Lesson video

In this lesson you will understand that the interior angles in a triangle add up to 180 and you'll also be able to solve problems involving unknown angles in a triangle.

Your first task then is a try this activity.

This triangle has been drawn between two parallel lines, how many of the missing angles can you work out? While so, whilst you're doing that, I want you to tell me or write down in your book the parallel line properties that you use.

Which of the parallel line properties did you use to solve these questions? Okay, let's work through our answers together.

I want to work out the missing angles which were the blue one and the orange one here.

So, if I'm looking at this 70, we've been told that we've got parallel lines, they lie between two parallel lines.

I know that 70 and these blue lines are alternate angles because they lie in the same region, which is the interior region, but they lie on different side of the transversal.

If you remember that this line that cuts across our parallel lines is called a transversal.

So 70 and the blue angle are the same because they are alternate angles.

I'm just going to write alt for alternate for short.

And this 30 degrees and our orange would also be the same because they are, exactly, they are alternate angles as well because, again, they lie in the same region, so it's still the interior region, but they on different sides of the transversal.

So this would be 30 degrees.

Right, we've worked out two, how can I work out this pink angle here? What property can I use to work out this pink angle? Right, angles on a straight, angles on a straight line add up to 180.

So angles on a straight line add up to 180, so I've got 70 and 30, what other number do I need to add to 100 to get to 180? It's 80, so here, this angle is 80 degrees.

So we've worked out our blue angle, our orange angle, and our pink angle.

Okay, we have our connect task.

There are two students here and one student is saying, "I can see that the interior angles of any triangle "must sum up to 180." And the other student is like, "But, how do you know?" And it's a very valid question, how do you know that the sum of any interior angle must sum up to 180? Looking at this, what can you tell me about this shape, anything? What can you tell me about this shape? Look at the angles, look at the length, look at the parallel sides, what can you tell me about the shape? I'm going to give you five seconds to think about it and then we're going to come back together to discuss it.

Okay, you could've probably seen from our try this task that this angle here is grey on my screen is the same as this angle over here.

Likewise, this blue angle is the same as this blue angle here, okay? So we've got two alternate angles and because we've got parallel lines, they are the same because alternate angles on parallel lines are equal.

Okay fine.

Now this pink line, again, from our try this task, what did we say about angles on a straight line? That they, yes, they add up to 180 degrees.

So let's try to explain to Yasmin now why the interior angle of any triangle must sum up to 180 degrees.

Say for example I choose this grey angle here to be 50, okay, we're just choosing any random value.

This is 50, because this is 50, the alternate angle on a parallel line, this angle inside our triangle must also be 50, okay? So we've got 50 degrees here.

If I choose this one to be 100, this blue angle here, if I say it is going to be 100 degrees, well, what this blue angle inside our triangle be? Yeah, it's going to also be 100 degrees, why? Well, because this 100 and this 100 are alternate angles so they are equal.

Right, if I've got 50 and 100, and I've got a straight line here and I know that angles on a straight line must add up to 180, I've got 50 and I've got 100, what's the value I need to make all the angles on this straight line add up to 180? Well, 100 plus 50 is 150 and 180 subtract 150 is 30 degrees.

And, let's add up the angles in our triangle, 100 plus 50 plus 30 is 180.

So we've proven that the sum of the interior angle in any triangle must be equal to 180.

Can you prove with another value? So pause the video now, can you prove using any other value that the sum of any interior angle would always be 180? Okay, we have two students, they are trying to find the remaining angles in this isosceles triangle.

Our first student is saying she's going to work out the blue angle first and our second student, he's saying he's going to work out the purple angle first.

Let's start with this first student, how can we help each student complete their working out? Well, what has this student done? She has drawn a line of symmetry on her isosceles triangle, and what do lines of symmetry show us? How many times the shape can be folded into each other.

So that means, the line of symmetry is like a mirror line, whatever I have on one side would be exactly reflected on the other side.

Or whatever value that I have here would be reflected on this side.

So let's try to work out the first half of the values.

So this is a right-angled triangle and right-angled triangles add up to, or right-angled triangles rather are equal to? 90 degrees.

So, and on the previous side we regathered or we proved that the sum of the interior angle in a triangle add up to 180.

So, these three angles must add up to 180.

At the moment, how much do we have? We have, 90 plus 70 and that is equal to 160 degrees.

So how many more degrees do I need to get to 180? 20, right? So this blue angle here would be 20 degrees.

Am I done? Not quite, 'cause I've worked out one half here, I need to put these exact values and reflect it onto the other side.

So that means this angle here will also be 20 degrees, and this angle here, well, we already know it's an isosceles triangle and the base angles of isosceles triangles are equal so this will also be 70 degrees.

Let's check, do they add up to 180? 70 plus 70 is 140, 20 plus 20 is 40, 40 plus 140 is 180.

Right, I'm going to change my pen, let's work through our other student's method.

He's going to work out the purple angle first.

Well, what do we know about the base angles of an isosceles triangle? They are equal, the base angle of isosceles triangle are equal, so this is 70, therefore this will also be 70, right? We've proven again that the sum of the interior angle of a triangle must always add up to 180, so this angle here must be 40.

So whether you use student A's method of splitting your shape in two and then reflecting your values on the other side or whether you use student B's method of I know that my base angles of an isosceles triangle are equal and then work out the missing values, whatever you decide to do, as long as the sum of your interior angle for a triangle add up to 180, whatever method you choose to do is totally fine.

Right, we have two questions on our screen.

First one, deduce the value of the marked angles, that means they want us to find this angle here and they want us to find this angle here and also this.

Let's start with A.

Well, how can I work out the value of A? What type of triangle is this? It is an isosceles triangle and what do we know about isosceles triangle? Well we know that two angles in an isosceles triangle are equal and the base angle of isosceles triangles are equal.

The base angle in this case is A and B, so I would take away 54 from 180 and that should give me 126.

Am I done? Not quite because these two are equal, 126 is for both A and B so what must I do to work out A? I must divide by two, exactly.

So I must divide by two, so the value for A is 63 degrees.

If the value for A is 63 degrees, the value for B must also be 63 degrees.

And we're just going to write it here, B is also going to be 63.

And we're just going to check, we must always check our work at the end, do my angles add up to 180? So at the top here, I am going to do 54 plus 63 plus 63 and you can tell me, do they add up to 180? They do, so I know that I have worked this out correctly.

180 degrees.

Okay, what about this one? This is quite an interesting question, isn't it? We've got a point here, we've got a vertex so to speak.

What do we know about when angles are opposite a vertex? Well we know that vertically-opposite angles are equal.

142 is vertically opposite to this angle here, so therefore this angle is going to be 142 degrees and I'm going to write my reason as VO which means vertically opposite.

Right, this is 142, we've got an 18 degrees, how would I work out this missing angle here? I should add 142 to 18, what would that be? It would be 160, wouldn't it? So, if this is 160 degrees, what would my, I'm just going to label this, M, because I can, what would M, what would the angle of M be? Well, it's going to be 180 take away 160 and that's going to give me 20 degrees.

Okay, I'm going to call this N, how do I work out the value for N? Now, your brain should be pinging at the moment, I can see something, I can see something.

Instead of working out 142 plus 20 then taking it away from 180, that's one way of doing it, what's another way of doing it? What can you see here? Well, I can see that we've found that this was 20 because they are alternate angles, so therefore N must also be 18.

So I can write N is 18 degrees because alternate angle and I could go further and prove it, I could go further and prove it.

So I've got 142, so I could write another way would be N is 142 plus 20, is what? Well, it's 162, isn't it? And 180 take away 162 is 18.

So we've found that there are several ways to work out an angle.

I could do it this way or I could look at my properties of a triangle to work out that because these two are alternate, they must be equal and because these two are alternate, they must be equal.

The most important thing here is that this angle here is 142 because it's vertically opposite to this.

Hello everyone, my name is Miss Jones and I just wanted to go through with you one more example before we do our independent task.

And what I'd like you to do here is take a problem and represent it with a bar model and we're going to be also asking you to do that in your task afterwards.

Why would we want to do that? Well bar model is a super useful way of taking something in context like these questions and showing the mathematical structure so that we can use that to visualise the problem and see what calculations we need to do.

So I'm going to do this for this first question here with an angle of 64 degrees.

I know that this is an isosceles triangle by the markings here, so A and B need to be equal.

So I've got three angles that need to add up to 180 and I know that one of them is 64.

So this is how I've drawn my bar model, I know that each bar is equal length which indicates that it's the same value so it's 180 and then this one is split into three parts, I know one of them is 64 and then I've got A and B.

Now if I know that A and B are equal value, what I can do here is easily see that I need to subtract 64 and then just divide by two.

So let's do that, 180, if I take away 64, I'm left with 116 and then I can divide by two, 116 divided by two is equal to 58 degrees, okay? So both A and B are equal to 58.

Okay, I want you to have a look at this second problem here and see if you can have a go at representing that using a bar model that shows how to calculate C and then work it out.

Okay, hopefully you've had a go at that and paused the video, let's look at it together now.

So in order to find C, I'll focus on this bottom triangle.

I know that if this angle is 136, this one should be 136 as well as it's opposite.

Now this is what my bar model looks like, so I've got 180 and then I've got three parts, I know 16, I know 136, so C is my unknown.

Again, I can use subtraction, if I subtract this and this from 180, then I can find out my missing value of C.

So I know that 16 added to 136 will get me 152 and I'm just going to annotate this time, so then C needs to be whatever's left from 180 which is 28 degrees.

180 take away 28 would leave me with 152.

C needs to be 28 degrees.

Okay, so I'll pass you back over to your teacher now who's going to explain today's independent task.

You now have your independent task now, so I want you to pause the video and complete your task and resume the video once you're finished.

Okay, welcome back, I hope you found the independent task challenging and engaging enough.

We're just going to go through the answers very quickly.

So I hope you've drawn your bar model.

Using the example, A, I hope you've been able to use it to draw the accurate bar model for each of the triangles.

But the missing angle for A is 96 degrees 'cause 50 plus 34, subtract 180 gives you 96 degrees.

For B, it's 118, for C, it's 63 degree.

If you disagree with any of this or if you got it wrong, pause the video and work out why you may have gotten that wrong and then resume to carry on.

So, we've got 67 degrees here, so therefore we are working out D.

So that will be 67 degrees 'cause it's an isosceles triangle, we add those together and subtract it from 180, we get 46 degrees.

In this case we get a decimal number and this is an equilateral triangle so each angle is 60 degrees.

What of this one? So, A, A and B would be the same because it's an isosceles triangle and then C, because this is 90 degrees, I can work out C because angles on a straight line add up to 180 and if I add 63 to 127, I would get 180, so that was what I used to work out what C was.

Now that I know C, and I know that this is 90 degrees, I can work out what my D value is because the sum of interior angle in a triangle add up to 180.

Here, starting with A, because that's the triangle that I have most information on, so I've got 53 degrees here, 90 degrees here, so A must be 37 degrees.

We've got 46 degrees here, so to work out B, I take away 46 from 180 and divide by two, so my B and C are the same.

D, I know that all the angles on the straight line would give me 180 degrees, so 53 plus 63 plus something is 180 and that something is 60 and since I know D, I can work out E as well.

Okay, decide if the following statements are always, sometimes, or never true.

The sum of the internal angles of three triangles is 540 degrees, that is always true and you can prove it by 180 times three is 540.

Two internal angles of a triangle will some to a value less than 90 degrees, that is sometimes true.

All the internal angles of a triangle are acute, well sometimes they are, sometimes they are not.

A triangle contains two right angles, it can never contain two right angles because 90 degrees and another 90 degrees takes us to 180 degrees and we know that the sum of interior angles in a triangle is 180 degrees using three different.

Okay, to find the missing angle on the triangle on the right it is 31 degrees because it's an isosceles triangle.

What is the internal angle of this shape? So, we've joined these two shapes together, what would be the internal angle of this shape? And it will be 360 degrees because we've just formed a quadrilateral by joining our two triangles together.

So it doesn't matter how many ways we join our shape, our internal angle would always be 360 degrees.

Right, your task for your exploration is to work out the missing marked angles, that means every single missing angle here, you are to work out.

So I want you to pause the video now and attempt this.

If you start and you figure out that you're really struggling, then carry on watching the video and I will provide you with some support.

So pause the video now and attempt this explore task.

Okay, my hint for you if you need some support is to start with the middle triangle because that is the triangle with the most information.

So start with this, what would this angle be based on everything we've learned in the lesson so far? Pause the video now and see if you can work out the rest by yourself.

Some answers for you.

So you would notice that these are vertically-opposite angles, these are vertically-opposite angles, these are vertically-opposite angles, and they would've helped us to work out the missing angles as well.

We've reached the end of today's lesson, very big well done for sticking through right til the end and completing your work.

I will see you at the next lesson.