Lesson video

Hello, everyone.

Welcome to another Maths lesson.

Thank you for joining me here for today's learning.

I'm Mr. Gratton and we'll be looking at using standard conventional terms and notations for all three linear paths, as well as all the different ways of notating an angle.

The three linear paths we will look at are lines, rays and line segments.

The definitions of these will become clearer as the lesson goes on.

Any two linear paths are parallel if they are the same non-zero distance apart.

You may think that a line is a line, but actually, there's more layers to it than that.

Let's have a look at what I mean by that.

Aisha has been asked to describe these three lines, but she says they're all just lines, so there's not really much to describe.

Pause here if you can spot any relevant differences between the three lines that are on screen.

In everyday language, each of these can be called a line, and that's okay.

Most people will understand what you mean by that.

However, each has a different set of mathematical properties and each one can be used in a different context.

Therefore, each has a different name mathematically to identify them.

Rather than saying they are all lines, we can say that they are part of the linear path family, linear path being the name of the collective group of all three different types of what you see on screen.

Each individual one is either called a line, a line segment, or a ray.

Which one do you think has which name? A line is a linear path that extends forever infinitely in both directions.

Whilst you cannot draw it going on forever, it is treated as extending beyond the limits of what can be drawn, specifically beyond the limits of any points marked on screen, such as this line which extends beyond the two points marked with a cross.

These marked points on the line only show the locations that the line just so happens to pass through.

These points never ever suggest an endpoint to a line as a line extends forever, and therefore doesn't have any endpoints.

A ray is a part of a line that extends forever infinitely, but only in exactly one direction with the other direction being the distinct start point of the ray.

In this example, the left hand point is the start point and the ray does not extend any further in the left direction.

However, the ray simply passes through the right hand point and then extends forever in that right hand direction.

A line segment, the third type of linear path is a part of a line that has endpoints in both directions at the two distinct cut of points at either side of the line segment.

In this example, one of the two endpoints exists as the leftmost point of the line segment, whilst the other endpoint exists as the rightmost point of that line segment.

You may have seen lines when representing linear equations on a graph.

For example, this straight line graph has equation of Y equals 0.

5x plus 4, and the line carries on forever in both directions.

To demonstrate this, this line continues to and beyond the coordinates -400, -196 and 10,000, 5,004 and beyond.

We just simply cannot draw it going that far because the graph paper or the screen just simply isn't big enough for us to do so.

And even if we could, it simply wasn't practical.

You will have seen line segments before when drawing a triangle.

Each of the three sides of the triangle is made from a line segment.

How many line segments are needed to draw a parallelogram? Well, four, because it has four sides and each side is a line segment.

Whilst you may not have seen a ray in a mathematical context before, you might have done in a scientific context.

A ray from the sun is a ray in exactly the same way as a mathematical ray.

The sun itself is the start point of the ray, with the ray of light then potentially extending forever in a straight line through any point in space.

For this first check, Sophia walks in a straight line from one end of the rugby pitch to the other, then shines a laser towards the sky.

Which word best describes the path she took to walk across the rugby pitch? And which word describes the path of the laser? Pause now to consider the definitions of line, ray, and line segment.

Sophia took the path of a line segment.

Where she started was one endpoint, and where she finished was the other endpoint.

The laser took the path of a ray.

The laser pointer itself is the starting point of the ray, but there is no defined endpoint as the laser's light will continue travelling up unto the sky.

Whilst a line has no start or endpoint, we can define the position and orientation of a line by stating two points on a plane that the line passes through.

We say that the line passes through A and B where A and B are two points.

You must describe a line through two points.

Through being the keyword here.

If you only use one point, then it may help define the location of a line but not its orientation.

For example, this line also passes through A but is orientated differently to the previous line.

We can also define the position and location of a ray, again, using two points.

The first point always defines its start point whilst the second point defines where the ray will pass through.

In this example, this ray starts from point C and passes through point D.

From is the key word for the start point and through defines a point with which the ray passes through.

And lastly, the two points which define a line segment describe its two endpoints.

The order of the two letters do not matter.

So, for example, this is a line segment from point E to point F.

And equally, it is a line segment from point F to point E.

We can also describe a line segment more simply as the line segment EF, or even just EF.

On screen is a summary of the three types of linear path that we've covered so far.

For this check, pause here to find which combination of these options best describes this diagram.

And the correct answer is a ray because it has a start point from point T and simply passes through point U.

And again, pause here to consider which combination of options best describes this diagram.

And the correct answer is line segment because there are two clear endpoints at Y and Z which can be referenced in either order.

Andeep asks if it is also possible to combine multiple linear paths into one diagram.

Well, we can start with line segment AB which has two clear endpoints, one at A and one at B, and then create a ray that starts at point A, which we've already drawn and continue that ray from A through any other point, this time point C like so.

Drawing more complex diagrams is possible as long as if a point is referenced in two separate linear paths, you must use the same point for both.

There cannot be two different locations defined as, for example, point A.

They must be the exact same location.

We can use descriptions of linear paths to check if a pair of points lie on the same linear path as a different third point.

And see how the answer may vary depending on the linear path chosen.

For example, we can draw a line through the point A and B that extend forever in both directions, far beyond the points A and B themselves.

This clearly does not pass through point X as point X lies to the right of that line through point A and B.

So, let's go on to the next description.

A ray is drawn from C through D.

And therefore, we start at point C and then we draw a linear path that extends forever in the direction of D as it passes through point D.

Again, this ray does not pass through point X.

So lastly, a line segment is drawn from C to A.

Again, this shorter linear path does not pass through point X.

And so, actually none of these three linear paths pass through point X.

Can you think of an instruction for any linear path that will pass through point X? For example, a ray from D through C will do as you can see here.

For this check, pause here to consider which of the following linear paths pass through point X? Both options A and C do as you can see here.

B is a line segment that is way too short, being exclusively on the left hand side of the diagram whilst X is pretty central.

For D, a ray from K through L goes in the wrong direction.

Had the ray been from L through K, then yes, it would've passed through point X as well.

And here's question one of Practise Task A.

For each of these instructions, draw either a line, a ray, or a line segment for each set of points.

Pause now to do this.

And for question number two.

Complete the table for each pair of points.

For the last two, you will need to figure out which points the options ask for.

Pause now to do this.

And well done for your work so far.

Pause here to have a look at the answers for question number one.

And pause here to have a look at the answers for question number two.

Well done if you've got all of these details correct.

From lines, rays, and line segments and onto varying ways we can label and describe angles from different notation forms. Let's have a look at them.

Incorrect description of angles may lead to misunderstandings.

For example, Sam asks Lucas to measure the size of the angle at D, but Lucas is confused by this as he thinks there are four angles around point D.

How could Sam better describe which angle they're asking Lucas to measure? Consistent with our previous cycle, points on a plane are represented by capital letters only.

In this diagram there are five points, D, E, F, G, and H with point D being the shared endpoint of four line segments.

Lucas is correct.

There are four angles around point D.

Angles, A, B, C, and D.

There are three ways of representing angles.

The first, if representing angles with an angle marker, you can represent an angle with a lowercase Latin alphabet letter such as this a degrees.

A degrees will represent the size of the angle at this location.

The second wave representing an angle is the common convention of labelling angles using the Greek lowercase letters such as this letter, pronounced theta, and this letter, pronounced alpha.

Using Greek lowercase letters is especially helpful as a substitute for our Latin alphabet when letters look similar in both lowercase and capital letter form such as the letters C or O.

The final way of describing an angle does not require any angle markers, and so is very useful in describing specific angles in a complicated diagram with lots of possible angle combinations.

An angle can be defined by three points.

Again, described using these capital letters that highlight the two line segments that bound an angle starting with this symbol that looks like an acute angle, which actually represents the word angle itself.

We start at point G.

Then, travel down to point D, and we have to focus on that entire line segment.

And then from point D, we travel to point F.

We now have two line segments that define the legs of the angle alpha.

And so, after learning about this new notation, Sam has adjusted their request.

Measuring the size of the angle, HDE.

Sam's new description clearly defines a unique angle, 67 degrees as Lucas has measured.

They could have also said EDH as both EDH and HDE represent the exact same angle.

For this check, pause here to give yourself time to place the named angles below into the table so they match with the values in this diagram.

And the answers are as follows.

For this next check, pause here to consider the size of angle NOP.

We start at point N, we travel to point O, and then from O we travel to point P.

We now have two line segments highlighted that are the angle legs of the angle 50 degrees.

And similarly for this check, pause here to consider the size of angle MOP.

What's the same and what's different about this question compared to the last one? And so, we start at point M, travel to point O, and then from O to point P.

Our angle legs now represent and bound two angles, the 35 degrees and the 50 degrees for a total of 85 degrees.

And so, angle MOP is the combined angle 85 degrees.

When defining an angle using this three letter notation, we default to it being the acute or the obtuse angle.

We need to explicitly state if we are looking for the reflex conjugate of this acute obtuse angle.

For example, angle ACB equals 80 degrees whilst the reflex angle ACB equals 360, take away that 80 degrees, giving a reflex angle of 280 degrees instead.

We can construct more complex diagrams by following angle descriptions written in three letter notation.

For example, in this diagram with a ray from A through D already given, we can draw a line through A and C like so.

And then, a line segment from B to D.

The triangle enclosed by these three linear paths is a right angled isosceles triangle.

And all angles in this diagram are either 45 degrees or multiples of 45 degrees.

Which of these angles is the biggest? Angle DBC? Reflex angle BDA? Or reflex angle BAD? Well, angle DBC is obtuse, which is automatically smaller than any reflex angle, so we can cross that one off our list.

Whilst angle BDA is 45 degrees due to it being one of the two base angles of a right angled isosceles triangle.

So, reflex angle BDA is gonna be 360, take away 45 degrees, giving a total angle of 315 degrees.

Finally, angle BAD is the 90 degree right angle of this right angled isosceles triangle.

So, reflex angle BAD is 360, take away that right angle of 90, giving a total angle of 270 degrees.

Therefore, reflex angle BDA is the largest of these angles.

For this final check of Cycle 2, pause here to calculate the size of reflex angle NOM.

We start at point N, travel to point O, and from O we travel to M.

And so, the non-reflex angle is 35 degrees.

And therefore, the reflex angle is going to be 360 degrees, take away 35 degrees, giving 325 degrees is the correct reflex angle.

And here's Practise Task B.

For question number one, write down the size of all six of these angles for this diagram.

Pause now to do this.

And for question number two, you first of all have to draw line segments, rays and lines with the instructions given, and then use a protractor to accurately measure these angles.

Pause now to grab a protractor and give these questions a go.

For the answers to Task B Question 1A, angle DCB is 73 degrees, angle EDB is 65 degrees, angle ACD is 73 degrees, angle theta is 27 degrees, angle EDC is 145 degrees, and reflex angle EDC is 215 degrees.

And for question two, angle CAB is approximately 40 degrees.

Angle CAD is approximately 112 degrees.

Angle EAB is approximately 140 degrees.

Angle CAE is 180 degrees.

And reflex angle DAB is approximately 288 degrees.

If you got within two degrees higher or lower than the angles on screen, I think you did pretty well with that.

Well done.

Using some of the notation we've looked at so far, we can develop a more thorough understanding of the properties behind parallel and perpendicular lines.

Let's have a look.

You may be familiar with give or take what a pair of parallel lines are.

Two straight lines are parallel if the shortest distance between them is always the same non-zero distance, no matter where you measure the shortest distance from on a pair of lines like so.

No matter where we measure the shortest distance between these two lines, the shortest distance is still five centimetres.

To show that two lines are parallel, we label the lines with arrows called feathers like this.

If there is a different set of parallel lines pointing in a different direction, then we can label the lines with a different number of feathers.

So for example, I've got a group of three parallel lines.

I can label each of these parallel lines with two feathers.

Two feathers per line shows they belong to a different parallel group than this group with one feather per line.

The property of parallelity that the shortest distance between two lines is consistent throughout the shape is relatively straightforward because two lines extend infinitely in both directions.

But what if we had line segments rather than lines? Can line segments that are far apart from each other also be considered parallel? Two line segments are parallel to each other if at least one line segment can be extended into a ray or a line such that the shortest distance between the extended line segments is always the same non-zero distance.

So for example in this diagram, EF and GH when extended into a ray or a line is that same consistent seven centimetres between both extended line segments.

The same method of extending line segments into rays and lines helps demonstrate if line segments are perpendicular to each other, even if they do not touch in and of themselves.

These two line segments are perpendicular to each other because the extensions of the line segments do intersect at a right angle.

As usual, perpendicular lines from extended line segments are still denoted by square angle markers.

We can analyse a more complex set of line segments by looking at the parallelity and perpendicularity of groups of these line segments.

For example, line segments AB and AC are definitely perpendicular to each other.

This is clear as they intersect at a right angle about the angle CAB.

AB and DE are parallel to each other as shown by the single parallel feathers.

And AC and FG are parallel to each other as shown by these double parallel feathers.

Even though DE and FG in and of themselves do not touch, they are still perpendicular to each other as the other line segments in their parallel groups are also perpendicular to each other.

This can be shown by drawing on those line segment extensions like so.

And for this final set of check questions, pause here to consider which line segments are parallel to DE.

The line segments parallel to DE are AB and FG.

This is because they all have a single feather to denote parallelity.

And pause here to consider which line segments are perpendicular to DE.

And the two correct answers are IJ and BC.

This is because AB is perpendicular to BC, and so DE is also perpendicular to BC due to them sharing a parallel property.

And here are the practise questions for Task C.

For question number one, write down line segments that are either parallel, perpendicular, or not parallel in this diagram.

Pause now to do parts A, B, and C.

And for part D, which line segment is perpendicular to AG? You must also give an explanation for your answer.

Pause now to do part D.

And here are the answers.

Pause now to check if your answers match any of those possible answers given on screen now.

And for part D, angle BFC is a right angle at 90 degrees.

This is because HE is a straight line and angles that meet at point F must add up to 180 degrees.

We are given that BFG and CFE are both 45 degrees, and so there is 90 degrees remaining at BFC.

Because BFC is 90 degrees, BF and FC are two perpendicular line segments.

If BF and FC are perpendicular, then AG and CF are also perpendicular because AG and BF are two parallel line segments.

And to everyone who has joined me for this lesson, thank you so much for all your amazing work on what was a technically rigorous lesson on naming conventions and notation as we look into defining the three linear paths.

The line, the ray, and the line segment.

By structuring a description by two points where each point is defined by a capital letter.

We've also looked into three different ways of describing an angle.

Lowercase letters, Greek letters, and three letter notation from three points on a plane.

And finally, we've looked at how extending line segments into rays or lines can help show whether there is parallelity or perpendicularity between those two line segments.

That is all for this lesson.

Absolutely well done for all the great effort you've put in today.

I've been Mr. Gratton, and until our next Mat lesson together, take care and goodbye.