Lesson video

Hello, and welcome everyone.

It is a pleasure for me, Mr. Gratton, to join you today for this maths lesson, where we'll be looking at classifying triangles by observing properties of their angles and side lengths.

The triangles we're going to look at today are scalene, whose three sides and interior angles are all different from each other.

Isosceles, which has a pair of angles that are equal and a pair of sides that are also equal in length.

We're also going to look at equilateral triangles, whose three interior angles are equal, and whose three sides are also equal in length.

Finally, we're also going to look at the right-angled triangle, which has exactly one interior angle of 90 degrees.

First on the list is the scalene triangle.

Let's look in more detail at all of its properties.

Before we look at any specific type of triangle, what does triangle in itself actually mean? Well, tri means three, and angle it means angle, so triangle means a two-dimensional shape with exactly three line segments.

Or more truthfully to its etymology, a two dimensional shape that has exactly three interior angles.

What do you know about the interior angles of a triangle? Well, the interior angles always add up to 180 degrees, no matter what they look like, how they are orientated, or what size they are.

A triangle can only have an interior angle sum of 180 degrees, such as in this triangle, 80 plus 66, plus 34 equals 180 degrees.

Any triangle whose interior angle sum is not 180 degrees has been incorrectly labelled, measured, or its sum incorrectly calculated.

Furthermore, the angles you must measure must be interior to the triangle.

Using any other angle on that triangle will result in a sum of something other than 180 degrees.

For example, in this diagram, this 100 degrees is exterior to the triangle, and so 20 plus 80, plus 100 equals 200, not 180, and so we know this is not the correct calculation.

Okay, here's a quick check.

Which of these diagrams shows that the sum of interior angles in a triangle is 180 degrees? Pause now to have a look through all five of these diagrams. Only triangles A and E show three interior angles that sum to 180 degrees.

In D and C there is one angle that is not interior that has been labelled, and in B there are well, no angles labelled at all.

Only three side lengths are given.

And for triangle E we have that angle marker which represents a 90 degree angle.

Now that we've looked at triangles in general, what are the properties specific to scalene triangles? Well, scalene triangles have three different-sized angles and three different-sized sides.

For example, in these two triangles, this triangle shows three angles which are all different, and this triangle shows three sides of different length.

Note, however, that both properties are true for both triangles.

That is to say, both triangles have three angles and three side lengths that are all different.

In this set of non-examples, the top triangle is not scalene, as two of its angles are equal at 34 degrees each, whilst on the bottom triangle it has two side lengths of equal length and so it is also not scalene.

Instead of an actual angle written with a numerical value, you may see scalene triangles represented by three different types of angle marker, one for each differently sized angle.

The angle marker of each differently-sized angle will have a different number of arcs.

It does not matter which angle has which number of arcs.

So for example, this distribution of arcs is equally as valid as the previous distribution.

Here's a quick check, which of these diagrams show a scalene triangle? Pause now to look at the side length and angles of all five triangles, B, C and E all show scalene triangles.

A right-angled triangle has exactly one interior angle of 90 degrees.

The angle marker for this type of angle is a square, rather than the arc for most other angles.

but a right-angled triangle must have exactly one right-angle.

In fact, it is not possible for any type of triangle to have two right-angles in them at all, and here's why.

90 degrees plus 90 degrees, that is two right-angles, already equals 180 degrees, which is the sum of all interior angles of a triangle.

However, because those two angles take up all 180 degrees of the interior angles, that leaves, well, zero degrees for a third angle, which isn't possible.

All angles interior to a triangle must have a value greater than zero degrees.

This means that a triangle with two or more right-angles simply cannot exist.

Many, if not most of the right-angled triangles that you'll meet are scalene, meaning that one angle will be 90 degrees and the two other angles will be both different from each other and also not 90 degrees.

Here we can see a right-angled scalene triangle labelled with one square angle marker, one angle marker with one arc, and one angle marker with two arcs.

This shows that all three angles are different from each other.

A right-angled scalene triangle can also be labelled numerically.

Note that the square angle marker always represents 90 degrees and so can still be omitted, even when all other angles are labelled.

In this non-example, this triangle is a right-angled triangle, but it is not scalene, as two of the other angles are both equal in length.

Whilst the rightmost triangle is a bit ambiguous, actually, this one shows three angles of different size, meaning it is scalene.

However, sometimes if all three angles are unknown, one of the three angle markers may actually represent a right-angle.

We cannot just assume its value.

If the size of one of the angles is later shown to be 90 degrees, we can either then label the angle as 90 degrees or replace the angle marker with a square one.

The side lengths and angles inside any triangle, scalene or not, remain invariant after a rotation or reflection.

This means that if you rotate or reflect a scalene triangle, the image will also be a scalene triangle.

In fact, it will be a scalene triangle whose corresponding angles are invariant between the object and the image, only the location of the angles will change.

And for a different transformation the angles inside any triangle remain invariant after an enlargement.

This means that if you enlarge a scalene triangle, the image will also be a scalene triangle.

So, in this example, all corresponding angles on the object and its image remain invariant, but the length of the sides each angle is adjacent to have changed.

For this check, this triangle is the object.

Five more triangles will appear on screen, some of which are images for this object after transformation has taken place.

Which of these diagrams shows a right-angled scalene triangle that has the exact same three angles as this object? Pause now to look through all of the details.

In fact, all of them are, except for D.

For some of the triangles it is the angle markers that indicate that it has the same angles as the object, and for others it is the numerically represented angle instead.

Here's a set of practise questions based on scalene triangles.

For question one, in which of these triangles have the angles been incorrectly measured using a protractor? Pause now to have a look at all five triangles.

And for question number two, each diagram shows a pair of scalene triangles where the object has been rotated, or reflected, or enlarged to become its image.

Correctly label with numbers or angle markers all missing angles on each of these scalene triangle images.

Pause now to do this.

Okay, onto the answers.

For question number one, both A and C have been correctly measured.

B and D are definitely incorrectly measured, because these two triangles should be scalene, but two angles are of equal size.

Whereas E has definitely not been measured correctly, because the three interior angles of this triangle do not sum to 180 degrees.

And for question number two, pause now to check that your images matches the ones that are on screen.

So far we've looked at triangles whose three interior angles are all different, but what if two angles are the same? Let's see what that's about.

Isosceles triangles have two angles that are equal in size.

And furthermore two sides are also of equal length.

For all isosceles triangles the side length adjacent to both equal angles is the odd one out.

This means that the two other sides are of equal length.

We can say that the two side lengths that are equal are the legs of the angle that is the odd one out.

We can show that two sides have equal length using a single hash mark.

The same number of hashes means that the sides are of equal length.

Furthermore, the two equal base angles of an isosceles triangle can be shown using angle markers made with the same number of arcs.

So, in this diagram we have two angles with two arcs as the angle markers, therefore these two angles are of equal size.

Also notice that the equal angles of an isosceles triangle, the two equal base angles they're called, do not need to be at the bottom of a triangle, as a triangle can be in any orientation.

These two equal angles are on the right of the shape whilst the triangle is currently in this orientation.

And the side adjacent to both of them is called the base side, meaning the other two sides of this triangle are equal in length, meaning that we can label them with a single hash mark each.

For this check, which of these three sides is different to the other two? Also, select the statement which correctly explains how you know this.

Pause now to answer both of these questions.

The side that is different from the other two is side A.

This is because the base side is always different from the other two and the base side is always adjacent to the two angles that are equal in size.

We know that side A was the base side of this isosceles triangle, because it is adjacent to the two angles marked with the double arced angle markers.

Onto this check, which of these isosceles triangles are correctly labelled? There may be more than one correct answer.

Pause now to look through all five triangles.

A, D, and E are all correctly labelled.

Triangle B has three angles of different size, whereas C, whilst it does have the correct hash marks, the two angles that of equal size are in the incorrect place.

We've seen that right-angled triangles can be scalene.

However, there is a specific type of right-angled triangle that is also isosceles, instead of scalene.

As we know, one angle must be the 90 degree right-angle and the two other angles must be equal to each other, because it is an isosceles triangle.

What size or sizes could these two angles be? Well, the interior angles in a triangle must sum to 180 degrees.

The right-angle is fixed at 90 degrees, meaning that there is 180, take away 90 degrees, equals 90 degrees remaining between the other two angles.

Since this is an isosceles triangle, both the remaining angles must be equal, therefore 90 degrees divided by two equals 45 degrees, and so each of those other two non-right-angle angles are always 45 degrees.

If a right-angled triangle is also isosceles, these two base angles must be 45 degrees.

There is no other pair of angles that are equal in size that sum to 90, and so the only triangle that is isosceles and right-angle is a triangle with the angles 90 degrees, 45 degrees, and 45 degrees.

Any other right-angled isosceles triangle will be similar to this one.

It is important to be able to sketch or draw a triangle from only information given.

Plain paper can give you a sense of the shape, but square paper can make the visual properties more accurate and easy to understand.

Isosceles and right-angled triangles can be drawn on square paper.

Firstly, draw a horizontal line to act as one of the two sides of equal length, like so.

Then from one endpoint of the line that you just drew, draw a vertical line with the same length.

This gives us two important properties of an isosceles and right-angled triangle, two sides of equal length and a right-angle made from the horizontal and vertical perpendicular side lengths.

Next up, connect the two lines to complete your isosceles triangle.

And finally, add the hash marks and right-angle marker to help show it is both isosceles and right-angled.

The process for drawing a scalene right-angled triangle is exactly the same with the one exception being that the vertical side that you draw must have a different length to the horizontal one.

The vertical side can either be greater or shorter in length than the horizontal one that you would draw first.

To draw a general isosceles triangle, rather than a right-angled one we start with a horizontal line to act as the base of the isosceles triangle.

Try to make this base side an even number of squares, such as this with a base length of four units.

Why? Because this makes finding the midpoint of that base side easier, and so your next step is to find the midpoint of that base side.

From that midpoint, count up as many squares as you want.

This will give you the third vertex of your isosceles triangle.

From there it is as simple as joining up this vertex with both endpoints of your base line segment to complete your isosceles triangle.

Adding the hash marks also helps to show that it is an isosceles triangle.

And finally, an isosceles triangle can be formed from two congruent right-angled scalene triangles.

If a right-angled scalene triangle is reflected where the line of reflection is one of the sides adjacent to the right-angle, the resultant combined shape is an isosceles triangle.

This is because the side used as the line of reflection is not a part of the perimeter of this now new combined triangle, and one of its sides has been reflected, creating a second side that is equal in length, one of the defining properties of an isosceles triangle.

As we can see, without the side used as the mirror line, the object and the image create an isosceles triangle.

This is because the other side length adjacent to the right-angle has doubled in length and also remained as one straight line segment.

If we did this exact same reflection at any other angle than 90 degrees, the result would not have one straight line, rather it would have two lines that meet at one angle.

We can also look at this process in reverse.

All isosceles triangles can be split up into two congruent right-angled scalene triangles.

For this check match the values with the angles to create an isosceles and scalene right-angled triangle.

Pause now to look through all possible options of these angles.

And here are the answers.

Note that angles C and D, these two angles must sum to 90 degrees.

This in addition to the 90 degree right-angle will then equal the 180 degree interior sum of a triangle.

Here's a set of practise questions by looking at the hash marks and angle markers, write down whether each triangle is scalene, right-angled and scalene, isosceles, or right-angled and isosceles.

Pause now to categorise all five of these triangles.

And for question number two, you have four pairs of congruent isosceles triangles.

For each pair correctly label the hash marks to show the size of equal length and the angle markers with a suitable number of arcs to show that the angles are equal in size.

Pause now to do this for all four pairs.

And onto question number three.

Each triangle here is labelled with either angle markers or hash marks and a partial description underneath of the interior angles of each triangle.

Label all three angles for each triangle in their correct position.

Pause now to do that for all four shapes.

Pause here for question number four, where you have to draw three triangles from three different descriptions.

And finally, pause here for questions five and six, which relate to right-angled scalene triangles, which create an isosceles triangle.

Okay, here are the answers.

For question 1A we have a right-angled scalene triangle, B is an isosceles triangle, C is scalene, D is also isosceles, and E is right-angled and isosceles.

Pause here to check for question number two, the angles on screen to the angles that you have labelled on your version of these diagrams. And pause here for question three.

Here are the angles that you should have labelled for these four triangles.

For question number four, here are some examples of the triangles that you could have drawn for these three descriptions.

For question number five, here are the lines of reflection that you should have drawn.

And for question number six, here are the completed isosceles triangles.

There's one final type of triangle to focus on, the equilateral triangle, which is a very particular triangle indeed.

Let's see why.

Equilateral triangles have three equal angles and three equal side lengths.

Note the etymology of the word equilateral, which can be broken down into equi, which means equal, and lateral, which means sides, meaning an equilateral triangle is a triangle with all equal sides.

Equilateral triangles can also be called regular triangles, in the same way that a square can be called a regular quadrilateral.

However, this title is far less common.

What size could these interior angles in an equilateral triangle be if they all must be the same? Well, again, we know that the interior angles of a triangle sum to 180 degrees.

Since the triangle is equilateral, all three angles must be equal.

Therefore, 180 divided by three equals 60 degrees for each angle.

Therefore, every equilateral triangle must have three angles of 60 degrees each.

There is no other possible interior angle in a triangle that is equilateral as no other number when multiplied by three gives you 180.

To show that an equilateral triangle has three equal side lengths we can label all three sides with a hash mark with the same number of hashes on each side.

And also, we can show the angles are equal by having angle markers each with the same number of arcs, preferably greater than one arc each to make it clear that the angle markers are denoting equally sized angles.

Who is correct and can you explain why? Jun, who says it is impossible for an equilateral triangle to also be right-angled, or Sofia, who says it is possible for an equilateral triangle to be right-angled.

Pause here to have a quick think and weigh the properties of a right-angled triangle and an equilateral triangle and see if they could work in tandem together.

Jun is correct.

It is impossible for an equilateral triangle to also be right-angled.

One reason why is a right-angle is 90 degrees, but all angles in an equilateral triangle must be 60 degrees instead, and so those two angles do not make sense together.

Furthermore, a different explanation is a right-angle is 90 degrees and all angles in an equilateral triangle must be equal, but interior angles in a triangle sum to 180 degrees, and 90 plus 90, plus 90 equals 270, which is definitely not possible for a triangle with an interior sum of 180 degrees to have.

For this final check, which of these correctly shows or describes an equilateral triangle? Pause now to look through all six options.

And the answers are A, D, E, and F.

Each of these four options either talks about three side lengths that are equal, or three angles that are equal at 60 degrees each.

Or in the case of F, both.

Okay, here's the final set of practise questions.

For question number one, categorise these six triangles into the table here.

Pause now to do this.

And for question number two, here are five different descriptions of five triangles.

Write down all possible names for each triangle.

Some descriptions actually don't actually make a real triangle.

Name these the impossible triangle.

Pause now to do this.

Okay, here are the answers to question number one.

Note that it is impossible for an equilateral triangle to also be right-angled.

And for question number two, A is an isosceles triangle, B is either isosceles or isosceles and right-angled, C is the impossible triangle.

This is because the sum of the angles is 190 degrees rather than 180.

For D, we have a scalene triangle, and for E we have an equilateral triangle or a scalene triangle.

However, it cannot be isosceles.

This is because if you have two angles of 60 degrees, automatically you have a third one of 60 degrees as well, making it equilateral rather than just isosceles.

And well done on all your work, effort, and attention in today's lesson about identifying and categorising triangles.

We have looked at scalene triangles, whose sides and angles are all different.

Isosceles triangles, which have a pair of angles and a pair of sides that are equal.

And equilateral triangles, where well, everything is equal.

We've also looked at right-angled triangles, which can either be isosceles or scalene, but not equilateral.

We've also looked at constructing isosceles triangles from right-angled scalene triangles, and how to label triangles with hash marks and angle markers.

Once again, thank you so much for joining me, Mr. Grattan in today's maths lesson.

That's all for today, and so until next time, have an amazing rest of your day.

Take care.