Lesson video

Today is our last lesson in this unit of missing angles and lengths.

Today we'll be calculating missing angles in quadrilaterals.

All you'll need today is a pencil and piece of paper.

Pause the video and grab your things if you haven't done already.

So here's our agenda.

We're going to find unknown angles in quadrilaterals.

You'll start with your quiz to test your knowledge from the previous lesson, then we'll work on representing angles, pictorially before calculating missing angles.

Then you'll do some independent learning, and a final quiz to test your knowledge from the whole of the unit.

So we'll start with our knowledge quiz.

Pause the video and complete the quiz, and then press restart once you're finished.

Great work, now let's get straight on with it.

We're going to be representing our angles pictorially.

So we know that the angles in a quadrilateral add up to 360 degrees.

We can use a bar model to represent this.

So the whole is 360 degrees, and the angles, a, b, c, and d add together to make the whole.

So as an algebraic expression, a plus b, plus c, plus d are equal to 360 degrees.

Now it's your turn.

Pause the video and represent this shape using a bar model and then an algebraic expression.

Angles were marked as c because they were the same.

In this case you actually know that those angles are 90 degrees.

So our whole is 360.

A plus b and c and c are equal to 360 degrees.

Algebraically that's a plus b, plus c, plus c equals 360 degrees.

Or you may have challenged yourself to use BIDMAS, and represent it using brackets.

Two lots of c are added to b and a to get to 360 degrees.

Now think about this shape.

What shape do you think this is, thinking back to our third lesson in the unit, and what do you know about the angles in the parallelogram? So you know that opposite angles are equal.

So we know that this angle is equal to this angle, and this angle is equal to this angle.

Let's go for our pictorial representation.

We know that a plus b plus c plus d are equal to 360 degrees.

We also know that a is equal to c, and b is equal to d.

We know that a plus b are equal to 180 degrees, and c plus d are equal to 180 degrees, and shown as a bar model, we can see that two lots of 180 degrees are equal to 360 degrees.

This will make more sense when we start to introduce some actual numbers.

So let's do that now.

We're going to calculate missing angles in quadrilaterals.

Here we have a quadrilateral.

This is a trapezium because it has one pair of parallel lines.

I can represent it using a bar model.

The angles in a quadrilateral add up to 360 degrees.

360 degrees is my whole.

And I know 45, 72 and 135 degrees are three of my parts.

My unknown part is d.

Algebraically that's 45, plus 72, plus 135, plus d is equal to 360 degrees.

So to find the missing part, we need to rearrange the equation using the inverse.

So 360 degrees takeaway off three known parts gives us our unknown part.

Now, here is where we come into some difficulty, repeated subtraction, if I'm going to set it out like this, like a column subtraction, I'm going to find that I can't subtract five, two, and five from zero.

I need to regroup from the tens.

So one 10 into 10 ones.

Now here I have a problem.

10 takeaway five is five, takeaway two is three.

I can't take away another five.

You can't subtract all of these numbers in one go.

So we've got two options.

And we talked about this yesterday.

Now option one is to subtract in stages, okay? So here is my subtraction.

I can either subtract in stages, so I can do 360 degrees subtract 45 degrees.

And I'll show you my working.

So it's 315.

So then I do this 315 subtract the next known angle, which is 72 degrees.

So now I'm on 243 degrees, and then I subtract my final known angle, 135 degrees.

So then I can see the angle d is 108 degrees.

Now my second option is to use my knowledge of BIDMAS, which can be more efficient in this case.

So I know that what I'm actually doing here is 360 take away 45, 72, and 135.

So I know that I'm doing 360 subtract, and I've added these altogether, 252, and that is equal to 108 degrees.

So you can think about which strategy you prefer, okay? It's up to you, but it's important that you know all the strategies available.

Let's do another one together.

This is a delta.

It has no equal sides.

It has one internal reflex angle.

So I can represent this pictorially.

360 degrees is the whole, and I've got these three known angles.

The unknown angle is p.

Remember that that's going to actually be a reflex angle.

So it's going to be greater than 180 degrees.

I can represent that algebraically.

The three known parts plus the unknown parts give me the whole of 360 degrees.

And then I rearrange it using the inverse to find the value of p.

So I'm doing the whole, 360 subtract 21, 49 and 83, which gives me p.

Remember, you can do repeated subtraction, or you can use BIDMAS.

In repeated subtraction you would do 360 takeaway 21, and then 49 and then 83.

And in BIDMAS you would do 360 takeaway 21, 49, and 83.

And once you've calculated all of those, you will find that p is equal to 207 degrees.

And I can sense check that.

I know that that is a reflex angle.

So that looks like it's possibly right.

And then I can double check by adding the four parts back together, and they will give me 360 degrees.

Now it's your time.

Pause the video and represent this shape pictorially and algebraically.

Then use your preferred strategy for calculating the missing angle.

You may choose repeated subtraction, or you might choose BIDMAS.

So your bar model will have looked like this.

360 degrees is the whole, and here you've three known parts and one unknown.

This one was 90 degrees.

So that's how I've known to put that one in there.

Here's your representation algebraically.

So the four part added together equal 360 degrees, and then rearranged using the inverse, 360 subtract the three known parts gives me the unknown.

You may have done repeated subtraction in which you found that f is equal to 80 degrees, or you may have chosen to use BIDMAS, either way you find that f is equal to 80 degrees.

Let's do one more together.

This is called an isosceles trapezium.

It's a trapezium because it has got one pair of parallel sides.

It's an isosceles because it's got these two sides.

The lengths are equal length, okay.

We know that it has got two pairs of equal angles.

So in this case, if we're only given one angle, we can work out the rest.

We've been told the angle d is 78 degrees.

Therefore we also know that angle c is 78 degrees.

So we can see this pictorially.

We know that 360 is the whole.

We know that a and b are equal size.

And we know two of the angles, when those two added together, give me 156 degrees.

So I know that 360 takeaway the known angles gives me a plus b, and that is 204 degrees.

So I know that a and b are equal.

So I know that I must divide 204 by two to find the value of a and b.

And they will be 102 degrees each.

Now it's time for some independent learning.

Pause the video and complete the task, and click restart once you're finished so that we can go through the answers together.

Okay, for question one, you were asked to calculate the size of angle d.

You will have drawn a pictorial representation because with 360 degrees as the whole.

So algebraically 360 degrees subtract 45, 127 and 81 degrees gives us d, or you may have represented that using BIDMAS.

And I know that those three angles added together gives me 253 degrees.

And therefore I know that angle d is equal to 107 degrees.

For question two, you were given two known angles or two unlabeled angles, but we knew this one was 90 degrees.

So we know that these four added together give us 360 degrees.

So 360 degrees subtract 144, subtract 79, subtract 90 are equal to f.

We know that we could have used BIDMAS to represent this.

360 degrees subtract 144, 79 and 90 gives us f, and I've added those three together to give me 313.

360 degrees subtract 313 degrees is equal to 47 degrees.

So f is equal to 47 degrees.

In question three, we had a parallelogram.

So we know that opposite angles are equal.

So we know that angle d is 110 degrees.

So we know that we have to do 360 takeaway 110, and another 110 will give us b and c, and b and c are equal.

So 360 take away 220 is 140.

So b plus c together gives us 140 degrees.

If I divide 140 by two, then I get the answer to one of those, 140 divided by two is 70 degrees.

So b is 70 degrees, and c the 70 degrees.

In question four, you were asked to calculate the sizes of angles, x, y, and z.

So starting with x, I can see that this is on a straight line with 84 degrees.

I know the angles on a straight line add up to 180 degrees.

So 180 degrees, the whole takeaway the known part, 84 degrees gives me x.

And I know that 180 takeaway 84 is 96.

So x is equal to 96 degrees.

Now I look at the quadrilateral, I've got three known parts and one unknown, which is y.

I know that 360, which is what the angles in a quadrilateral add up to, subtract, and I'm going to use BIDMAS, 138, 96, which we just worked out, and 67 will give me y.

I know that those numbers added together gives me 301, and 360 takeaway 301 is 59 degrees.

So y is 59 degrees.

Now I can work out z, y and z are on a straight line.

So the whole is 180, subtract 59 gives me 121.

Z is 121 degrees.

So that was a three step problem.

Great work if you persevered on that one.

Your final question, you were asked to find four unknown angles, or a four step problems. And I gave you the hint to work them out in alphabetical order.

Let's start with a.

A is on a straight line with 95 degrees.

So a straight line is 180, subtract 95 degrees is equal to 85 degrees.

So a is equal to 85 degrees.

B is part of a full turn with 309 degrees.

I know that full turn is 360 degrees.

So 360 subtract 309 degrees gives me 51 degrees.

So b is equal to 51 degrees.

Now I'll get to c.

I've just focused on my quadrilateral here, where I know that the whole is 360.

So I'm doing 360 subtract, 35, 85 and 51.

So 360 subtract these added together equal to 271.

And that subtraction is 89 degrees.

So c is 89 degrees.

Onto my last one, c and d are on a straight line, which equals to 180 degrees.

And I'm subtracting 89 to find c.

And that is equal to 91 degrees.

So to find d, so d is equal to 91 degrees.

Amazing work, it's time for your final quiz, Pause the video and complete the quiz, and click restart once you're finished.

In our next lesson, we'll be starting a new two week topic on coordinates and shape.

I'll see you then.