Lesson video

Hi, I am Mr. Chan.

And in this lesson, we're going to learn about the alternate segment theorem.

Let's begin with this example, we need to work out the size of the angle marked X.

And what we've got here is we can see that there's an angle at the centre and also an angle at the conference.

So the angle of the centre is twice the angle of the circumference We can say the angle of the centre is 104 degrees, that's twice the value of the 52 degrees there.

Now from the centre, we can also see that there are two radii drawn there.

So let's just put dashes there to indicate which ones am some talking about.

And because the two radii form a triangle, that makes the triangle isosceles, the angles in the isosceles triangle, the base angles are equal.

So that must make those two angles there 38 degrees Also because we've got a tangent drawn and it's meeting at one of the radius lines.

The tangent and radius are perpendicular at the point of contact.

So that means that we can work out angle X by subtracting 38 from 90 degrees.

So that must make angle X 52 degrees.

Now we started off with one angle of 52 degrees.

What do you notice? Let's pick up where the last example left off.

In this example, we have to work out the size of the angle marked y.

Now y, I can see far was part of the 90 degrees of the tangent and the radius make.

But there is a angle missing to stop me from figuring out y straight away.

So that missing angle I can say is 26 degrees.

Why is it 26 degrees? Is because there are three angles that we can see form part of the big triangle.

The triangle touches this circumference at three points.

So that must be 26 degrees.

That means that y is 64 degrees, because y makes a right angle as well as the 26 with the radius and the tangent.

So what's interesting with this circle theorem is the link between the 26 and the 38 on the segment, which is also 64 degrees.

What do you notice? Let's review what the examples have shown us.

This circle theorem is called the alternate segment theory.

What it tells us is that, the angle between the tangent and the chord is equal to the angle in the alternate segment.

So if we have an angle between the tangent and the chord there, the angle subtended by the same chord in the other segment is equal to the angle between the chord and the tangent.

Similarly, there is another chord created, which creates another angle with the tangent.

So this angle subtended by the same chord in the other segment will also be equal.

Here's some questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here in the first two answers, these first two questions really illustrate the alternate segment theorem, because if you have an angle between the chord and the tangent and an angle subtended from the chord upon circumference Subtended just means that the angle is drawn from that chord.

So if you have an angle subtended from the same chord point on the circumference, those two angles will be equal.

Here's more questions for you to try.

Pause the video to complete the task, resume the video once you're finished.

Here are the answers.

Let's look at part b in particular, where we're asked to find angle b.

Now angle b is drawn from a chord.

The chord has split the circle up into two segments.

So angle b, which is what we say subtended from that chord.

Now the chord actually makes an 85 degree angle with the tangent.

So what we can say about angle b is it must equal 85 degrees because the alternate segment theorem tells us that the angle that the chord makes with the tangent is the same as the angle in the alternate segment.

Here's some questions for you to try.

Pause the video to complete the task, resume the video once you're finished, Here are the answers for question four So in part a you're just practising using the alternate segment theory again, And in part b, hopefully you realised that there was an isosceles triangle in this question, where the base angles in an isosceles triangle are equal.

Here is a proof question for you to try.

The examples of the beginning of the lesson will help you with this.

Pause the video to have a go, resume the video once you're finished Here's the answer to the prove question.

What you'll notice about this prove is it relies upon other circle theorems to prove this alternate segment theory and also some basic angle facts, for example, angles in a triangle sump up to 180 degrees.

So hopefully you managed to get through this.

That's all for this lesson.

Thanks for watching.