Lesson video

<v Gratton>Hello everyone.

It is a pleasure to have you here today for this maths lesson.

I'm Mr. Gratton and I'll be your maths teacher while we will be using the angle facts of angles that meet at a point on a straight line, angles around a point that form a full turn, as well as other angle facts in order to calculate the size of missing angles.

Pause here to check the definitions of some of these important words and phrases that we'll be using during today's lesson.

First up, we'll look at understanding and presenting a simple and clear method to calculate angles in different locations.

Let's have a look.

If AB is a straight line segment, how can I find the size of angle x? Well, because AB is a line segment, any angles that meet at one single point on that straight line segment will sum to 180 degrees.

Therefore, the angles x and 67 degrees both sum to 180 degrees.

We can write this relationship as the equation, x plus 67 equals 180.

As with all linear equations with one unknown, we can solve this equation to find the value of x, in this case, by subtracting 67 from both sides, this gives us x equals 113, and therefore, the missing angle is 113 degrees.

This same method can be used even if there are more than two angles about that one point on a straight line.

Like with this diagram, which shows four angles that all meet about one point on the straight line segment AB, we can say that x, 62, 31, and 72 degrees all meet about one point on a straight line segment and so the sum of x, 62, 31, and 72 all equal 180.

As with before, we can write this relationship as an equation, x plus 62 plus 31 plus 72 equals 180.

We can sum the angles 62, 21, and 72, which are numerically represented angles, which all lie on the left hand side of the equation to get a total of 165.

After this, we can subtract 165 from both sides of the equation in order to solve the equation, giving us a value of x equals 15, and therefore the missing angle is 15 degrees.

We can apply the exact same process to calculate missing angles around a point that create a full turn, a total total of 360 degrees rather than the 180 from the straight line segment.

We can write this relationship as the equation, m plus 146 equals 360.

By subtracting 146 from both sides, we solve the equation to get m equals 214, and therefore, the angle m is 214 degrees.

Furthermore, we can also write the relationship between many angles around a point that create a full turn as an equation.

n, 89, and 113, as well as the right angle of 90 degrees, all sum to 360 degrees.

By first adding together the three numerically represented angles, we get a total of 292.

Then, by subtracting 292 from both sides of the equation, we can solve to get n equals 68, and therefore the missing angle n is 68 degrees.

Okay, let's look at two checks all at once.

Calculate the size of angles a and b.

Pause here to use your knowledge of angle facts for supplementary and conjugate angles.

Angles around a point that make a full turn sum to 360 degrees, and so a has a value of 314 degrees.

Angles that meet at a point on a straight line or a straight line segment sum to 180 degrees, and so angle b is 152 degrees.

For this check, Jun starts writing a method to find the size of the missing angle k.

What number or numbers would go inside the box? Pause now and there is more than one valid answer.

What would go inside the box are the three numerically represented angles, 67, 42, and 11 degrees.

However, it was also equally valid to put the sum total of these three numbers, 120, also inside that box.

Similar check, however, we're now focusing on the solving of the algebraic equation.

Izzy starts writing a method to find the size of the missing angle h.

What number goes inside boxes a and b? Pause now to give those two answers.

325 is the sum of the three angles that were represented numerically, the sum of 125 degrees, 103 degrees, and 97 degrees.

35 is the solution to the equation, 360 subtract 325 and is the value of h.

On two intersecting straight lines, all the angles that meet at the point of intersection will add to 360 degrees.

As they are all angles around a point that form a full turn, this shared point being the point of intersection between those two lines.

Furthermore, angles a and b lie on one of the two straight lines and meet at a single point.

Therefore, they are supplementary angles.

As with before, we can form and solve the equation knowing that supplementary angles sum to 180 degrees, therefore b plus a, which has a value of 34 degrees equals 180.

We can solve this equation to get b equals 146 degrees.

Similarly, both angles b and c also lie on a straight line this time, the other straight line of these two intersecting straight line segments.

Again, we can form and solve an equation to find the value of c, which in this case is 34 degrees.

Where have we seen the value 34 really recently? Well, we noticed that both a and c are 34 degrees, they are of equal size.

This is true because angles a and c are vertically opposite angles, and vertically opposite angles are always equal in size.

Opposite angles are formed from two intersecting line segments that meet at a common point where one of the two angles is bound by one part of each line segment and the other angle is bound by a completely different part of each line segment.

That is to say, they do not have and can never share the same part of either of the two line segments.

It's always helpful to identify as many angles as you can from rules that are familiar to you, as labelling some angles in this way may help you get closer to identifying other angles.

Let's demonstrate this by finding the values of angles a, b and c.

Starting off with b, we know that b is vertically opposite the angle of 138 degrees and because vertically opposite angles are equal, b also equals 138 degrees.

And furthermore, c is supplementary to 138 degrees because both those angles lie around one point on a straight line segment.

So 180 takeaway 138 equals 42 degrees, and so c has a value of 42 degrees.

And furthermore, a is supplementary as well to 138 degrees and so it also has an angle of 42 degrees.

Sometimes there are many different methods, many different routes that we can take to find the value of a missing angle.

Here's a different way of finding the angles a, b, and c.

As with before, b is vertically opposite to 138 degrees and so b has a value of 138 degrees as well, and we also know that angles around a point sum to 360 degrees, therefore a plus c plus b, which has 138 degrees and the other 138 degrees as well, all sum to 360 degrees.

We sum together the numerically represented angles, 138 plus 138, this gives us a total of 276.

We can then subtract 276 degrees to get 84 degrees, so a plus c has a total of 84 degrees.

However, since a and c are also vertically opposite each other, their angles are equal.

Therefore, we share the 84 equally between both angles, therefore giving us 42 degrees for each of the two angles, a and c.

In a more complex diagram with multiple intersecting line segments, it is sensible to clearly highlight the two straight line segments that intersect at the common point.

This allows us to more easily see which angles are vertically opposite each other.

By highlighting these two straight line segments, we can see that the 57 degrees is vertically opposite to a, and therefore a is also equal to 57 degrees, and crucially, this method also makes it very clear that the angle 123 degrees is vertically opposite the sum of both the angle b and the angle 59 degrees.

We can therefore set up an equation to calculate the angle b, b plus 59 degrees is equal in value to the vertically opposite angle, 123 degrees.

Therefore, b + 59 = 123, which when solved gives us 64 degrees.

Okay, for these two checks, find the value of the angles labelled f and g.

Can you also think of a reason why each angle has that value? Pause now to give yourself time to inspect both these diagrams. And the answers are as follows.

For f, we use the rule, the angles that meet at a single point in a straight line sum to 180 degrees, whereas for g, we use the rule, vertically opposite angles are equal in size.

And for this next check, XV is one of the two intersecting straight line segments in this diagram.

Which two letters define the endpoints of the line segment that passes through XV and not just touches it with one of its endpoints.

Pause now to inspect the diagram and answer this question.

And the line segment that passes through XV is ZY.

That line segment with the label W on it doesn't pass through XV, its endpoint just touches XV itself.

Here's another two checks in one.

Firstly, what is the value of angle m? Secondly, which statement could be used to justify the size of angle m? Pause here and there may be more than one correct statement.

Angle m is 63 degrees and the two statements that could be used to justify this are m is supplementary to 117 degrees and m is vertically opposite the 63 degrees in this diagram.

And for this check, pause here to consider which of these calculations can be used to find the size of angle n? And the answer is c.

Vertically opposite angles are equal and so 117 is equal to the sum of n and 87.

The two angles which compose this single angle that is vertically opposite to 117 degrees.

And here's the penultimate check, this one will require you to use a handful of different angle facts, one at a time, in different parts of the check.

You are given that b plus c plus d equals 320 degrees.

Using this relationship, find the value of angle a.

Pause here to do this.

We know that a plus b plus c plus d equals 360 degrees.

If b plus c plus d equals 320, then a must be 40, the remaining angle required to sum to 360 degrees, calculated by 360 take away 320 equals 40 degrees.

Now that we've calculated the size of angle a, pause here to find the value of angle d.

d and a are supplementary because they share a common point on a straight line segment.

Therefore we can write down the equation, d plus 40, the value of a, equals 180.

We can then solve this equation to get d equals 140 degrees.

Okay, well done so far! Onto some practise questions.

For question number one, fill in each stage of working by filling in each box to calculate the missing angle in each diagram.

Pause now to do all of this.

And for question number two, find the value of all of the angles marked with a letter.

Pause now to answer all six parts of this question.

For question number three, find the size of angles w, x, and y, knowing that AB and CD are two straight line segments that intersect.

And for question number four, you are given the equation, a plus 140 plus b plus c equals 350 degrees.

Knowing that RU and SV are two straight lines, find the size of all four angles a, b, c, and d.

Pause now to do both of these really challenging questions.

Okay, here are the answers for question number one.

Pause now to compare the answers on the screen to the answers that you've written for each part of this method.

And for question number two, a has a value of 113 degrees, b, 67 degrees, c, 45 degrees, d, 119 degrees, and e, 61 degrees, f has a value of 259 degrees, and g has a value of 21 degrees.

Well done if you got a handful of those correct.

And for question number three, y is vertically opposite 83 degrees and so also has a value of 83 degrees, w and 83 are supplementary angles and therefore sum to 180 degrees.

Therefore, w has a value of 97 degrees.

The x and the 37 both add up to 97 degrees because x and 37 combined are vertically opposite the angle w.

Pause here to match all of your angles, the angles for w, x, y, a, b, c, and d as shown on screen.

Well done so far for all of your effort on applying knowledge of algebraic equations and angle facts in order to calculate the size of missing angles.

However, it is a skill in itself to communicate these angle facts when calculating the size of an angle.

Effectively communicating what facts you are using and how, is especially important when dealing with more complex diagrams. Showing some calculations may not be enough to explain what part of the diagram you're performing calculations for.

Therefore further explanation must be given to make the purpose of any mathematical calculation more clear.

We will look into a basic structure and good practise of a well communicated response to finding the size of a missing angle and then develop these responses and adapt them to more complex diagrams. Step one is to state a fact that you'll be using in your explanation.

In this diagram, angles about a point on a straight line sum to 180 degrees, and then step two is to write down all calculations linked to the angle fact that you just mentioned relevant to the diagram that you've been given.

Angles that meet about a point on a straight line, the angle fact quoted in step one sum to 180 degrees and so we can form and solve a relevant algebraic equation about this straight line.

In this case, the angles p, 72, and 21 degrees all meet at the point on AB, and so we can form the equation, p plus 21 plus 72 equals 180.

We can then solve this equation to get p equals 87 degrees.

And therefore finally, step three, use any values you have calculated to show whether a statement is true or false.

In this case, we wanted to verify if p was an obtuse angle or not.

However, we have just solved an equation in step two and justified that p is 87 degrees, which is less than 90 degrees, and therefore p is acute and not obtuse.

We have explained and given a full reason to show that Lucas is wrong.

Simply saying that p is acute with no evidence to support that claim is not a mathematically rigorous practise.

Showing full method and explanation, however, is.

Here's a different type of context where fully justifying and evidencing your answer is important.

Is AB one straight line segment or two different lines that just so happen to meet at a point? Pause here to consider which angle facts you may need to use and how they can be used in order to answer this question.

Remember, step one is to state an angle fact in the context of the question.

If AB is one straight line, then the two angles about it should be supplementary, that means they should add up to 180 degrees.

Step two is to then form a mathematical equation from relevant information from the diagram.

In this case, adding together the two angles that meet at the point about AB to see whether they sum to 180 degrees.

85 plus 98 equals 183 degrees.

They do not sum to 180 degrees.

They sum to a greater value at 183 degrees instead.

Step three is then to use these values in context in order to answer the question given.

85 and 98 are not supplementary angles as they sum to 183 and not 180 degrees.

Therefore, we can say for certain that AB is not one straight line.

Sometimes you will have to state more than one angle fact, or apply it to two or more different parts of a diagram.

This means that you may need to mention more than one angle fact in step one, have more than one set of calculations in step two, each set of calculations referring to the same angle fact from step one and sometimes even needing to have multiple angle facts and multiple sets of calculations, almost like several different versions of step one and step two, which can then all be brought together to form a final conclusion, that conclusion, step three.

Here's an example of that, AB is a straight line segment.

Explain whether the angles r and t are vertically opposite each other.

I use one angle fact, if AB is one straight line, then t and 84 are supplementary to each other.

We can then form an equation based on this angle fact t plus 84 equals 180, and therefore t equals 96.

However, this is not enough information to answer the original question.

Remember, it is always sensible to calculate as many angles as possible as more than one angle may be needed to build a bigger picture of a diagram.

Therefore, I use another angle fact.

If AB is one straight line, then r and 76 degrees are also supplementary.

Whilst this is the same angle fact as previously used, it relates to a different angle and so must still be explicitly stated again to show its relevance to this different angle.

Forming and solving this equation gives r equals 104 degrees.

The conclusion then uses all angles calculated from this investigation.

r and t are not vertically opposite, as vertically opposite angles must be equal in size.

Note that the conclusion also explicitly references another angle fact.

This is necessary and absolutely fine as stating it here helps in justifying why the angles are not vertically opposite each other.

And for this check, Andeep measures the angles on this diagram using a protractor.

Choose whether he has correctly or incorrectly measure the angles on this diagram and select the appropriate explanation to justify why.

Pause now to look through all possible options.

Andeep has incorrectly measured the angles on this diagram.

This is because angles around a point that create a full turn should sum to 360 degrees or we could also say that a pair of conjugate angles sum to 360 degrees as well.

However, his two angles, 127 and 53, sum to 180 and not the 360.

Therefore, we know for certain, the angle has not measured at least one of these angles correctly.

In order to avoid some misconceptions, it is important to look at the details behind some of the angle facts that we are using.

For example, Aisha is incorrect in her conclusion that w plus x plus Y equals 160 degrees.

Let's have a look why, the rule is not just the angles on a straight line add up to 180 degrees, there is more detail than that.

It is more specifically, angles that meet about a single point on a straight line that sum to 180 degrees.

In this diagram, there are two points on the straight line with which the angles meet, both C and D are separate points on that straight line.

Because there are two points, the rule applies in two different places.

Angles about the point C sum to 180 degrees, and separately, angles about point D sum to 180 degrees.

As a consequence, let's have a look at the correct explanation to try and correct what Aisha has said.

Angles w and 20 degrees are supplementary angles about the point C, and so 20 plus 160 gives you the 180 degrees about that point on a straight line.

Angles x and y separately are supplementary angles about the point D on a straight line and so x plus y equals 180, so w plus x plus y, which is what Aisha wanted to find, is gonna be the 160 for w plus the 180 for the x and y combined.

Therefore, in total, w plus x plus y is 340 degrees, not 160.

Is there a quick way that you could have found out the answer to that question? Because it was also possible to do 180 plus 180 for the two separate angles around points on a straight line calculations, take away the one numerically represented angle of 20 degrees, so 360 take away 20 is also 340.

For this check, Sam's observation is definitely incorrect.

Which of these angle facts best starts an explanation to show why Sam is incorrect? Pause now to look through all three options and choose the correct one.

And the answer is b.

Angles about a common point on a straight line sum to 180 degrees.

And consequently, sticking with the same diagram, which of these statements are correct? Pause now to look through all three options and there may be more than one correct answer.

And the correct answers are a and c.

Because C and D are two separate points, the angle fact of angles around a point on a straight line sum to 180 degrees applies separately for point C and D.

And lastly for this check, which of these calculations to find the angles of j or k are correct? Pause now to look through all six options.

And the two correct answers are b and d.

This gives us, j equals 165 degrees, whilst k is 130 degrees.

Note that neither angle is 115 degrees itself.

This diagram shows a circle.

All points in the diagram are evenly distributed around the circumference of the circle with one point at the circle's centre.

The three angles each share the common vertex that is also the centre of the circle.

Two sectors are congruent if there are the same number of dots on the arcs of the sectors.

Congruent sectors have the same central angle as well.

We know that angles a and b are equal because they both have two dots on their arcs, whereas angle c is different, as it only has one dot on its arc.

Are any of the angles d, e, or f the same size? How can we explain the answer to this question? Well, the answer is no, as f doesn't have any dots on its arc at all, whilst e has two and d has three.

Next question though, can you add together any two of the angles so that the sum equals the third angle? Let's have a look.

The answer is yes.

e plus f equals that larger angle, d.

This is because combined, they have three dots each.

When adding two dots together, you must also count the dot on the radius adjacent to the two angles that you are adding together.

In this case, we must also include the uppermost of the three dots in the f plus e combined angle.

And one more thing to note about this diagram, both the angles d and the sum of the angles e and f lie on a straight line segment, which is the diameter of this circle.

This is because d plus e plus f equals 360 degrees, and the number of dots on d is equal to the number of dots on the combined e plus f, both at three.

If you've got two angles which sum to 360 degrees, which are both equal, then one of those two angles must then be 180 degrees, which creates a straight line around the centre of the circle.

Okay, onto a check, for the first question, which two angles in this diagram are equal in size? And for part b, which statement shows the correct reason for how you know these two angles are equal in size? Pause now to give both parts of this question a go.

And the two angles that are equal are x and z and this is because they have the same number of dots on the arc of the sector that bounds the angle.

Right, get ready for the last set of practise questions.

For question number one, which of these angle facts is the most helpful when trying to find the value of angle r? Pause now to choose the correct explanation and then write down a calculation to find the value of angle r.

And for question number two, complete the angle fact, calculation, and conclusion to find and justify the values of angles m and p.

Pause now to do both questions.

And for question number three, find the values of angles v, w, x and y, and for each, write down an explanation for how you know.

Pause now to give question three a go.

And pause here to answer question number four, which asks you to analyse angles with a vertex at the centre of a circle.

Okay, onto the answers for task b, for question 1A, a is the answer angles about a point on a line sum to 180 degrees, is the correct angle fact.

By writing down and solving a calculation, you should have then got r equals 63 degrees.

And pause here to check that your justification and calculations to find the size of angles m and p match those that are on screen now.

Pause here to look through the explanations and answers for angles v and w.

And pause here again to check the explanations and answers for angles x and y.

And for question 4A, angles a, b, and c are all equal in size because each of them have two dots on the arc of their respective sectors.

And for part b, RQ is not one straight line segment.

This is because d has three dots, whilst e has four dots on their respective arcs and they must have the same number of dots on their arcs in order for RQ to be one straight line segment.

This is because d and e then would both be 180 degree angles.

And finally, for part c, it is impossible to draw two vertically opposite angles whose shared vertex is also the centre of this circle.

This is because there is an odd number of dots on the circumference of the circle.

And thank you for all of your effort in communication for this challenging lesson that encourages mathematical rigour.

In this lesson, we have justified the size of a missing angle around a point, on a shared point on a straight line, and when angles are vertically opposite each other.

We've addressed a misconception about angles in a straight line and concluded that angles that meet about two different points in a straight line will sum to 180 degrees separately at each of those points.

We've also looked at properties involving angles that can be applied to equally spaced-out points on the circumference of a circle.

Once again, thank you so much for making the decision to join me today for this maths lesson and for all the hard work you've put in during this lesson.

Until next time, I'm Mr. Gratton and have an amazing rest of your day.

Take care.