Lesson video

Hello, everyone, and welcome to this maths lesson.

I am Mr. Gratton, and it is a pleasure for you to join me today in a lesson where we'll cover the introduction to what a transversal is and how transversals are used in the creation of groups of corresponding angles.

We will look in more detail throughout the whole lesson at what a transversal and corresponding angles are.

But very briefly, a transversal is a straight line, ray, or line segment that intersects through two or more of the linear paths at distinct points.

On the other hand, a pair of corresponding angles are angles that exist in the matching corner of their respective intersections on those same transversals.

So, first up, let's have a look at all the basics of what a transversal and corresponding angles are.

Here we have two line segments.

They can be of any length or orientation.

They may be parallel or perpendicular to each other, but this is not a requirement.

A transversal is a line, line segment, or ray that intersects through these two lines, like so.

Transversal means to traverse, to travel, or lie across other things.

And so, in a mathematical context, a transversal line means a line that lies across a group of other lines.

Transversals can have any length, direction, or orientation.

It will still be classified as a transversal as long as it passes through or intersects at least two other lines, like so.

And I say, "At least," because a transversal can intersect more than two lines, such as here.

This transversal intersects four lines.

Furthermore, a group of lines can also have multiple transversals, like this, four lines and two transversals.

Or maybe it's two lines with four transversals.

This interpretation is equally mathematically correct.

And further than this, sometimes every line segment in a group is a transversal to all of the other lines in that group, like so.

This is a transversal.

That's a transversal.

Oh, and that's also a transversal, so all three lines in this group are transversals to each other.

And here are some real-world images.

Can you spot where some of the transversal line segments are? Here are a pair of lines and the transversal that passes through them.

Furthermore, for the right-hand image, here are a pair of lines and a transversal, the starting line that passes through those two lines.

Okay, for this check, pause here to give the letter which identifies the transversal.

And the answer is A.

A is the transversal as it is the only line in this group that intersects through at least two, in this case, three, other lines.

Similar for this check, pause here to give the letter which identifies the transversal.

And I said letter, but actually, it's all three of them.

All three of these lines are transversals because each of them intersects through the others in the group.

Each of these diagrams shows a pair of line segments and a transversal.

Each pair of angles now marked on each diagram is a pair of corresponding angles.

Pause here to think of or discuss the possible relationship in the locations of each pair of corresponding angles.

The focus is on the transversal.

Sometimes, especially at the start of this topic, it is very good practise to identify and physically highlight with a highlighter pen or pencil, if possible, the transversal, so that you can see it and understand it more easily and use it practically throughout the lesson.

Now, still focusing on the transversal, imagine you were an insect walking across that transversal, like so.

Whenever the insect gets to an intersection between the transversal and another line, there are four angles around that point of intersection, above right, below right, below left, and above left.

This marked angle is on the right of the transversal and above or in front of this insect.

Make a note of that location.

And after making a note of the location of that angle, imagine the insect continued to move across the transversal from the previous intersection to the next one.

In which general direction would this other marked angle be? Ah, it's actually also to the right of the transversal and above or in front of that insect.

This is the exact same location as at the previous intersection.

A pair of corresponding angles must be on the same side of the transversal.

In this case, both of these marked angles are on the right-hand side of the transversal.

They also must be on two different vertices or intersections on that transversal, and they must also be in the exact same matching corner of those respective intersections.

On each of these diagrams are three pairs of line segments and three transversals with corresponding angles labelled on them.

Pause here to think about or discuss how you would describe the location of these corresponding angles labelled onto these three transversals using similar descriptive language as, for example, "Left of the transversal and above the intersecting line." Pause now to think or discuss.

For this first, leftmost diagram, the corresponding angles are to the right of the transversal and above each intersecting line.

For the second one, we have below the transversal and right of each intersecting line.

And for the third one, the corresponding angles are to the left of the transversal and above each intersecting line.

Okay, here's the next check.

Pause here to both identify the angle corresponding to angle x and choose the correct statements that justify the choice that you made.

There will be more than one correct statement that combines to make a full justification.

And angle x is corresponding to angle D.

This is because they are both on the same side of the transversal.

In this case, they're both to the left of the transversal and the angles are both above their respective lines that intersect with the transversal itself.

And for this check, pause here to identify which of these diagrams shows a pair of corresponding angles.

And the answers are A and C.

For A, both angles are above the transversal and to the right of each line of intersection.

For C, both angles are below the transversal and to the left of each line of intersection.

And finally, pause here to select the diagrams that show a pair of corresponding angles.

After five seconds, I'll open up a hint where I highlight the transversals.

And here are the transversals.

Pause again if you need some more time.

And the only two diagrams that show a pair of corresponding angles are A and D.

Some line systems, or groups of lines, may have more than two lines that are being intersected by the transversal.

Therefore, there may be a group of angles which are all corresponding to each other, like so.

This angle is to the left of the transversal and below the intersection line.

So is this angle, and so is this angle.

These three angles are therefore all corresponding to each other because they are all to the left of the transversal and below the line of intersection.

For the final check of this cycle, here's a beastly diagram.

Pause here to think through and find which angles are corresponding to angle x.

There will be more than one correct answer.

And the three answers are B, F, and G.

They are all above the transversal and to the right of the line of intersection.

Okay, here's the practise for Task A.

Question 1 asks you to grab a highlighter pen or pencil and highlight and label the transversal on each of these six diagrams. Pause now to do so.

And for Question 2, by first highlighting the transversal, identify which of these six diagrams show a pair of corresponding angles that's labelled.

Pause now to do this question.

And for Question 3, one of each pair of corresponding angles is labelled on these five diagrams. Label on the second angle in the correct location to create a pair of corresponding angles on each diagram.

For part F, create your own pair or trio of corresponding angles.

And on to the answers.

Pause here to match the transversals on screen to the ones that you have highlighted.

And for Question 2, A, E and F are all diagrams which show a pair of corresponding angles.

And for Question 3, here are the locations of where the corresponding angles should have gone.

For Part F, there are infinitely many possible solutions.

This is just one of them.

We've looked at transversals that intersect through lines in any two orientations.

But what about a pair of lines that are parallel to each other? Will the transversal reveal any interesting properties? Well, let's find out.

Here is a pair of parallel lines and a transversal that lies across them.

I measure this angle to be 53 degrees.

This can be done using a protractor.

I can then use angle facts to calculate the other three angles around this intersection.

I know that angles about a point on a straight line sum to 180 degrees.

So, 180 take away 53 is 127 degrees at this angle.

Vertically opposite angles are equal, so this angle is also 127 degrees.

And, well, this angle can be found using either angles on a straight line, angles around a point, or vertically opposite angles.

This angle has many different ways of being calculated.

Regardless, it's value will be calculated as 53 degrees.

Imagine the top parallel line were to be translated down the transversal without any rotations or change in orientation happening.

What would happen to the angles around that intersection? Let's have a look.

We can see that the translation preserves all four of the angles at that point of intersection between the line and the transversal.

This only works because the two lines are parallel to each other.

The angles will always change if the two lines were not parallel to each other and we did the same process.

On this diagram, which pair of angles are corresponding to each other? Well, this 53 degrees is corresponding to that 53 degrees.

They're corresponding 'cause they're both to the right of the transversal and above their respective line of intersection.

These two corresponding angles are both 127 degrees.

These two angles are corresponding, and so are these.

Notice how each pair of corresponding angles is equal.

This is always true as long as the transversal intersects through two parallel lines.

In fact, this is a key property of corresponding angles.

Corresponding angles on a transversal across a pair of parallel lines are always equal.

So, in this diagram, I've labelled the two corresponding angles both x because they are both the same sides.

Furthermore, a set of corresponding angles on a transversal across a group of three or more parallel lines are also all equal to each other.

So, these three corresponding angles are all the same y degrees.

On the other hand, corresponding angles on a transversal across a pair of lines that are not parallel to each other are never equal.

So, angle a in this diagram is not equal in size to angle z.

For this check, pause here to select the correct statement that completes the sentence.

For a pair of corresponding angles to be equal, they must.

For a pair of corresponding angles to be equal, they must lie on a pair of parallel lines.

B is the correct answer.

A is totally incorrect.

Corresponding angles must lie on the same side of the transversal.

C is also incorrect.

For corresponding angles to be equal, the lines with which the transversal intersects must be parallel to each other.

And whilst D, those two values were correct values for one example, corresponding angles can take on any value, depending on the lines and transversal we are dealing with.

For this quick check, pause here to find out which of these corresponding angles are equal.

Angles b and c are equal.

This is because the line that angle a is on is not clearly parallel to the other two lines.

And for this check, are line segments AB and CD parallel? Choose the correct answer from A, B, and C and select the statement that correctly justifies your answer from options 1, 2, and 3.

Pause now to do this.

No, they're not parallel, because the corresponding angles are not equal to each other.

If we know the lines are parallel, for example, it is either stated or further markers are labelled to show parallelity, then we can determine the size of one corresponding angle if we know the value for the other.

For example, find the value of angle x.

Well, x equals 66.

This is because both x degrees and 66 degrees are corresponding angles to each other across a pair of parallel lines.

Furthermore, we can use corresponding angles to find the value of other angles across a set of lines of which only some are parallel to each other.

In instances like this, it is yet again very important to highlight the transversal, like so, to visualise what is corresponding and what is not.

In this example, this is the relevant transversal even though there may be more than one in the diagram, because the angle we want to find the value of, angle y, is at the intersection of this transversal.

After we are confident that we've highlighted the relevant transversal, we can highlight any of the parallel lines in this diagram, and therefore identify the corresponding angles between the parallel lines and the relevant transversal.

Therefore, we can conclude that y is 123 degrees.

This is because they are both corresponding angles that lie on the two intersections between the transversal and the parallel lines.

Well, what about this 63 degrees? Actually, the 63 degrees just isn't useful.

It does not factor into any of our calculations when trying to find the size of angle y, as it is not on a line parallel to the line that Y is adjacent to.

For this check, pause here to write down an angle that finds the value of angle y.

Also, select the statement that correctly justifies your answer.

Here is the transversal and these are the two parallel lines that intersect that transversal.

Therefore, y is 61 degrees because both angles are corresponding across a pair of parallel lines.

Note that the angles labelled 78 is adjacent to a line not parallel to the other two, and so we do not need to use it in any of our calculations to find y.

Here's the next practise task.

For Question 1, write down the size of the angles marked with a letter.

Pause now to do this.

And for Question 2, using your understanding of notation and of corresponding angles, which of these six diagrams show a pair of parallel lines? Pause now to do this.

Okay, great work.

Angle a is 62 degrees whilst angle b is 136 degrees.

Both angles c and d are 99 degrees.

This is because we have three parallel lines in this diagram.

Angle e is 111 degrees and angle f is 82 degrees.

And for Question 2, B, C, D, and F all show a pair of parallel lines.

For diagram F, we know that this is a pair of parallel lines because the rightmost line is 150 degrees, whereas the leftmost line has two angles which sum to 150 degrees as well.

You may have seen a handful of angle facts across several different lessons, from vertically opposite angles to angles that meet at a point on a straight line, and now to corresponding angles on a transversal across a pair of parallel lines.

Let's look at some opportunities to compare when to use which and to use more than one with one diagram.

Sometimes it is necessary to use other angle facts alongside corresponding angles in order to calculate the size of a missing angle.

When going through multiple steps, it is important to say the rules that you are using and the calculations involved.

This is important in communicating what each calculation represents and justifying why it is the correct calculation to use in any specific situation.

For example, in this diagram, angles about a point on a straight line sum to 180 degrees.

Therefore, 180 take away the 64 degrees equals 116 degrees.

This 116 degrees is corresponding to the w degrees on a pair of parallel lines.

Therefore, w is also 116 degrees.

This is both a correct set of calculations and a set of angle facts which justify how the 64 degrees was used to get to the w equals 116 degrees as well.

Pause here to think of or discuss which other angle facts can be used alongside corresponding angles on this diagram.

The first angle fact that we can use is that vertically opposite angles are equal, and so this angle is also 127 degrees.

Furthermore, this 127 degrees is now corresponding to the x degrees on a pair of parallel lines, and therefore, x is also 127 degrees as well.

Once again, pause here to think of or discuss which other angle facts can be used alongside corresponding angles in this diagram.

The first angle fact is that base angles of an isosceles triangle are equal, and so this angle is also 67 degrees.

This new 67 degrees is corresponding to y on a pair of parallel lines, and therefore, y is also 67 degrees.

Here's the next check.

Write down the numerical value of angle y.

Pause here to do so and select the appropriate angle fact that justifies your answer y equals 55 degrees.

The two options for correct justifications are supplementary angles sum to 180 degrees or the angles about a point on a straight line sum to 180 degrees.

And for this final check, pause here to select the correct statement which justifies whether line segments AB and CD are parallel or not.

And the correct answer is C, no, because the corresponding angles of 55 degrees and 45 degrees are not equal to each other.

And for Question 1 of the final set of practise questions, find the value of the angles labelled with a letter either completing the sentences or creating your own sentences as justifications for each step of your process.

Pause now to fill in all of the details.

And here's the final question.

This diagram looks really intimidating, but all it requires to complete it are the angle facts that you've covered so far during this lesson.

By, first of all, finding corresponding angles to 119 and 65 degrees, find the size of all other angles labelled with an angle marker.

Pause now to give as many angles a go as you can.

And great work on your communication of the answers to Question 1.

For part A, a is 156 degrees because vertically opposite angles are always equal.

For part B, b equals 78 degrees because angles about a point on a straight line always sum to 180 degrees.

You could have also used that supplementary angles sum to 180 degrees.

And for part C, c equals 134 degrees because angles about a point on a straight line sum to 180 degrees, and d is also 134 degrees because c is corresponding to d and corresponding angles on a pair of parallel lines are equal.

And for Question 2, well done if you got any of these angles correctly labelled.

Well done for all the great work and effort that you've put into this lesson about transversals and angles.

We have covered a lot of unfamiliar information, such as what a transversal is, a line that intersects through at least two other lines.

We've also looked at corresponding angles, which are a pair or groups of angles which lie in matching corners of their respective intersection on a transversal.

Corresponding angles are always equal, so long as the transversal intersects a pair or group of parallel lines.

Corresponding angles are also never equal if the intersections occur with groups of non-parallel lines.

And well done in the first step in combining a wealth of knowledge of angle facts with your understanding of corresponding angles.

That's it for me for today.

I appreciate all of your time and effort.

So, until next time, take care, and have an amazing rest of your day.