Lesson video

Hello, I'm Mrs. Ashley.

And I'm really looking forward to working with you as we work through the lesson today.

So by the end of this lesson, we're gonna be working with line systems where we've got a transversal, some parallel lines, and what we know as supplementary co-interior angles.

So let's make a start.

Here is a keyword slide with many words that you have met before.

I would really suggest that you pause the video now just to refamiliarize yourself with them as we will be making use of them during the lesson.

So our lesson is about co-interior angles and co-interior is one of our new key words.

And we're gonna work up to that definition as we go through the lesson.

So our lesson has got three learning cycles.

The first one we're just gonna focus on what are co-interior angles.

Then we're going to in the second part, think about co-interior angles in parallel lines specifically.

And in the last part we're gonna think about identifying the rule.

So once we've now know what co-interior angles are, can we identify when to use it? So let's make a start at just thinking about co-interior angles.

Here we've got three line segments on the screen.

They're all different lengths.

They've got different gradients in terms of how steep they are and the way they're sort of orientated.

And we can arrange them.

So without twisting or rotating them, just translating them onto the page, we can arrange them like this, or I could do it like this.

Can you see which one's which with the three above and that arrangement? We could do it this way as well.

And finally I could do this way.

But there are more ways, this is not just the only way that I could do it.

But there are four examples there of how those three line segments can be arranged to be a system of line segments.

So here they're again those four that I came up with and one of them is known as the transversal.

So a transversal is a word that you've met before.

But just to recap with that, because it's important for our co-interior angles, the transversal is the line that intersects the other two at different or distinct points.

And so on the first three diagrams that's been made purple.

So it's the line that is intersecting or cutting, traversing is the verb, the other two line segments.

Why have I not coloured the last one? Why have I not indicated on the fourth arrangement where the transversal is? Can you think about that? Okay, and that's because actually they all would need to be coloured purple because they all traverse each other.

If you look, each line intersects the other two.

So on the screen there is a quite simple diagram and we've got lying three line segments once again.

We've got that transversal marked into purple, but it's the line that intersects the other two at two different points.

And all the angles that have been marked there, we would call them interior angles.

Think about why we would call them interior angles.

And then very similar to that, these are angles on the outside of the two lines and we therefore call them exterior angles.

So we've got the same three line segments there in a system of lines where we've got our transversal.

And here we've got four angles marked that are known as exterior angles because they're on the outside of the two lines.

If we go back to the interior, we have got two pairs.

We've got different colours there to indicate the pairs.

So we've got two pairs of co-interior angles.

And those pairs are on the same side of the transversal.

So the way that this diagram is orientated, we could talk about below and above.

So the darker angles, the greyed angles, are both below the transversal but are both inside the two lines.

So they are co-interior angles.

The white angles, the non-shaded angles, are both above the transversal.

So both of them are on the same side of the transversal and are both inside.

So they are co-interior angles.

So here we've got two pairs of co-interior angles.

Likewise we can talk about angles being co-exterior.

And it's just the only differences is that the angles are on the outside of the two lines.

They're both, but they are both on the same side of the transversal.

So co-exterior.

At the top of the screen, we've got examples of co-interior angles.

So just look through them.

We need them to both be on the same side of the transversal.

And then we need them to both be interior angles.

So within the two other line segments.

And then at the bottom half of the slide, we've got examples that, or non-examples of co-interior angles.

So the first one is not co-interior.

They are both interior angles but they're not co-interior because they're on different sides of the transversal.

Actually they are alternate angles.

The second diagram on the bottom half, the non-examples, is co-exterior.

So co-interior they need to be on the inside, but that one is showing you co-exterior.

And then the last one is the same diagram.

That above is marked as co-interior, but these two angles are in different positions and these are not co-interior.

So check on that, which show a pair of co-interior angles? So go through them.

Do they meet the requirements? Are they both on the same side of the transversal and are they both inside or interior angles? Pause the video whilst you're doing that and then when you come back we'll go through each one.

So the first one does show a pair of co-interior angles.

They are both interior and both on the same side of the transversal.

B is actually a diagram showing co-exterior angles.

C does show us a pair of co-interior angles.

And d is not showing us co-interior.

It's actually alternate angles there.

And the reason being that they're both interior but they are on opposite sides of the transversal that cuts the two lines.

So in this diagram, there are many pairs of co-interior angles.

Can you find any? Have a think.

So if we only focus on this part of the diagram, if we sort of ignore, don't rub it out but just disregard it for the moment.

If we only focus on the bit that's now been put into a rectangle.

Well a and n would be co-interior angles.

D and o would be co-interior angles.

And if we keep sort of moving that around the diagram, so just to focus ourself, 'cause we know that when we've got line segments and systems of lines, there can be many transversals.

So we just need to focus ourselves onto the part of the diagram that we really are working with at that point in time.

And then we would have the c and f are co-interior as well as d and e.

And hopefully you can see that they are both inside the lines and on the same side of the transversal.

Continue around the diagram then we can say that e and j are co-interior and h and k are co-interior angles.

And then lastly, that last sort of two vertices joined up, p and i and o and j.

Did you manage to find all of them? Or did you get any of those correct? Hopefully you did.

But this idea of just sort of putting yourself in one part of the diagram and not worrying about all of it at one point.

So really just focus your attention.

So here's to check.

Similar to that, it's a slightly simplified version.

So identify all of the pairs of co-interior angles in this diagram.

Press pause whilst you're doing that.

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

So here are all the pairs that hopefully you have got.

Remember if you've written i and a, that's the same pair as a and i.

It's still correct, so well done.

The order, again, doesn't matter.

So just go through and check that you've managed to find most of those pairs.

Hopefully you've got all of them and hopefully you tried that idea of just sort of isolating elements of the diagram.

So now you're gonna do some independent work on that idea of co-interior angles.

So on question one, you need to mark the angle that would be co-interior with the given angle there on the diagram.

Press pause whilst you have a go at question one and then when you come back we'll go on to question two.

Okay, so here's question two.

You need to add the labels to the diagram by using the given information.

So the given information is the four bullet points and there can only be one label at each angle.

So it might be that you think you know where it is.

And then the next sort of informational clue means it cannot be where you initially put it.

So think about if there is an alternative place.

Press pause whilst you're working through question two.

And then when you press play, we'll go through the answers of question one and question two.

Okay, so here's question one.

You need to mark on the co-interior angle.

So on a, b, and c, there was one position of where it could be.

It needed to be on the same side of the transversal, so identify where that transferal is and then they both need to be inside or interior angles within the other two lines.

On d, there were two positions that you could have marked it.

E, there was only the one position.

And F there was the only one position.

Even though the diagram, the system of lines was quite complicated on e and f, there was only one position for a co-interior with the given angle.

And now here's question two.

So you needed to label the angles.

We don't know what size they are.

We're not worried about what size they are.

There was three clues that used co-interior and then there was the one clue that used co-exterior.

So hopefully you got them in those positions.

So we're now up to the second learning cycle of this lesson where we focus on co-interior angles but this time with parallel lines.

So a specific case of the co-interior angles.

So angles a and b are adjacent at a point on a line.

So they sum to 180 degrees.

We can call them supplementary angles.

So supplementary angles sum to 180 degrees.

And we know that angles on a line at a point sum to 180 degrees.

If we translate the line, the angles are preserved.

So translation as a transformation only moves horizontally and vertically.

And by not rotating or reflecting, no angles are changed.

So the two lines become parallel.

So a translation of the first line and in this case it's moved down, is still the same angles.

What do you notice? So pause the video whilst you just look at that diagram.

Now that we've created a set of parallel lines, angles are preserved, what do you notice? Well the co-interior angles are supplementary.

So we saw that angles a and b sum to 180 degrees.

We can call them supplementary.

And our co-interior angles are also a and b.

So therefore the co-interior angles are supplementary.

Just to check, fill in the blanks.

Pause the video whilst you figure out what the blank should say.

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

So supplementary is the word and which means they sum to 180 degrees.

So co-interior angles on parallel lines are supplementary.

They sum to 180 degrees.

So here we've got another diagram, we've got parallel lines and we know they're parallel.

We're not just assuming it 'cause they kind of look like they are, we know they're parallel because they've got the mathematical notation of a feather, which is the the arrow.

So we've got parallel lines.

We've got two angles there that are on the same side of the transversal line.

And they're both between the parallel lines.

So we know we can describe these as supplementary co-interior angles.

They would be co-interior angles if the lines were not parallel, but because the lines are parallel then they are also supplementary.

And so because they're supplementary, that's a fact we can use to work out the value of x.

'Cause currently we know the size of of one of them, the 68 degrees, the other one is currently unknown.

But because we know there's a relationship between them, which is that they are supplementary, then we can figure out what the size of x is.

So we can set that up like a algebraic equation.

X plus 68, the value of x plus 68 equals 180 degrees.

The sum of those two angles is 180 degrees because they are supplementary.

And therefore, use an inverse, subtracting 68 degrees, we get 112 degrees.

So we now know that x is 112 degrees.

Here's the same setup, but this time we don't have any value to the angle.

But we have a relationship between them, which is that they are supplementary.

We also know that one angle is half the other, or we can think about that the other way round, which is one angle is double the other x and 2x.

But using the fact that they are supplementary, we can set that equation.

The two angles, the sum of those two angles is 180 degrees.

So x plus 2x is 180.

We can simplify, we can collect the like terms on the left-hand side of that equation.

So 3x equals 180, which would mean that 1x dividing both sides by 3, sharing that evenly would be 60.

And that would mean that 2x is 120.

Because we've said that one was half the other or one was double the other.

So a check.

Work out the size of angle h.

Pause the video whilst you do your calculations.

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

So the answer was 62 degrees, h was 62 degrees.

And that was used in the fact that we got parallel lines.

Co-interior angles in parallel lines means they are supplementary.

The two angles, that pair of angles needed to have a sum of 180.

We can think about that in the reverse.

Co-interior angles are only supplementary when in parallel lines.

So these co-interior angles, they're on the same side of the transversal and they're both inside of the lines, they have a sum of 180.

So if they have a sum of 180, then they must be parallel lines.

And we could then mark them with the notation to show that they are parallel.

Parallel lines come in loads of different places around your maths curriculum.

And one of the places is shapes, polygons.

So many shapes have sets of parallel lines.

And Andeep has said he can think of five quadrilaterals that have at least one set of parallel sides.

Pause the video and think about how many of the five that Andeep can think of that you can think of as well.

So you're looking for any quadrilaterals, so a four-sided shape that has got parallel sides.

As long as it's got one set, then that's fine.

Okay, so the square, the regular quadrilateral.

It actually has two sets of parallel sides.

The sort of opposite sides are parallel.

We can see that using the feathers.

We've got one arrow on a pair and we've got two arrows on a different pair.

Then the rectangle.

So the rectangle also has two sets of parallel sides and it's again the opposite sides.

Did you manage to find those two? Then we can think of the parallelogram.

I mean it says parallel in the word.

So parallelogram also two sets of parallel sides.

The rhombus two sets of parallel sides.

And the trapezium has to have at least one set of parallel sites.

So Laura has said, well all of these contain supplementary co-interior angles.

By definition, if we've got co-interior angles and then parallel lines, they are supplementary.

And here we've got transversal, but the transversal is one of the edges of the shape.

And then you've got your parallel sides which are the other two.

If you imagine extending the edges of these.

Extending the line segment slightly more, then you might see that sort of standard parallel lines diagram.

So here we're just gonna focus on the square and the rectangle for the moment.

We know also about squares and rectangles that the angles are all 90 degrees.

I've labelled them as a, b, c, and d.

And we're gonna talk about the pairs of co-interior.

So the supplementary co-interior pairs are a and d.

They would add up to 180.

They do add up to 180.

It's very clear they're both 90 degrees.

But the reason being that we've got co-interior angles in parallel lines.

A and b, so that's because there are two sets of parallel lines that we can sort of look at it from a different direction.

B and c.

And also c and d.

So that's true for the square and the rectangle.

If we focus on the parallelogram and the rhombus, this time it's not so clear to see.

Obviously two 90s added together makes 180 so that's obviously supplementary.

But with a parallelogram and a rhombus, they're not 90 degrees at all times.

So here we've got supplementary, a and d, a and b, b and c, and c and d.

So it's the same four pairs as the rectangle and the square.

Maybe think about why that might be.

And then lastly here we've got a trapezium.

It looks a little bit like an isosceles trapezium where we've got two equal edges, but this is true for all trapeziums. The supplementary co-interior pairs are, a and b, and c and d.

So because this has only got one set of parallel lines marked, then the other pairs are not here.

So on the rectangle, the square, the parallelogram, and the rhombus, there were two sets of parallel lines.

We could think about the transversal being slightly different places.

Whereas here, the edge that goes between the parallel lines is the transversal.

So check on shape and co-interior angles.

So which of the following shapes always have supplementary co-interior angles? Pause the video whilst you read through them and think about it.

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

So the rhombus will always have supplementary co-interior angles.

And the trapezium will always have supplementary co-interior angles.

So here we have a parallelogram.

And we know one of the angle's value is 130 degrees.

So p and 130 degrees are supplementary as they are co-interior.

So we would then be able to calculate that p is 50 degrees.

Q and 130 are supplementary as they are co-interior angles in parallel lines, hence q is also 50 degrees.

R and p are supplementary as they are co-interior angles in parallel lines.

So then r is 130.

And this shows the property that hopefully you are aware of.

The opposite angles in a parallelogram are equal.

P equals q equals 50 degrees and r equals 130 degrees.

So opposite angles in a parallelogram are always equal.

So here's a question for you for a check.

Given that a equals d equals 114 degrees, so that means they are both 114 degrees, work out b and c.

Pause the video whilst you're working out b and c.

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

So a and b are co-interior angles in parallel lines, so they are supplementary.

B would be 180 minus the 114, so 66.

And we can use exactly the same reasoning by at the other end of the trapezium that c and d are co-interior angles in parallel lines so they are supplementary.

And therefore c is 66 degrees.

As angle a equals d and as angle b equals c, this is an isosceles trapezium.

So when those angles are equal, the sort of base angles if you like and the top angles, then it is an isosceles trapezium.

There'd be a line of symmetry which would show that they are also equal.

So some work for you to do now based on co-interior angles in parallel lines.

Question one and two are on the screen.

So have a go at those first.

Press pause whilst you're doing it.

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

Here we've got a system of lines where you've got two sets of parallel lines and a fifth line that is not parallel to any of the others, two marked angles.

And your question is to try and work out all the angles you can on this diagram.

Press pause whilst you work through that question.

And then when you press play, we're gonna go through the answers to questions one, two, and three.

So question one was based on the given information, are there any parallel lines? Part a was no.

So 92 and 92, they're both on the same side of the transversal and they're both interior.

So 92 and 92 are co-interior angles.

But because they do not sum to 180 degrees then they are not parallel lines, they're not supplementary.

On b however, you've got co-interior angles that are supplementary.

So this is parallel lines.

And C, using the fact that it was an isosceles triangle.

So you've got the hash marks there to show that that was an isosceles triangles, means that you've got a 70 and another 70.

70 and 40 sum together is 110.

So 110 is that full obtuse angle.

110 and 70 makes 180, so it is parallel lines.

Work out the marked angles was what you needed to do in question two.

There were parallel lines which tells you there were supplementary co-interior angles to find.

So on the first one you needed to do 180 minus the 76 give you 104, that would give you a pair of angles that sum to 180.

Similar on b, that b is 97.

C was a little bit more complicated because of the way the diagram was sort of given more than anything, but c was 102 degrees.

Question three, you needed to work out all the angles you can on the diagram and actually you should have been able to find all of them.

So this wasn't limited to only using co-interior angles.

You could have also used vertically opposite angles on a straight line sum to 180 degrees.

So have a look, check through your answers.

But co-interior angles was definitely a way to move between the vertices from that 85 and then vertically opposite, et cetera.

So we're now to the last learning cycle of this lesson on co-interior angles and it's called identifying the rule because the more angle facts that you learn then the harder it can become to start to identify which one works where.

So that's the idea of this part.

It's being able to identify when we're gonna use co-interior or supplementary co-interior angles or whether we might use something like corresponding angles instead.

Just a very brief reminder about angles and parallel lines.

So in parallel lines there are equal corresponding angles, equal alternate angles, and now supplementary co-interior angles.

So the first diagram shows you equal corresponding angles.

So to identify corresponding, you're looking for angles on the same side of the transversal and they're in equivalent positions on the vertex.

For equal interior alternate angles, that's shown in the middle diagram.

Interior angles, so they are within the two parallel lines, but they are on opposite sides of the transversal.

And then lastly, the supplementary co-interior angles.

So they are on the same side as the transversal, but they are both inside of the parallel lines.

So we've got equal corresponding in parallel, equal interior alternate angles in parallel lines, and supplementary co-interior angles in parallel lines.

So this diagram has got many vertices labelled.

So we can use the letters to identify which angle we are talking about.

But just to be mindful of that.

So here it says, given that angle GIA equals 73.

So if you go to vertex G and then go from G to I, and then from I to A, we're sort of drawn the angle 73 degrees.

It's the two line segments that meet at the vertex I.

However that could be described, that same angle could be described as FIA.

So starting at vertex F up to vertex I and then to vertex A, that's still the same angle.

It could also be FIC.

Follow from F to I to C, you still make the angle 73 degrees.

So when I'm using the three letters, you may be thinking about it with a different three letters.

So just check we are on the same, we're talking about the same angle.

The other thing to mention, not just about the change of the letters but just might be the direction.

So given that GIA, well I could have said AIG.

If you started at vertex A then went to I then went to G, you still were drawing, you still making that 73 degree angle.

So anyway, given that that is 73 degrees, we can say that angle IGH and GIA are equal interior alternate angles, hence that angle is 73 degrees.

So we've got a set of parallel lines there.

The line segment IG or the line segment IF is the transversal.

So those two angles are on opposite sides of the transversal, but both within inside of the parallel lines.

So they are interior alternate angles and they are equal because of the parallel lines.

Using just the angle GIA, we could also say that FGD is 73 because they are equal corresponding angles.

So once again the letter order might change and you might use different letters.

So for FGD you could use FGE.

They are corresponding, they're on the same side of the transversal in the equivalent position on that vertex.

And then I can also mark onto that diagram, the angle DGI, which again could be IGD, could be EGI, but they all mean in the same angle.

So locating the same angle in the diagram would be 107 degrees because it is supplementary to angle GIA.

So we've got our co-interior, they're both in between the parallel lines and on the same side of the transversal.

So from that one angle, we can find three further angles out on this diagram.

And then if we use the other given fact, I've removed the ones, just the diagram doesn't get too complicated right now, but given the other angle which was DAC, we were told that was 114 degrees.

So we can then say that angle ADG is 114 degrees because it is an interior alternate angle, but it's more importantly it's an equal interior alternate angle because of the parallel lines.

Also we could say that angle FDE is 114 because it is an equal corresponding angle with that given DAC.

And then we could use supplementary co-interior to say the angle EDA is 66 degrees.

So all of those angle facts, the alternate angles, the corresponding angles, and now the co-interior angles gives you different angles around the diagram.

So here's a check you need to fill in the blanks.

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

Okay, so angle LJB is 98 degrees, that was given to you.

And then angle JBE and LJB, so I've just put the pink square there to indicate the line of working and which angles we're discussing are equal something angles, hence angle JBE is 98 degrees.

Well they're on opposite sides of the transversal, but they're both inside the parallel lines.

So I'm hoping you wrote interior alternate angles.

So those angles are interior alternate angles.

Then the second sort of justification line, angle ABC and LJB are equal something angles, hence angle ABC is also 98 degrees.

So again you've got the pink square, it's just slightly moved to show the angles we're talking about.

So they are on the same side of the transversal and they're in the equivalent position on the vertex.

So they are equal corresponding angles.

And then lastly, angle ABJ and LJB are supplementary co-interior angles, hence angle ABJ, it's not as easy as just writing 98 degrees, they're supplementary, they're not equal, is 82 degrees.

So just take a moment to look at this diagram that's on the slide.

There's quite a few line segments.

And we've got two sets of parallel lines, not two pairs because there's multiple lines, but there are two sets of parallel lines.

And we know the difference between them using the feather notation.

So using only the marked angles on the diagram, clearly there are many more angles but only the marked ones, what relationships are there between the angles? So maybe pause the video here and write down, jot down to yourself some relationships between them.

By relationships, I mean like are they equal? How do you know they're equal? Are they supplementary? How do you know they're supplementary? So think about the relationship between the marked ones only.

Had a go, you've had a at least thought about which, what sort of relationships there are.

So Jun says that a and c are equal exterior alternate angles.

So to be equal, we need parallel lines.

We can see there are parallel lines, so that's looking good.

Exterior alternate, so exteriors means they're both on the outside of the two lines that are being traversed which is true.

Alternate, they're on opposite sides as well of the transversal which is true.

And that's those two angles that Jun is discussing.

So Jun is correct, those two are equal.

Their relationship is that they are equal.

Why are they equal? Because they are exterior alternate angles in parallel lines.

Izzy said that b and c are supplementary angles on a line, those two angles.

So yeah, we've got a line, they're next to each other, they're adjacent to each other.

They will sum to 180.

So they are a pair of angles that sum to 180.

So Izzy's saying supplementary angles on a line is absolutely fine as a justification.

Jacob says c and d are equal corresponding angles.

So find c and d equal corresponding angles.

So yes, because they are on the same side of the transversal and they're in the equivalent positions at their own vertices.

And d and e are equal interior alternate angles.

So they're further apart on the diagram, but were still were that set pair of parallel lines.

And then they'll being traversed by another line.

They're on opposite sides of the transversal and they're both within the parallel lines.

So they are interior alternate angles.

And because of the parallel lines, they are equal interior alternate angles.

Aisha says f and g are equal exterior alternate angles and that's true as well.

I wonder how many you manage to get.

So a similar question.

And this time for your check, you need to write down all the pairs of co-interior angles.

So pause the video whilst you identify all the co-interior angles.

Press play and we'll go through the answers.

So a and b are co-interior.

They're on the same side of the transversal and both inside the parallel lines.

C and e, e and g, f and h.

Because of all of the parallel lines on this system, they're actually all supplementary co-interior angles as well.

So if I was to give you the angle of a, the size of a, you could calculate the angle b.

So we're into the last part of you to do some practise.

So question one, you need to complete the statements using this diagram.

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

Okay, here's question two.

You've got a diagram with parallel lines and some line segments between them as well.

So complete this method to find X.

So you've got some blanks to fill in.

Follow through this method to find the value of X.

Press pause whilst you work through that.

And then when you press play, we'll go back through the answers of question one and question two to finish this lesson.

You need to complete the statements for question one.

A and 91 are equal corresponding angles.

B and 91 are supplementary co-interior angles.

And c and 81 are equal alternates angles.

Question two, you needed to complete the method to find X.

X was 97 degrees.

So the value at the end was 97 degrees.

With many questions to do with angles, there might be a different method, a different journey through this diagram, but you needed to complete this method.

Once again with the angles, remember I've written angle PQV, you may have written VQP.

So just don't mark yourself as wrong, just check that we are discussing the same angle.

Angle PQV and QVW are equal interior alternate angles, hence angle PQV is 118 degrees.

Therefore, angle PQS is 83 degrees.

So angle PQV includes the 35.

So by doing a subtraction, you could work out the angle PQS.

Angle PQS and QSR are supplementary co-interior angles.

Therefore 180 degrees minus angle PQS gives you angle QSR.

And hence QSR which is X is 97 degrees.

The end of the lesson on co-interior angles.

So just to summarise, co-interior angles are on the same side of the transversal line and in between the two other lines.

When the co-interior angles are in parallel lines, then they are supplementary, which means they sum to 180 degrees.

Really well done today.

I've really enjoyed working through this lesson with you.