Lesson video

Hi? I'm Mr. Chan, and in this session, we are going to learn about the circle theorem, angles in the same segment.

Let's begin by looking at this example.

What is the size of angle x and how do you know? Well, from a previous circle theorem, the angle at the centre is twice the angle at the circumference.

That must mean that angle x is 65 degrees.

What about now? What is the size of angle y and how do you know? Well, for exactly the same reason, that the angle at the centre is twice the angle at the circumference angle y must also be 65 degrees.

So if both of these angles are 65 degrees, what do you notice? Here is another example.

Again, what is the size of angle x and how do you know? Well, if we drew these two line to help us we can use a circle theorem that the angle at the centre is twice the angle of the circumference.

That must mean the angle at the centre is 104 degrees.

So now that allows us to work out angle x.

Angle x must also be 52 degreess.

What do you notice? The circle theorem here is called angles in the same segment are equal.

Let's look at this circle theorem in more detail and help you find these angles in the same segment.

So in this example we are asked what's the size of angle x and angle y and how do we know? Well, it is important to know where these angles are being drawn from.

And we call that subtended by.

So I'm going to put a card into the picture.

There's not normally drawn few but the card that has split the circle up into two segments, a smaller segment and a larger segment.

So angle x has been subtended by the same card as angle y at 47 degrees.

And the examples are shown as the angles in the same segment are equal.

So that makes sense that the card has been circled between the two segments and all those angles are in the same segment and they are all equal.

But the important thing is the angles have all been subtended from that same card.

So what we can say now is that because angles in the same segment are equal, angle x must be the same as 47 degrees there as is angle y, y is also 47 degrees.

Here are some questions we need to try.

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Here are the answers.

So in this question you are just practising using the angles in the same segment circle theorem.

Remember angles in the same segment are equal and the angels are obviously subtended by the same card so hopefully you've thought about or drawn in a card into the questions I wouldn't have even thought.

Here's some more questions for you to try.

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Here are the answers.

These questions use the circle theorem angles in the same segment are equal to find other angles.

So for example let us look at part c.

Now in question c, I can see that angle c is in the triangle and I have only got one other angle in that triangle.

However, I know that angles in the same segment are equal, so the missing angle in that triangle must be 63 degrees.

So now that I know that the angle there is 63 degrees and also 94 degrees has been given, angles in a triangle add up to 180 degrees.

So I know the angle c must be 23 degrees.

Here are some problems for you to try.

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Here are the answers.

In part a, I can see that the angles x, y and z are in the triangle but are also told that angles x, y and z are in the ratio 3:2:4.

So I can share out the angles in the triangle which add to 180 degrees into that ratio to figure out what angle x,y and z are.

And then decide which angle, angle a leans to in terms of angles in the same segment.

In part b, I'm told that there is two angles that are equal there because they are angles in the same segment.

However, those angles are given to me using algebra expressions.

So if I make those two angles equal I've then created an equation I can solve, I can figure out what the angles are by solving the equation and then once I figure out what the angle is, I can figure out what angle b is again, by using the fact that angles in a triangle add up to 180 degrees.

Here is a question to try.

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Here is the proof for this circle theorem, that angles in the same segment are equal.

This proof begins by looking at the relationship between the angle at the centre and the angle at the circumference.

So it does rely upon you knowing that the angle at the centre is twice the angle at the circumference.

So that is where it starts of with this proof.

That's all for this lesson.

Thanks for watching.