the table to show the shapes that it could possibly be.

So for example, shape a quite a bit of that shape is covered over, so there are different possible shapes it could be.

Tick all the ones that it could be.

And then for shape B, you can see a bit more information.

C and so on.

Pause the video while you do this and press play when you are ready for question three.

And here is question three.

We have two pairs of students who are playing the game.

In each pair, one student draws a polygon and the other student asks three questions to find out what the shape is.

Based on the answers, think about the questions A, B, and C.

Pause the video while you do this and press play when you are ready for some answers.

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

Shape A is a rectangle.

It has four right angles.

Shape B is a trapezium, it has one pair of parallel sides which you can establish by looking for co-interior angles that sum to 180 degrees.

Shape C is a kite, we can see that two of the angles are equal to each other and they are across from each other, and the other two angles are different to each other.

And shape D is a parallelogram.

Question two, what could shape A be based on what we can see? Well, it could be any of these shapes and shape B, we have a bit more information here so we can narrow down what type of shape it could be.

We can see that it's going to have at least four sides, so it's one of these shapes.

And for shape C, you have all the information apart from a little tiny bit.

It could be either a rhombus or a parallelogram.

Then question three, what shapes could Alex have drawn? Well, we know it's a quadrilateral, that it has parallel sides, but it does not have any right angles, so it could be a parallelogram, a rhombus, or a trapezium.

What shapes could Sofia have drawn? Well, we know it has three sides, so it's a triangle.

We know it has a right angle, so that narrows it down to either an is isosceles triangle or a scalene triangle, and we know that all the sides are not equal, so hmm, that doesn't tell us anymore.

It's either isosceles right angle triangle or a scalene right angle triangle.

Which of Lucas's questions did not help him work out Sophia's shape? It was that last question there.

If Lucas has already established that there is a right angle, he already knows it can't be equilateral with all the angles being equal in that case.

He could have used this last question to establish if it was a scalene or isosceles by asking if it has a pair of equal lengths instead.

You're doing great so far.

Now let's move on to the next part of this lesson, which is using properties of shapes to find missing angles.

Missing angles can sometimes be found in polygons by identifying angles which are equal.

For example, here we have an isosceles triangle and one of the angles is labelled X.

Let's find the value of x.

There are multiple ways of doing this, but here is a solution that only uses reasoning about equal angles, and that is x equals 62 because base angles in an isosceles triangle are equal, and we can see that the x angle is equal to the 62 angle because both of those angles have one of the equal sides and the third remaining side.

Missing angles can sometimes be found in more complex problems by identifying angles which are equal.

For example, here we have a diagram that contains a pair of parallel lines and a isosceles triangle in between them.

We know that the line segment BE is equal to the line segment EF in length, and we want to find the value of x.

Once again, there are multiple ways to solve this, but here is a solution that only uses reasoning about equal angles.

That is angle EFB is equal to angle ABF, because alternate angles on parallel lines are equal.

And then once we know that, we can look at the isosceles triangle.

Angle FBE is equal to angle EFB, because base angles in an isosceles triangle are equal.

So with that solution we didn't need to do any calculations, we solved it by matching up equal angles.

Therefore x equals 62.

Missing angles can sometimes be found in polygons by identifying angles which are equal and angles which are supplementary.

For example, here we have a quadrilateral and we can see it's a parallelogram because it has two pairs of parallel sides and we want to find the values of X, Y, and Z.

Once again, there are multiple ways to solve this, but here's one solution.

We could say that x equals 132 because opposite angles in a parallelogram are equal.

So we've used a pair of equal angles to identify that one.

And then for Y, we could say a Y is equal to 180, subtract 132, which is 48 because co-interior angles in parallel lines are supplementary.

So we've used a fact about supplementary angles to work out that one.

And then for Z we could say Z is 48 because opposite angles in a parallelogram are equal and it's opposite the angle, which was Y.

Once again, we could work out Z using supplementary angles using either our answer from x or the 132 degrees, what was given to us.

Missing angles can sometimes be found in more complex problems by identifying which angles are equal and which angles are supplementary.

Take a look at this one, which contains two pairs of parallel sides and we want to find the value of x.

Once again, there are multiple ways to do this, but here's one solution.

Angle DHJ is equal to angle IHG.

That's because vertically opposite angles are equal to each other.

We could then say that angle JBD is equal to angle DHJ, because opposite angles in a parallelogram are equal to each other.

And once we know that, we can say that x is equal to 180 degrees.

Subtract 132, which is 48 because angles that form a straight line are supplementary.

I wonder if you would've done that in a different way.

Perhaps pause the video and think about any other ways you could do it before pressing play to continue with the lesson.

Okay, let's check what we've learned, which fact relates to the angles marked in this diagram here? You've got options A, B, and C to choose from.

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

Okay, the answer is B, co-interior angles in parallel lines are supplementary.

So we could use the 70 degrees to find the value of x.

Please do that.

Find the value of x.

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

The answer is x equals 110.

You can get it from doing 180, subtract 70.

So we've changed the diagram slightly here, which fact relates to the angles marked in this diagram? Pause video, why you choose from A, B, or C and impress play when you're ready for an answer.

The answer is A, base angles in isosceles trapezium are equal.

So we could use that fact with the 70 degrees to find the value of y, please do that.

Find the value of y, pause while you do it and press play when you're ready for an answer.

The answer is y equals 70, it's equal to the other angle.

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

This task contains two questions and here is question one.

You need to find the value of each unknown labelled A to P, pause video while you do this and press play when you're ready for question two.

And here is question two.

Find the value of each unknown labelled A to D and justify your answers with reasoning.

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

Okay, let's see how we get on.

Here are your answers to question one.

Pause video while you check these and press play when you're ready for question two's answers.

And then question two, A is 74 and as an example some reasoning you can give on the screen here.

Pause while you check this and press play when you're ready to continue.

B is 68 and there's an example of some reasoning you can give here.

Pause while you check this and press play when you're ready to continue.

C is 145 and here is an example of some reasoning you can give here and D is 116 'cause this is the same diagram, here is a continuation of reasoning from the previous answer.

Pause while you check this and press play when you're ready for the next part of the lesson.

Well done so far, it's now time to get practical because in the third part of this lesson we're going to investigate shapes with some paper folding.

Here we have Izzy, Izzy takes an A4 sheet of paper and fold it to make some shapes.

Let's see what Izzy does.

She says, "I'm going to fold the top right vertex over to part way along the left side so that it bisects a right angle in the top left vertex." And it's a bisected angle means to cut it into two equal parts so that angle will be 45.

Let's take a look at how do that, this is what it looks like.

What shape has Izzy now made? And how do you know she's made that shape? Pause the video while you think about these questions and then press play when you're ready to continue.

Well, Izzy has made a trapezium.

I wonder why it's a trapezium.

Izzy says, "It's a quadrilateral with one pair of parallel sides, so that makes it a trapezium." How do we know it has a pair of parallel sides? Well, she says, "I know that the left and right sides are parallel because it started as a rectangle and the parallel in that, also the two angles at the bottom are right angles because it started as a rectangle and because they're both right angles, they sum to 180 degrees, which means they are between a pair of parallel sides.

Izzy folds her trapezium to make another shape.

Let's see what Izzy does.

She says, "I'm going to take the bottom right vertex and fold it over so that the shorter side is resting along the slope side." Let's see that.

It looks like this.

She now has another shape.

Let's turn it over and take a look.

What shape has Izzy made? And how could she check this? Pause the video while you think about it and press play when your ready to continue.

Izzy has made an is isosceles triangle.

We can see it's a triangle because it has three sides, but how do we know it's isosceles? Izzy says, I could check that it has a pair of equal sides by folding it in half and seeing if the sides line up like so.

It looks like they do, she says, I could also check, that it has a pair of equal angles this way as well.

The angle in the bottom left, if they overlap so that they perfectly match up, then they're equal or they at least are approximate equal.

We'll come back to how to check a bit more carefully in a second, but it looks like based on these checks that it is an isosceles, so these are some practical ways to check and isosceles triangle, but how could we be a little bit more rigorous than that using mathematical reasoning? Izzy says, if I knew the dimensions of the original rectangle then I could prove that the two sides of the triangle are equal using Pythagoras' theorem.

Now you may know this, you may not, but the sides of an A4 sheet of paper are in the ratio of one to root two, so let's use Pythagoras' theorem now to work out the other side of this triangle which we think should be equal to root two.

Izzy says, "My first fold made a right angle triangle with a base of one unit and a height of one unit." So she says, "I could use Pythagoras' theorem to find the length of the side marked X." She would do that by doing x squared equals one squared plus one squared.

That would work out to get X squared equals two and then if you square root both sides you get X equals root two, which means yes, that side is root two.

Izzy says, "Therefore both sides are the same length." Okay, let's check what we've learned and this time it could be you who does some paper folding.

Take an A4 sheet of paper, hold it portrait so that the longer sides are going upwards and fold it with the following steps.

First, take the bottom right vertex all the way up to the top left vertex.

It should look like the image on the right where you have a diagonal fold, which does not go through either of the corners of this rectangle and you should have a right angle triangle on the top right and another right angle triangle sticking out on the left of it, pause video while you do that fold and press play when you're ready to continue.

Okay, now take each of those right angle triangles that are sticking out.

You can see the right angle triangle, the top right and the right angle triangle on the left.

Fold those over so that they are overlapping with the other two parts of that paper, which are overlapping as well.

In other words, you should see those angle triangles folded over with just little tiny bits sticking out from the edges like can see on the image on the right.

Pause the video while you do that and press play when you're ready to continue.

Okay, now open up your paper fully.

You should have your original rectangle but you can see now some very visible folds in it.

Pause while you do that and press play when you're ready to continue.

Okay now, with those folds you can see you have a right angle triangle in the top right corner of your paper and a angle triangle in the bottom left corner of your paper.

Fold each of those right angle triangles over its hypotenuse like you can see with the image on the right.

Pause the video while you do that and then press play 'cause I'm going to ask you a question.

Okay, now you've done that.

What shape have you made? Pause the video while you identify what shape it is, but consider how you know and consider how you could check by using your paper.

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

The shape you have made is a rhombus and the ways you can check that is you can fold your paper over it in various ways to see whether certain aspects are equal to each other.

For example, you could fold it over diagonally to see if the opposite angles, which are diagonal to each other are equal, or you can fold it over so that you have each side resting against each other side and see that the length of those sides are equal.

Either way, you've got a rhombus.

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

This task contains two questions and here is question one.

Once again, you're going to take an A4 sheet of paper and you're going to fold over using the steps below.

Steps one to four are what Izzy did earlier and then you are gonna be doing a little bit more after that.

In part A, you need to identify what shape you've made in step four and how you know and in part B, you need to identify what shape you've made by step seven and how you know.

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

Okay, question two, take a paper equilateral triangle and maybe your answer for question one might help you here and it using the steps below.

While you are doing that, identify what shape you make in each step from step two to step four and explain how you know each time.

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

Let's take a look at some answers.

Question one, what shape do you have in step four? It is a trapezium.

It's a quadrilateral with one pair of parallel sides, which are still in the same orientation as the vertical sides in the original rectangle.

We saw that earlier.

Then B, what shape do you have in step seven and how do you know? It's an equilateral triangle, you could check it by folding each vertex onto the others to check that the angles are equal.

You can also fold it in half each time to check that the lengths are equal as well.

Then with question two, you need to identify what shapes you have in steps two to four.

Well, in step two you have a trapezium.

It's a quadrilateral with one pair of parallel sides.

In step three, it's a rhombus.

It's a quadrilateral.

We have two pairs of parallel sizes.

The opposite angles are equal and all the sides are equal length.

You can check that by folding them onto each other and step four you have an equilateral triangle.

Again, it's a triangle of all the angles equal in size and all the sides are equal in length.

You can check it again by folding 'em onto each other.

Fantastic work today.

Now let's summarise what we've learned in this lesson.

Many shapes have similar properties and shapes can be classified according to the defining properties.

It's not always clear which shape is being described, so sometimes more details may be needed and no other properties of shapes can help you find missing angles and solve more complex problems. Great work today.

Thank you very much.