Lesson video

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

This lesson is about using everything you've already learned about angles in parallel lines, but bringing them all together.

So more complicated diagrams, but as long as there are pairs of parallel lines and a line that traverses them, then we can make use of all of the knowledge that we already have.

Using some key mathematical vocabulary through the lesson.

And they are words that you will have already met, but you may wish to pause the video now so that you can just re-familiarise yourself with all those definitions, the diagrams there just to remind yourself.

So our lesson has got two learning cycles and the very first learning cycle is about identifying any related angles in parallel lines.

So being able, because we now have a lot of information to our fingertips, being able to identify them.

The second learning cycle is then to find unknown angles by using some of those relationships that we've identified.

So let's make a start on that first learning cycle.

Here we've got a set of parallel lines which have been traversed by another line.

The feather notation indicates that those lines are parallel to each other, the arrow.

If we had more sets of parallel lines, then we would have other lines that would have two arrows or feathers on them to indicate they are different sets but are also parallel to each other.

And then the line that crosses or intersects the other two lines is the transversal.

You know that two lines are parallel if.

Read through your answers that are given here and make your decision on what's the end of that statement.

Press pause whilst you're doing that and then when you're ready to check it, just press play.

So I'm really hoping you went for C.

So it's the feathers or the arrows that indicate that lines have a relationship will be in parallel.

If there's the hash marks that's indicates the lengths are equal.

And nothing in maths is about the way that they just look, or if they just look that way, we can't assume.

So we need to make, even if they do look to be parallel, like they would continue and never meet, that the distance between them would stay the same.

We cannot assume that we must have either some way of being able to distinctly prove it or the feather notation.

So you'll have learned quite a lot of information regarding angles in parallel lines.

But we're just going to go back through, just to recap them quickly so that we're in a good place to continue with this lesson.

Jun has reminded us that corresponding angles will always be on the same side of the transversal.

And then on top of that, they need to both be either both above the line or both below the line.

So there are going to be in, they're in equivalent positions at the vertex.

They're also a different vertices.

And so B and F would be a pair of corresponding angles.

And because of the system of lines being parallel, then we also know they are equal in size.

So corresponding angles in parallel lines tells us that those angles are equal to each other.

So Jun just identified the angle, B and F were a pair of corresponding angles.

I'd like you for this check to write down a pair, a different pair of corresponding angles from this diagram.

Pause the video whilst you are finding a different pair of corresponding angles, and then when you're ready to check press play.

There wasn't just one answer.

So there are some options here.

So you may have chosen A and E.

They're both on the same side of the transversal and they're both above the other lines.

C and G, they're both on the same side of the transversal, they're both below.

D and H, same side of the transversal, both below.

And the fourth pair was the one that Jun had already given the B and the F, they were both on the same transversal and above the two parallel lines.

So once again, we've got that same diagram where we've got para parallel lines and a transversal.

And Jun is reminding us of another sort of angle theorem.

So alternate angles are a pair of angles both between or both outside the two lines and they are on opposite sides of the transversal and therefore B and H would be an example of alternate angles.

And more importantly, they are exterior alternate angles because they are both on the outside of the two lines.

Because it is a system with parallel lines, then they are equal.

So B and H are equal exterior alternate angles.

A similar check to the one you just had.

So write down a different pair of alternate angles from the diagram.

You can't use B and H because Jun has already used it.

Press pause whilst you are writing down a pair.

You might want to find more than one pair and then press play when you want to check.

So once again, there was more than one correct answer here.

So you could have chosen A and G.

Note, they are on opposite sides of the transversal and they are both on the outside of the two lines.

One is above and one is below.

So they are exterior alternate angles.

C and E, they're on opposite sides of the transversal and they're both within the parallel lines.

C and E are interior alternate angles.

D and F, so they are interior alternate angles.

And then the fourth pair was the one that Jun had identified previously, the B and the H.

They're all in parallel lines, so these pairs are all equal.

And then the last of reminder from Jun is about co-interior angles.

So co-interior angles are on the same side of the transversal and they are in between the two other lines and that's why they are interior.

So co-interior, they're on the same side of the transversal and both in between the two lines.

So C and F would be an example of co-interior angles, because they are parallel lines, then they are supplementary as well.

So co-interior angles are supplementary when in parallel lines.

So there is only one answer to this one.

So write down the other pair of co-interior angles from the diagram.

Press pause whilst you are identifying that pair of co-interior angles and then press play when you're ready to check it.

There was only one answer as I said, D and E.

So we were looking for the other pair on the other side of the transversal 'cause it still had to be in between the parallel lines.

So we just used the word supplementary with our co-interior angles in parallel lines and supplementary angles sum to 180 degrees.

Jun has said that A and B are supplementary as they are adjacent.

So if you look at them on the diagram, they are adjacent to each other on a line.

And we all know that angles on a line add up to 180 degrees or sum to 180 degrees.

So we can say that that pair of angles are supplementary.

So if supplementary angles, sum to 180 degrees, which of the pairs are not supplementary? So it's a check for you.

So go through the pairs that are given here and decide which ones do sum to 180.

But you are looking for the ones that do not.

Pause the video whilst you're working through those.

And then when you press play, we'll go through the answer.

So C was the only none supplementary pair there.

A and B were supplementary 'cause they were adjacent on a line, F and E were supplementary 'cause they were adjacent on a line.

H and E they were adjacent on a line, A and D adjacent on A line, D and C adjacent on a line, C and F, supplementary co-interior, A and H are actually co-exterior.

Co-exterior also are supplementary when there are parallel lines.

But G and E are vertically opposite.

So they might be supplementary if they were both 90, but they have to be equal because they're vertically opposite.

So C, G and E not supplementary.

So systems of lines can get fairly complicated.

So here we've got a diagram where we've got sort of six distinct intersections and we've got a set of parallel lines, a trio of parallel lines and then we've got a pair of parallel lines.

We can see the feather marks there.

So within this system there are only equal angles or supplementary angles.

And if we restrict our focus to a certain section, it's a good way to make a start.

So if you sort of ignore parts of the diagram, 'cause sometimes it can be quite overwhelming that they're quite complicated and you might just think, "How do I start? There's just so much to do." But if you just restrict your focus, it's a good way in.

So what I mean by that is if I just focus on the angles within my rectangle and I know that we've got supplementary or equal angles, well I can use vertically opposite angles are equal just to mark those there.

I can use the alternate and the correspondent angles to get from one vertex to another.

I can use vertically opposite angles are equal, but on the single vertex.

I can use the idea of supplementary adjacent angles on a line and supplementary co-interior angles.

If we now change our focus, so we've added to our diagram, if we now change our focus we can identify equal and supplementary angles within this area of the diagram.

So once again, I'd get A, B, B and A here.

And I can again do corresponding angles to get my A on the left hand side onto the right hand vertex because they are on the same side of the transversal and in sort of equivalent positions.

And then I can use supplementary angles and I can use vertically opposite angles, et cetera.

And you can work through the whole system in this same fashion of just focusing on two vertices at one point.

And we end up with a diagram that looks like this.

So because we've got intersecting parallel lines, sets of parallel lines obviously, 'cause otherwise they wouldn't intersect.

All of these vertices have actually got all the same angles marked on them.

So here's a check.

Is this diagram correctly labelled? If you work through each part, if you restrict your focus, would you be labelling them like this? Pause the video whilst you work through that and then when you're ready to check, press play.

So no, this has been labelled as if the other two lines, the other set of lines were parallel.

So it's not the same diagram as the one we just worked through.

We've got a trio of parallel lines, but the other two are not parallel to each other.

So we don't have the same idea of supplementary and equal angles.

So we're now at the task part of the first learning cycle.

So on this diagram I would like you to colour if you've got lots of different colours or label all of the equal angles, 'cause there's going to be different angles that are equal to each other.

It might be that you, if you only have the one colour, you could do some that are sort of with lines.

Someone can be fully coloured.

Find a way to make sure that you know which ones you are saying are equal to each other.

Pause the video whilst you do that.

And then when you come back, we'll go to question two.

The screen has changed, it is question two, but this time you are labelling or colouring once again all the co-interior angles on the diagram.

So again, either use a labelling system, so the same letters or different colours to identify the pairs of co-interior angles on this diagram.

Press pauses whilst you get on with that question.

And then when you're ready, come back for question three.

So here is question three.

It's got three parts within it.

So using this diagram, write down the pairs of equal alternate angles, for part B equal corresponding angles and for part C supplementary co-interior angles.

Pause the video whilst you are writing down all those pairs.

Use the letters on the diagram for the marked angles.

And then when you press play, we'll go through the questions one, two, and three.

Question one, you needed to colour or label the equal angles.

So you needed two labels or two colours.

So we've got A's and B's on my diagrams. You could have used any letters or notation to indicate the ones that were equal.

So we can sort of move between the Bs from one parallel line to the next by using corresponding angles, we can use vertically opposite angles are equal, we can use alternate angles are equal.

And then we can then also see from this diagram the supplementary kind of relationships as well that A and B are supplementary because of angles on a straight line, but also because co-interior angles are supplementary in parallel lines.

Okay, so question two, I found it quite difficult to show all of the pairs of co-interior angles on one diagram.

So I've split them into six diagrams. The two that I've labelled A, I've got one as A1 and one as A2.

So that indicates a pair of co-interior angles.

And then likewise we've got B1 and B2 and that's another pair of co-interior angles.

They're not equal necessarily, they could be if the transversal was perpendicular to the parallel lines, but otherwise they're just going to be supplementary.

Okay, so on question three you need to write down the pairs, just to note the order of your pair doesn't matter.

So on part A I start with A and G.

If you've written G and A, they are still a pair of equal alternate angles.

So the equal alternate angles from this diagram are A and G, B and H, C and E, D and F, M and L, N and K, O and I, P and J.

Just to note, B and P are interior alternate angles.

However, they're not equal because they're not on parallel lines.

So B and P are alternate angles, but they're not equal alternate angles.

And there are other examples of that as well, like F and K.

Okay, so onto part B, equal corresponding angles.

So we've got A and E, B and F, C and G, D and H, M and I, N and J, O and L, P and K.

Once again there are other corresponding angles on this diagram, but they're not equal.

Finally we've got part C, which is where you are looking for supplementary co-interior angles.

And they were D and E, C and F, P and I, and finally O and J.

So we're about to start the second learning cycle of this lesson, which is about finding unknown angles.

So we're still dealing with parallel lines and all of those facts that we've just recovered to make use to find unknown angles.

So in the first learning cycle we were identifying relationships, and so this is where we're going to really make use of that.

So by looking at that diagram, I can see there is a relationship between A and 67 because I know they are corresponding angles.

But not only do I know they're corresponding angles, I know that they are equal because they are in a parallel lines.

So A is equal to 67 degrees.

I therefore know that the two angles are both 67 degrees.

Angle B is going to also be 67 degrees.

I could use the A from the previous slide and say, oh these are vertically opposite and therefore equal.

But if I hadn't have done A first, if I'd started with B, these two angles are exterior alternate angles in parallel lines, so therefore they are equal.

They're on opposite sides of the transversal and they're both outside of the two lines.

And so I know that B is also 67 degrees.

In this diagram, I can say that these are supplementary angles and they are supplementary co-interior angles.

The only reason they're supplementary is because of the parallel lines.

They're co interior because they're on the same side of the transversal and both inside of the two lines.

By being supplementary, I need to then do a calculation, 180 subtracts 67 leaves 113 degrees.

So D would be 113 degrees.

So on this diagram, which of these are correct? Press pause, read through them, decide which one you think is the correct answer.

And then when you're ready to check press play.

So the answer was C, it is an equal angle.

They are 92, A would be 92 degrees, but not for alternate angles.

They are on the same side of the transversal.

So if they're on the same side of the transversal, we really should be only thinking about corresponding or co-interior.

They are corresponding angles because they're in the same equivalent position on their own vertex.

At times there may not be a direct link between the angle that you've been given and the angle you actually want to know.

And this is where adding other angles to the diagram can be really helpful to get to the missing angle that you're trying to work out.

So maybe you could pause and think about what other angles could you quickly add to this diagram from that given information.

(indistinct) this is that X corresponds to that other position, which would also be X.

And then we can now see that 71 and X are supplementary because they're adjacent on a line.

And then we just do a quick subtraction, 180 subtracts 71 gives us 109 degrees.

And 109 degrees in that position means that X is also 109 degrees in the original position because they are equal corresponding angles.

What if our diagram's slightly more complicated? So here we've got work out the value of Y.

So Y is the angle, EHD, that's one way I could describe it or DHE.

Okay, we've got two other angles given to us.

So Sofia is going to walk us through her way of getting through this diagram, adding other angles to it so that she can get the calculation for Y.

So angle BHD and angle HDC equal 180 degrees because they are supplementary co interior angles.

So hence BHD is 54 degrees.

So she's using the given 126 degrees and noticed a relationship between that angle and angle BHD that they are supplementary.

And then she says, well AE is a line to line segment, so angle AHB and BHD and DHF must sum to 180 degrees.

Angles at the same vertex or at the same point sum to 180 degrees on a line.

And therefore Y is 75 degrees.

So you can sum your other two, 51 and 54.

That makes 105.

And then look to be how do we get from 105 up to the 180? How do we complete the angle? What angle do we need to turn 75 degrees? So here we have another diagram, parallel lines once again so we can think corresponding, we can think alternate, we can think supplementary co-interior and we need to work out M.

We've got two given angles, we've got the 40 degrees and then we've got the symbol for a 90 degrees and there are angles that you could add from this information.

So you could put vertically opposite angle is 40 degrees, vertically opposite angle is M.

We could add that information, we could work out the adjacent angles to the 40 because of angles around a point or because of that angles on a line summing to 180.

So here vertically opposite, pairs of angles are supplementary on a line, but it doesn't really at this point feel like we've made any kind of work towards the value of M, which is seems on the diagram seems very far away from the information we've just added.

So when it comes to diagrams, often a way through a question, a problem question is to add a line, a very well thought about line gives you a lot of information.

So here if we add a parallel line that passes through the vertex at the right angle, it then allows us for using corresponding angles and alternate angles and co-interior angles.

Because we haven't worked with any diagrams where the third vertex in between the parallel lines.

So let's make that vertex be on a parallel line and that would mean that that angle there is 40 degrees, it corresponds to the original given 40 degrees, it's an alternate angle to the vertically opposite 40 degrees.

It's a supplementary co-interior angle with 140 that we worked out.

So we can now sort of translate our 40 to a different position, but we knew it was a 90 degree, we were told that.

Which means that on the other side of that parallel line is 50 degrees because 50 add 40 is 90.

Does that help? Well I can then use interior alternate angles are equal in parallel lines to get my 50 degrees from that position at that vertex.

I've now got 50 degrees at the same vertex with M, and M therefore must be 130 because 50 and M are two angles on a line.

So angles on a line add up to 180 degrees and therefore M is 130 degrees.

So by adding that parallel line on the vertex that passes through the vertex of that other angle that we were given, we then create lots of alternate angles and corresponding angles, which sort of allows us to see angles moving towards the vertex of our unknown.

So a check where would you draw an additional parallel line on this diagram so that it's helpful? Pause the video and then when you're ready to check press play.

I'm hoping you placed a parallel line through the vertex of the other angle between the set that we already had.

And that would be really helpful because we then can see alternate angles, co-interior, supplementary co-interior angles, corresponding angles.

So we're now up to the point where you are going to do some practise about finding unknown angles using alternate angles are equal, corresponding angles are equal and supplementary co-interior angles when there are parallel lines.

So question one, you've got some unknown values to work out.

So work out the size of the marked missing angles, press pause whilst you work through each diagram.

And then when you're ready for question two, press play.

So question two, show that triangle BEF is isosceles.

So my hint to you would be just start by adding angles on the diagram that you know and write down the justifications as to why.

Then think about how you know a triangle is to be isosceles and it's probably that you've already done it really.

So my hint was if you're not sure where to start, start by adding on angles to the diagram that you are confident or correct and think about the justifications.

Press pause once you get through that question.

And then when you press play, come back for question three.

So here we've got question three, it's got two parts to it.

And again, by adding lines, those lines are to be parallel lines.

Work out the missing angle in each diagram.

You could add more than one parallel line if it's useful, if it's helpful.

So press pause whilst you work through those questions and then press play when you're ready to go through the answers to questions one, two, and three.

So I'd go through each diagram and check that you've got the same angles.

The green angles are the sort of working out angles, the ones that I've worked out on my way, the part of the journey towards the one I was trying to work out.

Sometimes you may have done it in a different way, so you may have worked out other angles along the way.

So check your final answer first and then if you've got it wrong or you are not hundred sure how it worked, then maybe go back and look at the sort of green angles as well.

So question two, you needed to show that triangle BEF is isosceles.

So first of all, triangle BEF is the triangle that you can sort of see within them parallel lines.

To be isosceles, it needs to have at least two equal angles and at least two equal edges.

We're dealing with angles here, so we're not going to be able to prove that the edges are equal.

So we are going to have to look to prove that there are a pair of equal angles.

So as I said, you may want to just start by adding angles to the diagram that you know through corresponding angles, alternate angles or co-interior angles.

So I went for angle CBF equals angle, BFE as they are equal interior alternate angles.

So I can add 64 degrees to the diagram.

So I've just used alternate angles there are equal.

And then I've added in the angle BEF by using adjacent angles at a point on a lines need to be 180 degrees.

So I can use 128 degrees, add 52 degrees equals 180.

So I've got the 52 degrees.

So then I can see that angle DEB is a equal interior alternate angle with angle CBE and angle CBE as sort of being split by the edge of the triangle.

We don't know it's been split in half.

It has been, but it's not guaranteed to be split in half.

So don't just assume that.

But we know that 128 is the total of angle CBF and angle FBE.

And therefore FBE is 64 degrees.

And then hence triangle BEF is isosceles because we've got two equal angles within it.

And so I've added the hash marks to show its an isosceles.

The two equal angles are the angles at the base of the equal edges.

So it's those two edges that would be equal.

Question three, there was two parts, adding lines, adding parallel lines at the vertices is where you're going to really help yourself out.

So on the first one I've added two parallel lines and then you were given an angle, but we could split it into its parts using the alternate angles.

And so we've used sort of equal alternate angles working through the diagram to get 76.

And on the second one we've got one parallel line.

I can use, the answer is 120 degrees.

And where did that come from? Well, it came from an alternate angle with the 37 making that part of it.

And then I could have used, there was two ways you could have done this.

We needed to use, we had a reflex angle 263 and angles around a point add up to 360 and a pair of angles that summed to 360.

We could say a conjugate pair.

So we could get the 97 degrees.

I've then used or equal alternate angles to move the 97 degrees or to find another 97 degrees.

But you could have done it using co-interior, supplementary co-interior to get from the 97 to the 83.

And then to find Y, which is the full obtuse angle was the sum of 37 and 83.

So 120 degrees was the answer.

So we've come to the end of the lesson, which was angles on parallel lines traverse by a straight line.

So diagrams containing sets of parallel lines means the alternate, corresponding and co-interior angles can be identified.

So sometimes the start to a problem is just identifying relationships between angles.

Then by using the facts about angles in parallel lines, any missing angles can be calculated.

This sometimes comes from multiple steps.

You might need to work out an angle along the way.

Additional lines or extensions of lines can sometimes be necessary to support finding the missing angles.

So don't be afraid that if you've got a diagram, actually adding or extending a given line might make it a little bit clearer.

Really well done today.

I've enjoyed working through this lesson with you.