Lesson video

Hello there and welcome to today's lesson.

My name is Dr.

Rowlandson and I'll be guiding you through it.

Let's get started.

Welcome to today's lesson from the unit of angles.

This lesson is called Forming Equations of Angles.

And by the end of today's lesson, we'll be able to form and solve equations using our knowledge of angles and polygons.

Here are some previous keywords that will be useful during today's lesson, so you may want to pause the video if you want to remind yourselves the meanings of these words and then press play when you're ready to continue.

This lesson is broken into two learning cycles, and we're going to start with forming and solving equations with angles.

Angle facts can be used to form equations when solving angle problems. In some cases, some equations can be formed by adding angles together and then equating it to what those angles should sum to.

In other cases, some equations can be formed by finding angles that should be equal to each other and then equating those.

The approach you take may depend on the information that you are presented with.

Let's take one of these situations, the fact that some angles can be formed by adding angles together and equating them to what those angles should sum to.

And here's an example of where we might do that.

Find the value of x.

Let's start with what we know.

We have three interior angles of a triangle and 4x degrees is one of them.

Now we know that those angles in a triangle should sum to 180 degrees, so we could add together these three angles and write it as an equation that is equal to 180 degrees.

That'd be 4x + 70 + 50 = 180.

We now have an equation that we could simplify and solve to find the value of x.

If we simplify it by adding together the two angles we know, we get this, and then if we start to solve it by subtracting 120 from both sides and then divide by 4, we get x = 15.

Now, in this question, only one of the angles is algebraic.

The other two, we have the actual numbers for.

But this question could be made more complex by using algebraic expressions for more of the angles.

For example, what we can see here.

Each of those three angles has an algebraic expression but we could still do the same thing.

We know that all three of these angles, whatever they are, should sum to 180 degrees.

So if we add together these three expressions, they should equal 180.

So we can write an equation that says just that.

We can then simplify this equation to get 11x + 15 = 180 and solve it by subtracting 15 from both sides and then divide in both sides by 11 to get x = 15.

Let's check what we've learned.

Here we have a diagram where some angles are labelled with algebraic expressions and others are given the exact numbers for.

Which angle fact could be used to form an equation for x? Is it a, angles in a triangle sum to 180 degrees? Is it b, angles forming a straight line sum to 180 degrees? Is it c, angles in a quadrilateral sum to 360 degrees? Or is it d, angles around a point sum to 360 degrees? Pause video while you choose and then press play when you're ready for an answer.

The answer is c, angles in a quadrilateral sum to 360 degrees because that's what we have here.

So with that in mind, use the fact that angles in a quadrilateral sum to 360 degrees to write an equation for x.

Pause video while you write down an equation and then press play when you're ready to see what it is.

Well, our equation could be written by adding together all four of those angles and point it equal to 360.

It could look like the equation at the top or you could simplify it to the equation at the bottom, 12x + 228 = 360.

With that in mind, now find the value of x.

Pause video while you do that and press play when you're ready for an answer.

The answer is x = 11 and here's your working for how to get it.

Let's look at the other situation now where some equations can be formed by finding angles that should equal each other and equating those.

For example, here we have an isosceles triangle where one angle is labelled 4x degrees and we want to find the value of x.

Let's think about what we know.

We know that base angles in an isosceles triangle are equal.

Therefore, that angle which is 4x degrees would be equal to the angle which is labelled 60 degrees.

So we can write an equation that says just that, like this, and then we can divide both sides by 4 to get x = 15.

This problem could also be made more complex by using algebraic expressions for both of the equal angles.

For example, let's find the value of x here.

Each angle we know is equal because it's an isosceles triangle and these base angles are equal to each other.

That means 5x - 15 is equal to whatever 2x + 30 is, so we can write an equation that says just that.

It looks like this.

We can then solve this equation by subtracting 2x from both sides and then add 15 to both sides, divide both sides by 3 to get x = 15.

Let's check what we've learned.

Here we have a diagram that has two angles labelled with algebraic expressions.

Which fact could be used to form an equation for x? Is it a, the alternate angles and apparel lines are equal? Is it b, fact corresponding angles in parallel lines are equal? Is it c, fact diagonally opposite angles in a quadrilateral are equal? Or is it d, fact vertical opposite angles are equal? Pause video while you choose and press play when you're ready for an answer.

b, corresponding angles in parallel lines are equal.

So with that fact in mind, write an equation for x.

Pause video while you do that and press play when you're ready to see what it is.

Our answer is 5x + 16 = 7x - 4 because those two angles are equal to each other.

So find the value of x.

Pause video while you do it and press play when you're ready for an answer.

The answer is x = 10 or 10 = x, and there's your working to solve it.

Okay, it's over to you now for Task A.

This task contains two questions and here is question 1.

In each question, you have a diagram contain angles where some are algebraic and some are numerical.

What you need to do is find the value of x in each question.

Pause the video while you do this and press play when you're ready for question 2.

And here is question 2.

You have three polygons and an angle labelled with an algebraic expression.

What you need to do is find the value of x in each question.

Pause video while you do this and press play when you're ready for answers.

Okay, let's take a look at some answers.

Question a, the answer is 46 and the fact you use is that interior angles of a quadrilateral sum to 360 degrees.

For b, the answer is 35.

The fact you use is that opposite angles in a parallelogram are equal.

In c, the answer is 38 and the fact you use is a co-interior angles between parallel lines sum to 180 degrees.

For d, the answer is 12.

You use a fact for angles on a straight line sum to 180 degrees.

In e, the answer is 14.

You use the fact for angles around a point sum to 360 degrees.

And with f, x is 6.

You can use the fact that vertical opposite angles are equal.

Then question 2.

In part a, x is 60.

You can use the fact angles in equilateral triangles are equal and the sum of them is 180 degrees.

Or you might just know fact the interior angles of a equilateral triangle are 60 degrees.

For part b, x is 54.

The way you get that is fact the interior angles in a regular polygon are equal.

Each of those interior angles is 108 degrees, so you can half that to get a 54 or you can solve the equation 10x, which comes from doing 5 lots of 2x, equals the sum of the interior angles, 5 - 2 multiply by 180.

For part c, x = 50.

You can use the fact that exterior angles of a regular polygon are equal and sum to 360 degrees.

Wonderful work so far.

Now, let's move on to the second part of today's lesson, which is expressing one variable in terms of another.

When angles are unknown, algebraic statements can be used to express connections between angles.

This can sometimes be done by identifying the sum of angles.

For example, here we have a triangle where one angle is given to us, 40 degrees, but we have unknowns for the other two angles.

One's expressed in x, I wonder if one is expressed in terms of Y.

What we want to do now is express the connection between x and y.

Express y in terms of x.

Well, we could start with what we know.

We know that angles on a triangle sum to 180 degrees, which means these three angles add up to 180.

We could then simplify this by subtracting 40 from both sides and I want to write y in terms of x, so I want to say y equals something with x in.

So I could subtract 3x from both sides to get y = 140 - 3x.

So in this situation, we use the same strategy as we used earlier.

We use a fact about the sum of angles to create our initial equation and wrote our final answer as y in terms of x.

We can also use the other strategy we used earlier instead.

That is, by identifying angles which are equal.

So in this situation, we want to express y in terms of x.

Well, we have an isosceles triangle.

We know that the base angles in isosceles triangle are equal, which means those two angles we can see highlighted are equal to each other.

We can write it like this, y = 3x.

It can sometimes be done by using a combination of facts.

For example, here we have an isosceles triangle but this time, it is the angle at the top which is labelled y.

That's the angle which is different to the other two.

Let's express y in terms of x this time.

Well, we know that base angles of isosceles triangle are equal, so the other angle on the bottom right must be equal to 3x degrees.

And then we know that angles are triangle sum to 180 degrees; therefore, we could write this and then we could simplify it, subtract 6x from both sides, and now we have y written in terms of x.

So let's check what we've learned.

For which diagram is y equal to 4x.

Pause the video while you choose and then press play when you're ready for an answer.

The answer is b.

In this one, we have the angle which is labelled 4x degrees.

It's vertically opposite the angle which is labelled y degrees, which means y must be equal to 4x.

So with this diagram, which fact could be used to express y in terms of x? Is it a, angles around a point sum to 360 degrees? Is it b, angles forming a straight line sum to 180 degrees? Or is it c, vertically opposite angles are equal? Pause the video while you choose and press play when you're ready for an answer.

The answer is a, angles around a point sum to 360 degrees.

Here we have a diagram.

Please express y in terms of x.

Pause video while you do this and press play when you're ready for an answer.

y is equal to 90 - 4x.

We get it by first using the fact that angles form in a straight line sum to 180 degrees, which means y + 90 + 4x = 180.

We could subtract 90 from both sides and then subtract 4x from both sides to get our answer.

Sometimes you may need to express other unknown angles algebraically before you can write one variable in terms of another.

For example, here we have isosceles triangle with one of its edges extended.

It says x degrees is an exterior angle of this triangle, so express y in terms of x.

Let's think about what we know and consider different ways we could do it.

Well, one method could be to express unmarked angles in terms of y until we get to the point where we can connect x and y together.

For example, we could write this as y degrees because it's equal to the other angle.

We could then work out the third remaining angle of that triangle in terms of y.

That would be 180 - 2y.

And then we know that the angle on the inside, 180 - 2y degrees, and the angle on the outside, exterior angle, x degrees, they sum to 180 degrees.

So we could do this.

180 - 2y, the interior angle, + x, the exterior angle, = 180.

We could then simplify it, rearrange it, and simplify some more and divide both sides by 2 until we get y = x over 2.

Another way we could have done this is to express your no marked angles in terms of x instead.

So let's start off the fact we have, an exterior angle, which is x degrees, which means the interior angle will be 180 - x degrees, and then we could work out this angle in the top right corner here of the triangle by doing 180, which is the sum of the angles in the triangle, subtract the angle we have there, 180 - x, to angle we just worked out and then divide it by 2.

And that will give us both of the remaining angles in terms of x.

That gives us x over 2.

And we know that those two angles are equal to each other, y and x over 2, so we can write it like this.

Now, both those methods we've seen so far have some parts which are easier than others, which means a third method could be to express some unmarked angles in terms of x and some in terms of Y.

In other words, express each angle in the easiest way using algebra.

For example, if we know the exterior angle is x degrees, then the interior angle is found by doing 180 - x.

We also know that isosceles triangles have two equal angles, so one angle is y, the other angle is y, and then we can start to connect these together.

We could say that the sum of those three angles is 180 degrees, so y + y + 180 - x = 180 degrees, and then we could simplify our equation, rearrange it by adding x to both sides, subtract 180 from both sides, and then divide both sides by 2 to get the same answer again, y = x over 2.

Some problems may require many steps before you can express one variable in terms of another.

For example, with this diagram here.

We have two isosceles triangles joined together so that the base of each triangle form a line, and we want to express y in terms of x.

Now, there are multiple ways you can do this.

Let's do it by expressing unmarked angles in terms of y.

We could right that this angle is y degrees because it's equal to the one we're given because it's an isosceles triangle.

We could work out the third remaining angle on that isosceles triangle on the left as 180 - 2y degrees, and then because the angle to the right of it forms a straight line with it, we can do 180 subtract the angle we just worked out and that'll give us 2y.

And then we know that the angle on the far right is equal to that one because it's an isosceles, so that's 2y.

And now we have all three angles in that right hand triangle labelled algebraically, so we can write 2y + 2y + x = 180.

We could simplify it to get 4y + x = 180, subtract x from both sides, divide both sides by 4, and we get y = 180 - x over 4.

Let's check what we've learned.

Here you've got a diagram with an isosceles triangle and x is an exterior angle.

Express y in terms of x.

Pause the video while you do this and press play when you're ready for an answer.

Well, x and y add together to make 180 degrees, so y = 180 - x.

How about this time? I've changed the question ever so slightly, could you please express y in terms of x here.

Pause video while you do it and press play when you're ready for an answer.

The answer is y = 2x - 180.

And there are different ways you can get that answer, but here's one example of some working.

You can work out the interior angle in terms of x as 180 - x.

The other angle at the top left corner would be equal to that and all three of them would add together to make 180, which you can simplify and rearrange to get y = 2x - 180.

Okay, it's O2 now for Task B.

This task contains two questions and here is question 1.

In each part, you need to express y in terms of x.

Pause the video while you do this and press play when you're ready for question 2.

And here is question 2.

Same thing again, express y in terms of x but this time, it's a little bit more complicated.

Pause video while you do this and press play when you're ready for some answers.

Part a, y = 180 - 5x, which you can get from doing angles in a straight line sum to 180 degrees.

Part b, y = 90 - 5x.

Your starting point there is that angles inside a triangle sum to 180 degrees.

Part c, y = 300 - 5x, which you can get by using the fact that angles around a point sum to 360 degrees.

In part d, y = 3x.

Your starting point there is that diagonally opposite angles in a parallelogram are equal.

And part e, y = 180 - 6x.

Your starting point there is a co-interior angles between parallel lines sum to 180 degrees.

And part f, y = 102 - 2x.

Your starting point there could be by interior angles in a quadrilateral sum to 360 degrees.

Then question 2, part a.

You can get y = 270 - 6x.

Your starting point could be to find the exterior angle where the 90 degree angle is, which is also 90, and then the fact that exterior angles always sum to 360 degrees.

Part b, you can get y = 54 - x over 10.

Your starting point there could be that interior angles of a pentagon sum to 540 degrees.

And part c can get y = 2x - 36.

This could take quite a few steps, but one thing you could do is you could work out the other two interior angles of a triangle in terms of y and then notes fact the interior angles in a regular pentagon are all 108 degrees and then use that to connect the two variables together.

Fantastic work today.

Now, let's summarise what we've learned during this lesson.

Missing angles can be found in polygons using knowledge of exterior and interior angles.

A missing angle can be represented with an unknown, and angle facts can be used to form and solve equations to perhaps find that unknown.

Equations can be made using facts about what angles sum to, and also equations can be made by identifying angles which equal each other.

Also, algebraic statements can be written about missing angles in polygons by expressing them in terms of a variable.

Well done today.

Have a nice day.