Lesson video

Hello, welcome to today's lesson on internal angles in a quadrilateral with me, Miss Oreyomi.

You will need a paper and a pen for today's lesson as usual.

So if you need to pause the video to get these, then please do so and then resume when you are ready.

Okay.

In this lesson, you will be able to use triangle to deduce the sum of the interior angles in a quadrilateral.

And you will also be able to find our known angles in a quadrilateral.

Let's look at our try this task, I am going to read the words out for you.

It is possible to make an octagon by overlapping two squares.

What other polygons can be formed by combining the two square? So let's, these are the two squares overlapped, delete the overlapped bit, and then you have an octagon.

Remember what we said previously, a polygon is a 2D shape with straight sides.

So this is an octagon, it's not a regular octagon, but it's an octagon because it has one, two, three, four, five, six, seven, eight sides.

So how many other polygons can you make by combining two squares together? So pause the screen and attempt this.

And once you're done, come back and see the possible answers we could've come up with.

Okay, I hope you had fun doing that.

These are some of the examples you could have come up with.

So we've got hexagon over here, it's having two squares linked in this way, overlapped in this way, delete the overlap bit, and you can have hexagon.

You can have hexadecagon.

How many sides will a hexadecagon have? What about tridecagon? How many sides do you think it's got? And you could also have a nonagon, an octagon, a heptagon and a decagon as well.

Which ones did you come up with? Let's think about these two quadrilaterals on the board.

Quadrilateral ABCD is split into two triangles, ABD, and BCD.

Our first student is saying, I can see that the interior angles of any quadrilateral must sum up to 360 degrees.

And our second student is saying, how do you know? It's always a very good question to ask, how do we know, how did we come up with this conjecture.

So let's think about why is student a saying that he can see that the interior angle any quadrilateral must be 360? Well, it says, it's been split into two triangles.

What do we know about the sum of interior angles in a triangle from our previous lesson? Exactly, we know that the sum of interior angle in a triangle adds up to 180.

So I split my first triangle and I have 180 and my second triangle, how many degrees would I have in total? 180 degrees as well.

So, if I add my 180 degrees from my fast triangle, and my 180 degrees from my second triangle, the total sum of interior angle for a quadrilateral is therefore 360 degrees.

So the sum, of interior angle for a quadrilateral is 360 degrees.

'Cause I have two triangles in a quadrilateral.

Now a question to ask is, would this work for any quadrilateral? If I draw any quadrilateral on my screen right now, if you pause the video and draw any quadrilateral on your book, can you always, is it always possible to split the quadrilaterals into two triangles? Why don't you pause your screen now and attempt this and see if that is always the case? Okay, hopefully if you paused the screen and attempted this, you should have been able to deduce to come up with the fact that you can always split a quadrilateral into two triangles.

So I took the liberty of drawing two quadrilaterals on my screen, and let's see, can I always draw two, can I always split a quadrilateral into two triangles? So I could split it from here, so that'll be triangle one.

And that would be triangle two.

What of this one? I could split it here, that would be triangle one and that will be triangle two.

So yes, I can always split my quadrilaterals into two triangles.

Therefore, that means the sum of every interior angle in my quadrilateral must add up to 360 degrees.

Let's look at this one then.

I want to find the value of a in this first quadrilateral, of b in the second quadrilateral, and of c in this third quadrilateral.

Student a say, I use the bar model to represent each situation.

And student two is saying, I formed equations.

Now we know that we can use both and would come up with exactly the same answer.

So are going to now do some examples.

So let's look at the first one, we have 100, 120, 80 and a.

The total must be equal to? We just established that, must be equal to 360 degrees.

Okay.

So, what do I already have? I have 120 and 80, that gives me what? 200.

And I've got 100 here.

So what I currently have is 300 degrees.

What must I add to 300 to get to 360? So I'm on 300 and I want to get to 360, I need to add 60 more, exactly.

So my a would be 60 degrees.

Okay.

So that is using the bar model way.

Let's compare using equations.

So I know that I've got 100 plus 80 plus 120 plus something else.

There's something else which we don't know.

We know that it's a, but I do not know what a is yet.

Plus a, must give me 360.

So I have taken all the angles in my quadrilateral 100, 120, 80, and a, and I know because all the sum of the interior angle must be equal to 360 degrees, I have made the angles all equal to 360 degrees.

Now I am going to collect like terms. I've got three numbers here, 100 plus 80 plus 120, that gives me 300.

300 plus a, which again, we don't know what a is yet.

300 plus a is 360.

To get 300 away so that a is by itself, I am going to subtract my left hand side, I'm going to subtract 300 from my left hand side and do the same for my right hand side.

So I'm going to take away the 300 from both sides.

Over here, I am left with just a, because 300 take away 300 is zero, I'm left with just a and on my right hand side, I am left with 60.

So using either the bar model or the equation method, I worked out what my missing angle is, which is 60 degrees.

For this one, we have a kite in front of us.

Now we know that a kite has one line of symmetry and I could split my quadrilateral into two triangles, work out one triangle, and then reflect my answer onto the other side.

If I do that, I would know that the sum of the interior angle for my triangle is 180 degrees, but I'm not going to do that, you could do that actually.

You could pause the video now and do that.

If you split your triangles into two, would you get the same answer as if you just work it out straight as a quadrilateral? I am going to explain how to work it out straight as a quadrilateral.

So, I've got 132, plus 40 and then I've got b and b.

So I've got two unknowns in this quadrilateral, two unknown angles.

Now, 132 plus 40 is 172.

Again, the total angles have got, the interior angle in my quadrilateral is 360 degrees.

So at the moment, I'm on 172, and I'm trying to get to 360.

Now if you look at the squares for b and b, they're exactly the same.

So that tells me that my answer for b, they are the same.

Whatever answer I get for b, two bs must be equal to the same thing.

Again, think about the line of symmetry.

Whatever answer I get here is exactly reflected here.

So 360 subtract 172, is 188.

So these two, these two bs is equal to 188.

So the two bs together is equal to 188.

If I just want to work out the value of one b, what am I going to do? I'm going to divide by two.

If I divide by two, one b is going to be 94 and the other b is going to be 94.

So I have worked out my two unknown angles.

Let's do it using the equation method then.

So here, I've got 132 plus 40 plus two bs because I do not know what two of my bs are.

This b and this b, one b plus one b is two b.

So plus two b.

Again, they must all be equal to the sum of the interior angle in a quadrilateral, which is 360 degrees.

So again, this gives me 172 plus two b is equal to 360.

Okay.

Now, I want the bs to just be on their own.

What am I going to do? I don't want the 172 to be in front of the two b, so what am I going to do? Yes, I will subtract 172 from both sides.

If I subtract 172 from both sides, I will be left with two bs on my left hand side and 188 on my right hand side.

Now, I don't want two bs, I don't want the value of two bs, I just want the value of one b.

So what am I going to do to get rid of that two? I'm going to divide both sides by two.

So over here, I'm going to divide by two and on the right hand side, I'm going to divide by two because whatever I do to the left, I must also do to the right.

So here I'm left with b, and over here, I am left with 94.

And that's how you work out the value for your unknown angle.

Why don't you pause the video for this one and see if you can attempt this yourself? Can you do this one yourself? Pause the video now and then once you had to go, come back and we'll go through it together.

So hopefully you came up with something like four c, I'm putting four at the top 'cause is the largest number of cs.

So we've four cs, three cs, then two cs, then one c over here.

And then total I'm trying to get to is 360, okay.

Now I've got four cs, three cs, two cs and one c, what the number of cs that I have, yes, I have 10 cs and 10 cs is equal to 360.

How do I even work out what one c is? Well, I could draw another bar model.

I know that the total amount of my cs, the total number of my cs, we've 10 is 360.

So, what is one c going to be? I've got one, two, three, four, five, six, seven, eight, nine, 10.

The total amount of all these cs must give me 360, so what would one c be? One c would be 36, because I could either do 360 divided by 10.

And that would give me one of my cs as 36.

Now if one of my cs is 36, I've got 36 here, two slots of c would be what? 72, three slots cs would be 108 and four slots of cs would be 144.

Hmm, does these equal to 360, let's check.

144, you can do this as well by the way, taking 144 plus 108 plus 72 plus 36, does that give me 360? I'm doing some mental math at the moment in my head whilst writing this and the answer is, guess.

So that is how I can work out what my missing angles are.

So over here, I would write 36, over here I would write 108 degrees, don't forget your degrees.

Over here, I would write 144 degrees and for two c, I would write 72 degrees exactly.

So that is how you can use your bar model to work out missing angles.

Let's quickly do the same using equations.

So I've got 10 cs.

I'm just going to add it up together.

So four c plus three c three c, plus two c plus c, again, I know that it is equal to 360 degrees.

So, adding all my cs together, we know that I'll get 10c is equal to 360 degrees, and then c, to get c on its own I'm going to divide both sides by 10, divide this by 10 divide this side by 10, I am left with, just c on this side and 36 on that side.

So I know that one c is 36, and I'm going to do the same thing as I did over here to work out what two c is, three c is and four c is, then add them all together to check that it's equal to 360 degrees.

You now have your independent task, so pause the video now and attempt the questions.

Once you're done, resume and let's go through the answers together.

Okay, let's go for the answers very quickly.

So for this one, I've got 115 plus 120 plus 80, and then I'm going to subtract that from 360.

And I believe all of these added together is 315.

Going to subtract that from 360, I am left with 45 degrees, so check in you work.

For this one, we've got 70 plus 170 plus 70 again, so that takes us to 310, subtract that from 360, b is 50 degrees.

Okay.

So check in your work.

So here, I've got 34 plus 140 plus 29.

That takes us to 203, subtract that from 360, and we have 157 degrees.

And if you've drawn a bar model for each one, that is perfectly fine as well.

This is an example of the one that your bar model should be looking like for each of these questions.

Okay, let's think about this one.

We've got, there we've got a kite and got a line of symmetry telling us that whatever we have on one side, is equal to what we have on the other side.

So I've got 30 here.

So this angle would also be 30 degrees.

I've got 94 here, these two angles are equal in a kite.

So this will also be 94 degrees.

Now, how can I work out, what, that's 94 degrees.

How can I work out what this angle would be, and what this angle would be? Well, I know that the sum of interior angle in a triangle are 180, so I'm going to add these two together and subtract from 180, and that is 56.

So this will be 56, and therefore this will also be 56.

If you add all your angles together, you should get 360 degrees.

Okay, let's look at the second one.

52 so this angle will also be 52 degree because it is an isosceles triangle and base angles in an isosceles triangle is equal to 360.

52 plus 52 is 104, subtract that from 180 and I am left with 76 degrees.

So my angle here is 76 degrees, so I've found one.

Again, I've got 35 here so this will also be 35 degrees for the same reasons, isosceles triangle have two equal angles.

35 plus 35 is 70, take that away from 180, I have 110 degrees.

Next one, we know that the line of symmetry means that whatever is on one side is reflected on the other side.

So I've got 50 degrees here, so this will also, this will also be 50 degrees, right.

Again, this triangle is an isosceles triangle.

So this will also be 50 degrees.

If this is 50 degrees, what must my top angle here be? 80 degrees because 50 plus 50 is 100, and I need 80 to get me to 180.

And all I have to do now is just reflect my angle on the other side.

So this will also be 50 and this will be 80 degrees.

Okay.

Okay, let's move on to this one.

This is testing all your knowledge of angles, so alternate angle, corresponding angle, angles on a straight line, angles around a point.

So let's look at the first one.

I've got 108 and, is asking for this one.

This is going to be 72 degrees.

Why? Well, because angles on a straight line, this is a straight line here, add up to 180 degrees.

I already have 108, and 72 plus 108 gives me 180 degrees.

Another way of thinking about it is 72 and 72 are corresponding angles.

So these two angles are corresponding.

If I know that this is 72, can I use the same knowledge to work out what this angle here would be? Again, because I straight line add up to 180, and I already have 72, therefore this one would be 108 degrees.

Can you see how this 108 is equal to 108 because they are alternate angles.

So I've got 108, I've got 72, I want to work out what this missing part is, right here.

What would it be? Well, I already have 288, what shape does this form? A circle, right? And we know the angles around a point is equal to 360 degrees.

So, 360 takeaway 288 is 72 degrees.

So I've got three, out of my four angles.

How can I work out or what would this angle be essentially? It would be 108 because opposite angles are equal.

Now, let's look at this trapezium right here.

Because we said these two angles are equal in a trapezium, in an isosceles trapezium, so this will also be 72 degrees.

If I know that this is 72 degrees, these two angles must also be equal.

So I need to add these two together.

72 plus 72 would give me 144, 144 subtract from 360, okay, let's think about this isosceles trapezium.

We know that in an isosceles trapezium, the bottom angles are equal and the top angles are equal.

So if my bottom angle here at this corner is 72, this angle here would be 72 degrees.

And the two top angles are the same.

And I know that angles in a, the sum of interior angles in a quadrilateral is 360 degrees.

So I'm going to do 360 subtract from 144, which is 72 plus 72.

That gives me 216, that 216 degrees is for both z and y, but I don't want both z and y, I want y and z.

To get y and z, I'm going to divide 216 by two, and that takes me to 108 degrees.

So y is 108 degrees, and z is 108 degrees as well.

Okay.

Let's think about explore task.

I'm going to read what's on your screen out for you, so pay attention.

A points has been moved to different positions to make triangles within a quadrilateral.

So, from here has been moved to there to make triangles.

So it's got two triangles here in quadrilateral ABCD.

Here we have three triangles, and here we have four triangles.

How can we deduce the sum of the interior angles of the quadrilaterals using other ways of splitting.

So at the start of this lesson, we established that quadrilaterals can always be split into two triangles.

But for some reason, students have split this triangle into three, and this triangle into four.

This task wants you to explain, how you can deduce the sum of interior angle that it would always be 360, using other ways of splitting.

So pause the video now and see if you can attempt this task.

If you're stuck, then carry on watching video and provide you with some more support.

Okay, let's look at the first one.

I've got 180 here, I've got 180 here.

So, this is perfectly fine because 180 plus 180 is 360 degrees.

So that fits into what we deduce at the start that the sum of angles in a quadrilateral, sum of interior angles in a quadrilateral add up to 360 degrees.

What is happening here? So they've drawn three triangles.

What do you see happening here? What's the angle fact we know about this.

What type of, firstly, what type of line is this? Secondly, what is happening around this point? Hint, this is a straight line and angles on a straight line add up to? Based on that hint, can you work out how to get 360 degrees from here and how to get 360 degrees from here? So pause your video now, and attempt this.

When you're ready, press play to resume.

Okay, let's go through this we know that the two angles, two triangles here, triangle one and triangle two, and again, triangles add up, sum of interior angles in a triangle add up to 180, so this is perfectly fine.

Here, there is three times 180.

So 180 in this triangle, 180 in the second triangle and 180 in the third triangle, which gives us 540 degrees.

However, we've proved that the sum of interior angles in a quadrilateral is 360.

So the additional 180, is coming from these three angles here because we know that angles on a straight line add up to 180.

So for example, this student has added the extra 180, they haven't subtracted it from 540 to give them 360.

And here again, we have four triangles, one, two, three, and four.

And four times 180 is 720, for them to get 360 and to prove that the sum of interior angles in a quadrilateral add up to 360, they need to subtract 360 here, because angles around a point is 360 degrees.

So they've got extra 360 degrees.

Take away this extra, 360 degrees from 720, and you will again be left with 360 degrees for the sum of interior angles in a quadrilateral.

We have now reached the end of today's lesson and a very big well done for sticking through and completing it.

And I hope your knowledge on the interior angles in quadrilateral is now excellent.

You can always rewatch this video if you just need to go over some things and I will see you at the next lesson.