Lesson video

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

So during the lesson we're going to demonstrate and prove that the interior angles of a triangle sum to 180 degrees.

On the screen, there are some key words that we'll use during the lesson.

You may want to pause the video now just so that you can re-familiarise yourself.

So the lesson takes two parts.

The first learning cycle is missing angles in parallel lines, and then the second learning cycle is proving the sum of interior angles in any triangle.

So we're going to make a start on that first learning cycle to do with missing angles in parallel lines.

So when we're working with parallel lines, there can be equal corresponding angles, equal alternate angles, and supplementary co-interior angles.

The three diagrams there show you them three types.

The first one there is the equal corresponding angles.

We've got two angles on the same side of the transversal and then they're in equivalent positions at each vertex.

In the middle we've got equal interior alternate angles.

You can have equal exterior alternate angles as well, and they are equal because of the parallel lines.

One angle is on either side of the transversal and in this case the interior angles because they're both between the parallel lines.

And then lastly, we've got an example or a diagram to show supplementary co-interior angles.

So co-interior angles are supplementary if in parallel lines, which in this case they are.

We know that because of the feathers and they are supplementary so they sum to 180 degrees.

So a check here, we've got parallel lines and the question is why are these angles equal? So pause the video and think about the different scenarios in which you find equal angles in parallel lines and which one matters here.

Press play when you're ready to check.

So these are corresponding angles in parallel lines and that is the reason they are equal.

If the two lines were not parallel, then they would be corresponding angles, but they wouldn't be equal.

Another check.

Why are these two angles equal? So again, pause the video and then when you're ready to check your answer, press play.

So they are exterior alternate angles in parallel lines and again, therefore they are equal.

So exterior, because they're both on the outside of the parallel lines and alternate angles have to be on opposite sides of the transversal, which these two 59 degree angles are.

Why are these angles equal? So another check for you.

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

So they are vertically opposite angles, and vertically opposite angles are always equal regardless if there are sets of parallel lines or not.

How do we know that the angle is 121 degrees? Well, adjacent angles at a point on a line sum to 180 degrees.

I'm sure you're very familiar with the fact that angles on a line sum to 180 degrees.

They have to be at the same point.

Okay, so angles that have a common vertex will, if there is a line, then they will sum to 180 degrees.

But that's not the only justification of why that is 121 degrees.

You could say that it is a supplementary angle with the 59 degrees, as they are co-interior angles.

So the two vertices, we can use the 59 degrees on the bottom vertex.

And the fact that this angle of 121 degrees would be supplementary to it, because they are co-interior angles in parallel lines.

There are other 121 degree angles that we could add to that diagram and they can be found using a variety of different rules and reasons.

So here we can see them.

So equal exterior, alternate angles, supplementary co-interior angles, equal corresponding angles, vertically opposite angles and also about angles on a line summing to 180 degrees.

So when we've got a system with parallel lines, a transversal moving between the different angles, there's often many reasons and justifications as to why and how you can get there.

So using equal alternate angles, equal corresponding angles and supplementary co-interior angles, we can work out missing angles when we've got a system of lines that include parallel lines.

So here we've got the line segment ED and we've got a line segment FH that passes through the point G.

So angle E, D, G.

Remember when you're using the notation for angles or the the naming of angles using three letters that the order doesn't really matter in terms of the direction of the letters doesn't really matter.

So EDG is the same as GDE.

So if we follow the letters, it shows you the line segments that create the angle.

So E to D, that line segment, and then D to G, that line segment.

So angle EDG, and that's that one there on the diagram, is 62 degrees and we know it's 62 degrees as it is an equal alternate angle with the angle DGH, which was given on the diagram.

It's an interior alternate angle.

Angle EGH, which is this obtuse angle here, is 120 and it's the combination of the two given angles.

So if you think about angle being a turn, if we turn from the line segment GH, until we are facing the direction of GE, we will have turned 120 degrees.

We've turned through 62 degrees and then we turn a further 58 degrees.

Angle DEG, which is noted with that pink square is 60 degrees and we can find it to be 60 degrees as its supplementary co-interior angle with EGH, which we know to be 120.

Supplementary means they're going to add up to 180.

So 120 plus the 60 gives us 180 degrees.

Let's work through another one of these, again, we've got some line segments and we are using parallel lines so we've got alternate angles and corresponding angles and co-interior angles to look out for.

So angle TWU.

So follow from vertex T to vertex W to vertex U.

Then we create that angle there is 28 degrees and we know it's 28 degrees because it's an equal angle within an isosceles triangle.

So the one thing I didn't speak about was the hash marks that were on that diagram to indicate that the triangle TWU is isosceles.

By definition an isosceles has got equal angles or at least a pair of equal angles.

The angle WTU would be 124 degrees because it is an alternate angle with the given 124 degrees at angle XWT.

So we can find missing angles using all of these rules that we have for angles in parallel lines and any other sort of properties of shape that we know from our prior knowledge.

So here's a check, another diagram here with parallel lines and June has said that m is 32 using supplementary co-interior angles and there's the working out of Jun's there.

And Jacob has said the m is also 32 and there's a different reasoning and rationale as to why.

So who is correct? Pause the video whilst you read through them, check against the diagram and make a decision.

Press play when you're ready to check that.

So actually they both are.

So in many cases when it comes to angle problems, angle questions, there might be actually more than one way of getting to the correct answer.

So their justifications are valid for their working out.

So you're now going to do a bit of practise working with angles, missing angles in parallel lines.

So question one, there are two parts.

So on question part A, you need to fill in all the missing angles on that diagram.

Given 47 degrees on part B, you need to fill all the missing angles again, but this time it's a different diagram and the given angle is 81 degrees.

So pause the video whilst you are adding all of those angles onto those diagrams and then press play when you're ready for the remaining parts of question one.

Okay, so here's part C, D, and E.

So they're a continuation on question one.

So the diagrams from part A and B have been combined.

Use your previous answers to fill in as many angles as you can.

Part D, what do you notice about the three angles in the triangle and part E, where else can you see these three angles next to each other? So press pause whilst you're working through those three parts and when you come back we will go through the answers to all five of them.

So here are the missing angles that you should have added to your two diagrams from part A and part B.

You were told 47 degrees on the part A diagram, so you could use vertically opposite angles are equal, corresponding angles are equal, alternate angles are equal.

And then you could use angles on a straight line sum to 180 degrees to work out the 133 degrees.

And then in the same way vertically opposite angles are equal, alternate angles are equal, et cetera.

You could have also used supplementary co-interior angles to work out one of the 133s.

And then exactly the same sort of process for B except from the angle that was given was 81 degrees.

So its supplementary value was 99 degrees.

And then for part C, D and E, well part C said these are the two diagrams that sort of been overlaid on top of each other.

They'd been combined.

So we could add on the 47s and the 133s and the 99s.

But we then also needed to sort of recognise when they had overlapped when the 133 had a part of it, which was 81 degrees and that would leave 52 degrees around that top vertex.

And you've got two of them, they're vertically opposite each other.

Then part D says, what do you notice about the three angles in the triangle and the three angles in the triangle sum to 180 degrees.

It then says, where else, part E, where else can you see these three angles next to each other? And those three angles could be seen around the point on the line.

So again, on that top vertex, you could see those three angles that were within the triangle have found themselves sort of next to, adjacent to each other.

So we're now going to go onto the second learning cycle, which is about proving the sum of interior angles in any triangle.

The real basics, a triangle has three vertices, and at each vertex there is an interior angle.

So we've got a diagram of a triangle here.

We've labelled the vertices A, B, and C.

Each interior angle has got a different colour.

And if all three angles were equal, then we would call it an equilateral triangle.

If two of the angles are equal, then we would say it's an isosceles triangle.

And if all the angles are different, then it would be a scalene triangle.

This triangle, and we're going to tear off the corners, so the vertices and arrange them, we can demonstrate the sum of the interior angles.

So we're going to tear off the corners of this triangle.

And you could do this as well.

You could draw yourself any triangle that you wish to and then cut it out very close to its edges and then tear off.

I'm going to demonstrate that now.

So if I tear here that vertex A, that corner of the triangle, just put that to the side.

Tear off B and tear off C.

I'm trying to demonstrate the sum of the interior angle.

So actually, how large and how much area your triangle or my triangle takes up is not the important part here.

So that central part is not needed anymore for my demonstration.

It's just the three vertices and the angles, the interior angles at each one.

And so now taking them and arranging them, I can demonstrate the sum of the interior angles and they can be arranged into a line which shows they sum to 180 degrees.

They were the three vertices or the three interior angles of my triangle that I'd torn off and then arranged so that they lie.

So they sort of all meet at one vertex and they are arranged into a line.

Does this prove that this is true for all triangles? If I've done one, if you do one, if you do another one, if I do another one, have I proven that this is true for all triangles or no? It demonstrates that the sum of the three interior angles are 180 degrees, but it doesn't prove it for all triangles, it only proves it for this particular triangle.

So here's a check, true or false.

Proving something works for one example means it works for all examples.

Pause the video, decide if that's true or false and then when you come back we'll go onto its justification.

Okay, we just saw that that is false.

So now what's the justification for it being false? So A, a proof can contain previously proved facts or B, a proof needs to hold for all cases, and one example does not show it for all.

Again, pause the video, rethink about that and then when you're ready to check, press play.

Okay, so the justification should have been B.

It is true that a proof can contain previously proved facts, but it's not the reason why working for one example shows it for all.

So it it doesn't.

So that's not true.

So here we've got another demonstration of the sum of the interior angles in a triangle is 180 degrees.

So we've already seen one, which is where you've got one single triangle, you tear off the corners and you can arrange the interior angles to be 180 degrees, to be on a line.

And here we've got three copies of the same triangle are placed together.

So you've got a rotation by 180 degrees to be that central one.

And then again, sort of a translation of the original one.

And that middle vertex on the bottom has got A, B and C.

So the three angles within the triangle, the three interior angles are all on aligned.

So again, we've demonstrated that this works.

So each of the three angles are at this point and they are on a line so they sum to 180 degrees.

Here I've got a GeoGebra profile which is a dynamic software tool to identify and look at this for any triangle.

So we've got a triangle here with three, is scalene as we can see from the markings of the angles and it allows me to move and change my triangle as I would like to.

Put it up there.

So five decimal places the size of those angles.

So quite accurate measure of each of those angles.

And you can see the sum is 180 degrees.

So for this one triangle it is 180 degrees.

If I move my triangle, the angles at each of those vertices has changed, but the sum is still 180 degrees.

So that's me showing you for two triangles.

But remember we're trying to get to a proof, we're trying to prove the sum of the interior angles for any triangle is 180 degrees.

So I'm going to remove the sum there and take away the showing the angles.

And we're going to try and set up and look at the three congruent triangles and the way they're arranged to see if we can actually get to a proof.

So the first or the second copy was a rotation by 180 degrees around the midpoint of that edge.

And the feathered line segments means they are parallel and that's because the interior alternate angles are equal.

So if you look, we've got the two, the angles with the two arcs are equal, we know that from the rotation and therefore those two edges must be parallel because if they are equal and alternate then there are parallel lines.

And then that third copy can come from a rotation by 180 degrees around the midpoint of the edge of the second copy.

And once again, this rotation has created a line segment that's parallel to the previous parallel line segments because of the interior alternate angles being equal.

And as it shares a point with one of them, they form a straight line.

So angles on a straight line about a point sum to 180 degrees.

Again, if you focus on that sort of middle vertex, you've got the one arc, the two arcs, and the three arcs.

So each of the three angles within the triangle is there at that vertex.

And we know it's a line because those two line segments are parallel and meet at a point.

So now we're going to look at angles in parallel lines and again go through a proof.

So if vertex A is on a line segment OD, which is parallel to BC, so that's what the diagram shows us.

We've got the angle OAB marked with is in purple and with three arcs.

The angle BAC has only one arc and is the sort of aquamarine colour and DAC is the blue with two arcs.

So we know that angle OAB, angle BAC and angle DAC sum to 180 degrees as they are adjacent angles at a point on a line.

So that is something that has been proved previously and we are using it here.

So those three angles are adjacent at a point on a line.

So therefore they sum to 180 degrees.

Angle OAB equals angle ABC.

So we've got the same markings there as they are equal interior alternate angles.

Again that is something we have shown to be true previously, so we can use that here.

Lastly, angle DAC is equal to angle ACB as they are also equal interior alternate angles.

So we're making use of the parallel lines.

And therefore angle ABC plus BAC plus angle ACB is 180 degrees.

If we know that the purple, the three arcs, the greeny aquamarine, the one arc angle and the blue, the two arc angles sums to 180 because when they are adjacent to each other on a line based onto 180, then if the triangle can be shown to have those three angles, then they must also sum to 180 degrees.

Aisha just wants to let you know that those three dots in a triangular formation is a shorthand notation that means therefore.

So our proof has gone through and then says "Therefore, the three angles in the triangle sum to 180 degrees." So for this check, I'd like you to complete the blanks in this proof.

You've got the diagram there so that you can reference which angles it is discussing.

So press pause whilst you're filling in the blanks.

And then when you're ready to come back and check, press play.

So firstly you should have said 180 degrees.

So again, that is talking about the top three angles, the angle OAB, the angle BAC, and the angle DAC.

They would sum to 180 degrees because they're adjacent.

Then the next line says the angle OAB and angle ABC are as they are equal interior alternate angles.

So the line segment AB is the transversal here.

And so you've got an angle on either side and they are both within the parallel lines.

Angle DAC is equal to which angle, because of interior alternate angles? Well hopefully you went for ACB.

If you wrote BCA, that is equivalent.

So that is correct as well.

And therefore, remember Aisha told us what that symbol means.

Therefore angle A, B, C plus what plus ACB is 180 degrees? Well it should be the angle in the centre, BAC.

Again, if you wrote CAB, that is the same angle.

So we've seen a couple of proofs now of that the sum of interior angles of any triangle is 180 degrees.

And so we're just going to look at another one here to show that this triangle in the centre of this set of lines, and in this case there are three pairs of parallel lines, would also show us that the sum of those three angles, interior angles of that triangle are 180 degrees.

So on this GeoGebra file, we've got that diagram set up to be dynamic.

So we can move the vertices and see that we have always got the trio of angles adjacent on a line and we know that angles that are adjacent at a point on a line, some to 180.

And those three angles are also the interior angles of the triangle.

So regardless of where the vertices are, the angles will change size, but their some will always be 180 degrees.

So hopefully you saw there that although the angles were changing size at each point you still had a one arc, two arc and a three arc angle.

And within the triangle the interior angles were one arc, the two arc, and the three arc angles as well.

We've seen it dynamically, we're seeing it move, we understand that that's the case.

But let's try and work out why using justifications.

So angle A, angle B and angle C are adjacent angles and aligned.

So we know that A plus B plus C, the sum of those three angles is 180 degrees.

Angle E and angle F are adjacent angles on a line.

So they would also be, D plus E plus F would also add up to 180.

And finally, angle G, angle H and angle I are adjacent.

So here I've got the labels to be different, they're marked differently.

We've got the same colour situation and we're going to try and justify why they are correctly coloured using reasons.

So angle A is an interior alternate angle with H.

So we've got a set of parallel lines, they're the ones with the three feathers and then we've got a transversal line.

And so A and H are equal, 'cause they are interior alternate angles.

But it's also true to say that H and F are an interior alternate angle.

And that's because again, you've got a set of parallel lines and that's the one with the one feather.

And then you've got a transversal.

So if A equals H and H equals F, then it's also true to say that A equals F or A equals H equals F.

And we know that A plus B plus C is 180 degrees because they are on that line.

So we can substitute A for H because we know that A and H are equal.

And similarly you could substitute the A for the F because you know that A and F are equal.

So all three of those statements about summing to 180 are true.

Continuum of a similar theme, but this time we're looking at angle B.

Angle B and angle D are interior alternate angles with each other.

They are equal.

And angle B is also an interior alternate angle with angle I.

So because B is equal to D and B is equal to I, then that means that D and I are equal.

So again, we can substitute, we have the H plus B plus C from the previous slide that because A equals H, then H plus B plus C equals 180.

Then we can swap the B for a D and we can also swap the D for an I because they are all equal.

So we're just writing lots of sums to 180.

And then finally, if we look at C, angle C, that is an interior alternate angle with E and angle E is an interior alternate angle with G, and more importantly they're equal because of the parallel lines.

So C is the same size as E is the same size as G.

So from before, H plus B plus C equals 180.

So H plus B plus E would be 180.

And those three interior angles in the triangle are labelled H, B, and E.

So we now know that if A plus B plus C equals 180 and D plus E plus F equals 180 and G plus H plus I equals 180.

And because of all of the equal alternate angles that are on this diagram, then H plus B plus E equals 180.

The interior angles add up to 180.

Well done if you followed through with that, that was quite complicated.

Lots of letters being said.

But using alternate angles, if you had just started with A, B, C labelled and then marked on the other angles that you know, using alternate angles, you'd gradually build up to the idea that the interior angles sum to 180 degrees because they involve the same angles, A, B, and C.

So to finish this lesson, you're onto your last task.

So first of all, finish the sentence.

The interior angles of a triangle sum to 180 degrees, which is the same as what? So what other mathematical facts sum to 180 degrees? Part two or question two, can a triangle have these three angles, 34 degrees, 62 degrees and 84 degrees? So pause the video and press play when you're ready for the remaining question.

Question three, using corresponding angles and vertically opposite angles, prove that the sum of interior angles of any triangle is 180 degrees.

So pause the video whilst you write down your proof.

And then when you're ready to check your answers to question one, two, and three, press play and we will go through them.

Okay, so question one, you needed to finish the sentence.

So adjacent angles on a line sum to 180.

Co-interior angles in parallel lines sum to 180, they're supplementary.

A half turn is 180 degrees.

Two right angles is 180 degrees.

You may have come up with some other ones, but they were the ones that I was thinking of.

Question two, can a triangle have these three angles? Well, if the three angles sum to 180 degrees, then it is a triangle.

So adding them up, summing them, we get 180.

So therefore, yes, the answer is yes.

You can have a triangle with those three angles, it would be scaling, 'cause they are all different, but it is a triangle.

And then question three.

So using corresponding angles and vertically opposite angles, prove that the sum of interior angles of any triangle is 180 degrees.

With this proof, the top four statements need to be said, but the order is not very important.

It just might be the way you approach the question.

So check that you've got those four statements and you finish, you conclude your proof to say, "Well therefore I've shown that these are equal.

I know that there are ones above are 180, so this is 180." So the top four statements, the order is not that important, but your conclusion of the proof does need to be at the end.

So here we've got angle BDC plus angle ADB plus angle ADG is 180 degrees.

So those are those given marked angles on the diagram as they are adjacent angles on a line.

Then using corresponding angles, and we know that they are equal corresponding angles because of the parallel lines.

So angle BDC corresponds with angle DFE.

You might have said BFE.

And also, angle ADG corresponds with DEF.

So those two angles or those two statements about equal corresponding angles did need to appear.

Whether you did one before the other doesn't matter.

And then angle ADB, which was given, is vertically opposite angle FDE.

So you know that they are equal because they are vertically opposite.

The whole point of the proof.

We've taken the three given angles that you know have a relationship is of that they sum to 180 into interior angles of that triangle that therefore must also have the same relationship of summing to 180 degrees.

So to summarise this lesson, the sum of interior angles in a triangle is 180 degrees.

So regardless of the type of triangle, whether it's scalene, isosceles, equilateral, whether it has a right angle, whether it has an obtuse angle, it will still sum to 180 degrees.

A proof shows that it holds for all triangles.

Whereas a demonstration only shows for a particular triangle.

So we saw some demonstrations, but when we also went through and you did one for yourself, a proof to show that this is true for all triangles.

Well done today and I look forward to working with you again in the future.