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Circle Theorems: Angle in a semi-circle

# Circle Theorems: Angle in a semi-circle

### Circle Theorems: Angle in a semi-circle

We can apply universal rules to solve geometrical problems involving circles, where there are tangents, secants or chords present.

The properties of circles have been been studied for centuries, and there are a range of generalised results which are true for circles of any size.

First, let's define some lines!

1

This is a secant

A secant moves through two points on the circumference of a circle, and keeps going

2

This is a tangent

A tangent touches the circumference of a circle at a single point

3

This is a chord

A chord is a straight line which connects any two points on the circumference of a circle

4

What type of line is this?

Let's go through the first two circle theorems!

1

The angle in a semicircle is $90\degree$

The line moves through the centre of the circle. No matter where you move the yellow point, the angle is always $90\degree$!

2

What is the sum of the other two angles in this triangle?

1

The angle in the centre is twice the angle at the circumference

By drawing two radii and two chords, we form a situation where there are two angles. Try dragging around the green points - the angle in the centre is always twice the angle at the circumference.

2

What would the angle at the circumference be here? Write the number of degrees.

3

The angle at the circumference is $51\degree$

This is because the angle would be half the angle at the centre, so $102\degree\div2=51\degree$

4

What is the angle at the centre here? Write the number of degrees.

5

The angle is $156\degree$

This is because the angle in the centre is double the angle at the circumference, so $78\degree\times2=156\degree$