Lesson video

Hello, I'm Mrs. Lashley and I'm really looking forward to working with you as we work through the lesson today.

Okay, so in the lesson we're going to use the fact that the interior angles of a triangle always sum to 180 degrees to find missing angles in a variety of problems. Some words we're going to be using during today's lesson is about interior angles and also the sum.

You may want to pause the video so that you can read them and check that your understanding is okay before we move on.

So the lesson's got two learning cycles.

The first one is to find missing angles in triangles, and the second is again to find missing angles but in more complex problems. So we're going to make a start with that first one about finding missing angles in triangles.

So I'm sure you are familiar with the fact that the sum of the interior angles in any triangle is 180 degrees and we are going to really make use of that fact during this whole lesson.

Lucas has suggested that we need to have two out of the three angles in a triangle in order to find out the third.

Alex doesn't think we need to have any numerical value for any of the angles in a triangle and Andeep has agreed with both, but saying in slightly different situations.

What's your opinion on that? How can we make use of the sum of interior angles in a question about finding missing angles in a triangle? Is there a certain amount of information you do need to be able to answer it? Or is Alex right that we don't even need to know the size of any angle and you can still work it out? So Andeep has said, if we have no given information, the only thing we could do would be to use a protractor and measure the angles.

So that would be measuring them rather than calculating them.

So in this case, all three angles are missing.

You need to be told that this has been drawn accurately so that you could use a protractor to measure them.

Andeep goes on to say that, like Lucas said, if you don't have the numerical value, the numerical size of two of the angles, then we can calculate the third by using the sum.

So here we can write an equation because the sum is to add.

So X, the unknown angle plus 126 degrees plus 45 has to equal 180 degrees because this is a triangle and we know that the sum of the interior angles is 180 degrees.

We can simplify, we can add together the two numerical values within our equation and solve it to see that X would be nine degrees.

So for this to be a triangle, for the sum to be 180 degrees with those two given angles, then the third angle must be nine degrees.

So check for you.

Calculate the missing angle in this triangle.

So pause the video whilst you're working through that and then when then when you're ready to check whether your a value is correct, press play.

So I'm hoping you started by writing an equation, simplifying it and then solving it.

So a needs to be 27 degrees in order for this to be a triangle.

So Andeep has now shown us another situation that agrees with Alex that we have no numerical size of any of the angles, but in this case we know the ratio between them.

So we've got three angles marked on our diagram, A, B, and C.

But we're also told that the ratio between them, so A to B to C is one to 14 to five.

And Andeep is saying that with that information you actually can work out the three angles.

So here we've got a bar model and the full bar is 180 degrees.

That is the sum of our triangle and we've split it into an amount of parts.

How many parts? Well we've split it into the total parts that our ratio is given.

So one, add 14, add five.

So there are 20 equal parts within that ratio.

If I now colour code them, this bit represents A, this bit represents B and this represents C.

So if we know that these 20 equal parts has a total or a sum of 180, then we can work out what one equal part is worth in degrees and that would just come from a division because they are equal parts.

So each equal part is nine.

So all of those boxes can have a nine, nine degrees.

And now we can see that A would be nine degrees because there is only one part 14 of those parts makes the angle B.

So 14 times nine is 126 degrees.

And lastly we've got five times nine, which is 45 degrees.

So we started with no numerical value of any of our angles in our triangle, but we did have a relationship between them.

So by knowing the relationship between them, actually we can figure them all out.

Andeep goes on to say, or if we know the type of triangle, then that can sometimes add information.

So this is an isosceles triangle and we know that because of the hash marks that two of the edges have been marked to be equal in length.

So we also know if it's an isosceles that not only does it have two equal edges, but it's also got two equal angles.

So we can, they are the base angles of the triangle and so here we're going to call them M.

We could have chose any letter there, but we've gone for M.

So now we've only got one angle.

So we've worked out when we had two angles, we've worked out when we had none but a relationship between them.

But now we've only got the value of one of them.

However, we do know that the other two are equal in size.

So we can set up an equation, two M, the sum of the two equal angles plus the 140.

That top angle or the third angle must equal 180.

If this is a triangle, then it must sum to 180 and then we can just solve this linear equation.

So we know that two M plus 140 is 180, then two M must equal 40.

How can that sum be 180? And then we can divide them both by two.

We can split them evenly because we know that they are equal and therefore M is 20 degrees.

So Andeep has shown us that actually sometimes with working out missing angles in triangles, you might get quite a limited amount of information given to you, but there might be other information that really helps you along the way.

So this time we're going to do an explanation and a check.

I'm going to do an example model for an example and then you'll do a similar one yourself.

So here, in an isosceles triangle, the largest angle is three times the size of the smaller equal angles.

Work out the size of the largest angle.

I'm going to use a bar model here again.

So because we've got sort of a relationship, right? We had the ratio previously, we have got a relationship between these angles.

We know that the largest angle is three times the size of the smaller angles.

So the total is 180 degrees because it is a triangle.

But why do I have five boxes? Why have I split this bar model into five equal parts? Well that's because the largest angle is three times the size of the smaller equal angles.

So if the smaller equal angle is one box, then the largest is three boxes three times.

So here would be our largest angle and then equal smallest angles.

Because there are five, because it's split into five equal parts, then I can work out that each of those equal boxes is 36 degrees, each of our equal parts.

The largest angle is three times and the question is for the size of the largest angle.

So I need to do 36 times by three, which is 108 degrees.

So this would be an is isosceles triangle with 108 degrees and a 36 and a 36 degrees.

The largest angle is 108 degrees.

So a similar idea for you.

So in an isosceles triangle, the largest angle is four times the size of the smaller equal angles.

Work out the size of the largest angle.

So pause the video whilst you're working through that and then when you're ready to check press play.

So you can do this with a bar model as well, but yours, the difference from yours to mine is that you would need six equal parts because your largest one was four times the size.

So if you thought about this algebraically, if you called your smallest angle x, then the largest one would be four x four times larger.

And then if you've got another x, because you've got two equal angles by it being an isosceles then how many x's do you have in total? You have six x's.

So that's why there are six parts to the bar model.

The largest angle would take four of those equal parts.

And then you've got your two equal angles, which are the smaller ones.

Dividing by six, sharing that 180 degrees into six equal parts means that each equal part is 30 degrees and therefore your largest angle was 120 degrees.

So this isosceles would have 120 degrees as the sort of top angle and the two base angles that are equal would each be 30 degrees.

So now let's think about angles in triangles, but also with parallel lines.

So work out the size of the angle marked A.

So we've got a set of parallel lines, we've got two given angles and then we've got this unknown angle, this missing angle that's marked as A.

So adjacent angles on a line at a point sum to 180 degrees, we're probably feeling quite familiar with that.

So the angles on a line at a point sum to 180 degrees.

So I can use that fact to work out that that angle is 73 degrees so that it totals of 107 to make 180.

And then because of the parallel lines, I can say, well this angle is also 73 degrees because it is an equal corresponding angle because we've now got parallel lines which allows equal corresponding angles.

And then finally the interior angles of a triangle sum to 180 degrees.

I've got two angles within the triangle so I can add them up and find out how much left to get to 180 and A would therefore be 35 degrees.

You can probably find a different way around that diagram.

I went for working out the 73 and then equal corresponding angles, but you may have had an alternative sort of method to get to the same answer.

So we've got another system of lines here where again with parallel lines, a few more additional line segments as well.

Laura thinks that the triangle HBE, so the triangle that's sort of created by all those line segments is isosceles.

So if we know the interior angles of a triangle, then the type of triangle can be determined.

So if you know that two of the angles are equal, then you'd say yes, it's isosceles.

If all of them were 60 degrees, all of them were equal, then you'd say yes, it's an equilateral.

Working through our diagram if using all angle facts that we know, let's see if Laura is correct.

Angle HBE is 52 degrees as it is vertically opposite angle ABC and angle ABC was given to be 52 degrees.

Angle IHE is 132 degrees because it corresponds with angle GEF and they would be equal because of the parallel lines.

Angle EHB is 48 degrees because it is adjacent to angle IHE and angles on a line at a point sum to 180 degrees.

We can then calculate that angle HEB is 80 degrees as the interior angles of a triangle sum to 180.

So was Laura correct? No, actually it was a scalene triangle.

So looking here we've got three different angles so therefore it's a scalene triangle.

So a check.

What type of triangle is ZQX? So triangle ZQX, what type of triangle is it? Pause the video whilst you work out other angles to help you with your decision.

Press play when you're ready to check that.

I'm hoping you came up with scalene.

So you can get to 62 degrees and 77 degrees in a whole variety of ways using the two pairs of parallel lines.

But if you work out those other interior angles of the triangle, you can see that they are, none of them are equal, so therefore it is scalene.

So on the screen we've got what looks to be a triangle and Sofia says "what's wrong with this diagram?" So pause the video and just have a ponder and think about that.

What's wrong with this diagram? The issue is actually that the angles just do not sum to 180 degrees so it's impossible, it cannot be a triangle.

But at a first glance, it looks to fit the definition of an isosceles, it's got the hash marks to indicate that it's an isosceles and the two angles at either end of those equal edges are marked to be the same.

The issue is that if we sum the angles, they don't add up to 180 degrees.

Sofia wants to know if this is an impossible triangle.

So again, I'd like you just to pause and come up with your own answer to Sofia before I go through it.

So note we've got markings of an isosceles triangle.

The base angles are equal in their given numerical value and if you sum them, it equals 180 degrees so it is an isosceles triangle.

Don't be put off by the decimal value of those angles.

We use a protractor which is to one degree accuracy, but you can measure angles to a much higher degree of accuracy and therefore decimals or fractional parts would be included.

So a check, which of the following are possible triangles? Pause the video whilst you make your decisions.

When you're ready to check them, press play.

So A and C are possible triangles, they sum to 180 degrees whereas the central one, triangle B does not sum to 180 degrees.

So onto your first task of the lesson.

So question one, you've got five parts and you need to decide which of the following are not possible triangles.

Pause the video whilst you're doing that and then when you're ready for question two, press play.

Okay, so on question two there are four triangles and you need to work out the missing angles in each of the triangles that has enough information.

So they may not all have given you enough information for you to be able to work anything out.

Press pause whilst you have a go at those questions and then when you're ready, come back and we'll move on to question three.

Okay, so question three we've got parallel lines.

So this time you're going to also be using equal turner angles, equal corresponding angles.

You could use supplementary co-interior angles, vertically opposite angles, angles on a straight line summing to 180 degrees.

So you've got every angle fact that you know that you can use in these diagrams. But remember you also know that the interior angles of a triangle sum to 180 degrees.

Press pause whilst you have a go at those two questions.

And then when you're ready to check the answers to all of the questions you've done so far, press play.

So question one, you need to tell which of the following are not possible triangles.

Well, it was B, C, and D.

So B, if you add them up, if you sum them, they don't total 180, neither does C and neither does D.

Question two, you needed to work out the missing angles in each of the triangles.

So on the first one, A would be 52 degrees.

You were given two of the three angles.

So it was just a case of adding those together and finding the difference from 180 degrees.

B was 38 degrees and you can calculate that by the fact that it was an isosceles triangle.

So although there was only one angle given, you actually knew two because they were the equal angles within the isosceles triangle.

C, D and E, it was a ratio.

So again, you could have used the bar model concept, work out what one equal part would be and in this case it was 7.

5 degrees and then multiply them up.

The last one F though, there wasn't enough information, you wasn't told explicitly that it was isosceles, you wasn't told any other information.

So you just only had one angle and that wasn't enough to work out F.

Question three, again, as I said, you could have used any facts you know about angles as long as you worked in the circumstance that you were trying to use them.

So the missing angle on the first one, x was 83 degrees and y was 85 degrees.

Okay, so we're now on to the second learning cycle, which is again to work out missing angles but in more complex problems. So let's have a go at doing some complex problems. Izzy has said to us that in an isosceles triangle, one of the angles is 45 degrees.

What does that mean the other two angles are? Sam suggests that that means that they're both 67.

5 degrees to make it an isosceles triangle.

Lucas has suggested actually one could be 45 and the other could be 90 and it would still be an isosceles.

So we've got two different answers to Izzy's question one from Sam where they have made the base angles equal and they'd be 67.

5 degrees using the 45 as the other angle.

And Lucas using the 45 degrees as one of the equal angles and working out the other.

Izzy has put another question out there now.

A triangle has the angles, 30 degrees and 60 degrees.

What type of triangle is it? Sam says, well "the third angle would be 90 degrees so it is a scalene triangle" they're all different so it's a scalene triangle.

Lucas says "the third angle would be 90 degrees so it is a right-angled triangle." Actually both Sam and Lucas are correct.

It would be a right-angled scalene triangle.

So a check for you.

Which of these could not be an isosceles triangle? So pause the video whilst you're working through those and then press play to check your answer.

So firstly, B could not be an isosceles triangle.

It's got two equal angles but they're both obtuse and so automatically by being both obtuse it sums to over 180 degrees.

If it's a triangle, the sum has to equal 180 degrees so that one totals more than 180 degrees therefore those three angles could not create a triangle but also C, because although it's sums to 180 degrees it isn't isosceles.

You need to have at least a pair of equal angles for it to be an isosceles.

So this part of the lesson is all about more complex problems. So one way of making things slightly more complex is when we don't have the value of any of the angles, but instead we've got sort of a connection between them and we've written this in algebra.

So what do you know about this triangle? Well you know it's a scalene triangle because the three algebraic expressions all include one x and one of them is just x, the other is x add five.

So it is five degrees more than x and the other one is 25 degrees more than x.

So therefore they are all different in size.

Can we calculate the size of each angle? What do you think about that? Is algebra enough? Can we get to the value, the size if we've only been given algebra? Well yes, they sum to 180 degrees, which allows us to write an equation.

We've got angle one, angle two, and angle three.

The three angles, the three interior angles of the triangle must sum to 180 degrees.

We can collect the like terms, we've got an x in each of those, that's three x and we can add together five and 25.

So three lots of x plus 30 equals 180, which means that three x must be 150 and therefore X is 50.

So our three angles are 50, 55 and 75 and we can see that's a scalene triangle.

They are all different in size.

So what do we know about this triangle? Well this one is an isosceles triangle.

We don't have the markings, we don't have the hash marks to indicate it's an isosceles.

We do have two equal angles.

Can we calculate the size of each angle? Yes, they sum to 180 degrees.

So if we write this one as an equation, angle one, add angle two, add angle three, the three interior angles sum together equals 180 simplifies to five x equals 180 and therefore we can share 180 into five equal parts and we get the x is 36 and the three angles of the triangle would be 36, 36 and 108.

It's an isosceles triangle, it's got two equal angles.

So a check for you, calculate the size of each angle.

Pause the video whilst you're working that out and then when you're ready to check it, press play.

I'm hoping you started with an equation.

It may have been that you've simplified already.

Okay, so you may have written this line first.

That's absolutely fine.

So the sum of the three angles equals 180 and then that means that 10 x equals 180 and therefore x is 18.

So the size of each angle, we're not finished yet, that is the size of one of the angles that happens to be marked x, but you haven't worked out the size of all three angles.

So the three angles are 18, 36 and 126 degrees.

So make sure that you did go all the way to giving me the size of each angle.

We're moving up the complexity of diagrams now, there's not just a single triangle, it's a little bit more complicated.

So this is a parallelogram with an isosceles triangle.

We've got the hash marks there, we can see this as an isosceles triangle.

What angles can be calculated using the given information? So we've got a 70 degree angle, which is one of the interior angles of the parallelogram.

And then we've got a 12 degree angle which is not an interior angle of the parallelogram nor is it an interior angle of the isosceles triangle.

So firstly we can use the knowledge of opposite angles in a parallelogram being equal we can mark that that opposite angle is also 70 degrees.

And then the base angles are equal in an isosceles triangle so I can calculate them because the top angle or the third angle in the isosceles is the combined total of 70 and 12.

The interior angles in a triangle, sometimes 180.

And where the edge of the parallelogram goes across the triangle, we actually find two further triangles.

So we can work out the 61 degrees using the 49, the 70 to get the 61 and we can work out the 119 using the 12 and the 49.

So there's two further triangles within this diagram and we know that the angles would sum to 180.

Equal interior alternate angles.

And I've not got the feather markings on the diagram, but a parallelogram has got sets of parallel sides.

So we can use equal interior alternate angles.

Supplementary co-interior angles can also be used because of parallel lines and the parallelogram properties.

So that would be 110 and vertically opposite angles are equal.

So I'd know that that one's 119, I could have got that 119 by using adjacent angles on a line sum to 180.

So actually all of the other angles could have been found.

So what angles can be calculated using the given information on this diagram? So what do we actually have going on? Well we've got parallel lines 'cause of the feather marks.

Well one angle of 38 degrees and we've got a right angle as well.

We've got hash marks to indicate there's an isosceles triangle.

So equal interior alternate angles means that the 38 degrees that was given can also be written in a different position.

The interior angles in a triangle sum to 180.

So this is using the right angle triangle and the sum and the fact that angles in a triangle sum to 180 degrees.

The two base angles are equal in an isosceles triangle.

So 180, subtract the 38 and then dividing it both by two will give you 71 degrees, which means that the angle that's not got any other line cutting through it is 71 degrees and the other side is 71.

But you know that part of it is 52 degrees so you can calculate that the other part is 19 degrees.

Adjacent angles at a point on a line, sum to 180 degrees.

So we can also add that 142 degrees right at the sort of top of the diagram.

So often there is a lot of information that you can get out of the diagram from a very small amount that was given to you and it's sometimes to do with properties of the triangle or the quadrilateral and then lots of it will be any other facts you already know.

So here is a check and you need to complete the reason.

So look at the diagram, consider the diagram and then complete the reason.

Pause the video so that you can have a real think about that and then press play when you want to check the answer.

This missing word was isosceles.

So A is 58 degrees because the base angles of a isosceles triangle are equal and the interior angles sum to 180 degrees.

So there was a lot going on there.

There's actually two isosceles triangles within that diagram and you just needed to focus on the one with the 64 degrees that was given and that was one of the base angles.

Here we're looking at a problem where Andeep has got some triangular tiles and he realises that they can be arranged like this on the right.

So sort of like a windmill fashion.

What does that mean for the angle r? So the angle, the triangle as far as we can tell is scalene that you've got an angle p and angle q and an angle r and he realised that it's that with using more than one tile, one of these tiles and arranging it in that fashion that it fits like that.

So what does it mean for angle r? Maybe pause the video and have a think about that before I go through it.

Well it means that six fit around that middle point.

So each angle must be 60 degrees.

So if there are six fitting nicely tessellating together around that point, then each of the angle r is 60 degrees.

So we now I know a little bit more about that tile that he has found.

They also can be arranged like this so that the p is in the centre.

So what does that mean for angle p? Well this time five of the triangles fit around a point and that would mean that p is 72 degrees because 360 is the angles around a point.

And if each of them is p, then 360 divided by five is 72.

So our tile has now got a 72 degree angle and a 60 degree angle.

Can they fit with all the q angles in the middle? Well we can calculate q by using the sum of interior angles in a triangle.

So 180 minus the 60 minus the 72 leaves us with 48 degrees.

So we know that q has a size of 48 degrees, 360 divided by 48 gives you 7.

5.

So no, we can't fit them all around the point with no overlap or gap, there would be a gap and you can see that in the diagram.

So they're not going to fit round the point exactly.

So what angle is left that gap that has been created? Well we've got seven lots of q and we know that Q is 48 degrees, so that gives us 336 degrees.

We also know that angles around a point add up to 360.

So the remainings angle is 24 degrees.

So a check thinking about the tiles that Andeep found or similar tiles to what Andeep found.

Which of these triangular tiles could be arranged around a point? Pause the video whilst you're going through that and then when you want to come back and check, press play.

So A would fit and you've got a 90 degrees and 90 degrees is a factor of 360 and you've also got TWO 45 degrees.

You can work out the missing angle, it'd be 45 degrees and 45 is a factor of 360.

So it would fit exactly around using all three of the vertices.

B would also fit, 100 is not a factor of 360, so you couldn't arrange them around that Vertex 35 is not a factor of 360 degrees, so it wouldn't be arranged around that vertex.

But the missing angle, if you calculate the missing angle, it is 45 degrees and we know that 45 is a factor so that around that vertex you could get the tiles to arrange like a windmill.

And C.

C fits as well.

60 degrees is a factor.

So you could arrange it around that 60 degrees.

52 is not a factor and the missing angle would be 68 degrees, which is also not a factor of 360, but that one angle, that one vertex with the angle 60 degrees would arrange exactly around a point.

So onto the task element of the lesson.

For question one, you need to work out the missing angles in each triangle.

So pause the video whilst you're working through those questions and then press play for question two.

Question two, you need to work out all the angles that you can on this diagram.

So think about alternate angles that are equal, corresponding angles that are equal, isosceles triangles, the properties of that, angles around a point, angles on a straight line, angles in a triangle sum to 180 degrees and work out all the angles that you possibly can on that diagram.

Press pause whilst you're working through that and then when you're ready for question three, press play.

So question three is a question like Andeeps triangular tiles.

So triangle A, B, C has been rotated about angle A to form this windmill pattern.

It could also be rotated about angle B or angle C and it would fit a whole number of rotations without overlap or gap.

How many other triangles can you find where all the angles could be the centre in a windmill pattern? So each of the three angles within the triangle need to be able to fit around to make a windmill pattern.

Pause the video whilst you're working through that question.

And then when you press play, we're going to go through the answers to all the questions.

Scalene is the solution for A, so you were told algebraic expressions for the three angles and so we can set that up as an equation and the three angles would be 20 degrees, 60 degrees and 100 degrees.

It would be a scalene triangle.

For part B, once again, you need to set yourself up an equation, collect like terms, simplify it and solve it.

You work out that L is 30 and which would mean that your three angles are 50 degrees, 60 degrees and 70 degrees.

Again, another scalene triangle, but you should have known that already using the expressions.

And part C, this one was an isosceles triangle so you needed to recognise it was isosceles and that there would be two equal angles of two m plus five.

So set up the equation, we can expand the brackets, simplify it down, and m was 32, which then means that the three angles are 32 degrees and then we have two 74 degrees because it's isosceles.

Question two, work out all the angles that you can on this diagram.

And so it was making use of isosceles triangles or turner angles being equal.

You could have also added on the reflex angles at each of the vertices as well if you wanted to.

So well done if you did.

So here's just a reminder of the question for question three.

So we were looking for three angles of any triangle that could be put together in this sort of windmill fashion around a point.

So here are the answers for the integer solutions.

So an equilateral triangle because they would all be 60 degrees then they're all factors of 360 and therefore that would work at every vertex.

A triangle that is 90, 72 and 18, that sums to 180.

So it's a triangle and all three of them independently are factors of 360, which it means there would be no overlap or gap using those tiles.

If you think you found a different set of angles, potentially you used decimals rather than just integers, then you can check that they work by doing two things.

Firstly, check they sum to 180 degrees because if you found three angles that do not sum to 180 degrees, then you haven't found a triangular tile.

So that's the first check.

And secondly, if you divide 360 by each of your angles, you should get an integer value each time.

If it's not an integer then it wouldn't complete the windmill pattern.

You would have a gap between them or you'd have to overlap.

Okay, so if you do feel like you found any additional sets of angles, then check them using those two ways.

So to summarise today's lesson, which was about using the fact that the interior angles of a triangle sum to 180, so the sum of interior angles of any triangle is 180 and missing angles can be calculated using the sum and other triangle properties.

Really well done today and I really hope you've enjoyed it and I look forward to working with you again in the future.