## The **8 Circle Theorems.**

You should be familiar with all **8 circle theorems** to the point where:

a) You can **identify** when they should be used.

b) You can **describe** which one you’ve used with appropriate language.

Make sure you are happy with the following topics before continuing

**Note:** Circle geometry problems often require knowledge of all the basic geometry rules in order to solve them.

## Rule 1

Angles in the **same segment** are **equal**.

\textcolor{red}{x} = \textcolor{red}{x}

Triangles drawn from the same chord will have the same angle when touching the circumference.

## Rule 2

**Opposite angles** in a **cyclic quadrilateral **add up to ** 180\degree.**

\textcolor{purple}{w}+\textcolor{red}{x}=180\degree

\textcolor{blue}{y}+\textcolor{limegreen}{z}=180\degree

This is a 4 sided shape with every corner touching the circumference of the circle.

## Rule 3

The **angle at the centre** is **twice** the** angle at the circumference**.

The angle formed at the centre is exactly twice the angle at the circumference of a circle.

## Rule 4

The** perpendicular bisector** of a **chord** passes through the **centre of the circle**.

A line perpendicular and in the centre of a chord (a line drawn across the circle) will always pass through the centre of the circle.

## Rule 5

The **radius** will always meet a **tangent to the circle** at 90\degree.

A tangent (a line touching a single point on the circumference) will always make an angle of exactly 90\degree with the radius.

You can say that a tangent and radius that meet are **perpendicular** to each other.

## Rule 6

The **tangents** from **the same point** to a circle are **equal in length**.

AB = BC

Two tangents (a line touching a single point on the circumference) drawn from the same outside point are always equal in length.

## Rule 7

The **angle inscribed in a semicircle** is always a **right angle.**

A triangle drawn with the diameter will always make a 90\degree angle where it hits the circumference.

Another way of saying this is that a diameter ‘subtends’ a right-angle at the circumference.

## Rule 8

**Alternate Segment Theorem**: The angle between the **tangent** and **the side** of the triangle is **equal** to the **opposite interior angle**.

\textcolor{limegreen}{x}=\textcolor{limegreen}{x}

\textcolor{red}{y}=\textcolor{red}{y}

The angle between the tangent and the triangle will be equal to the angle in the alternate segment.

(This is the hardest rule and can be tricky to spot).

**Example 1: Angle in a Semi-Circle**

Below is a circle with centre C.

BD is a diameter of the circle, A is a point on the circumference.

What is the size of angle CBA?

**[2 marks]**

If a question says “show our workings”, you must state what circle theorem/geometry fact you use when you use it.

BD is a diameter of the circle, we know that triangle BAD is confined within the semi-circle.

So we can use **Rule 7**, the angle in a semi-circle is a right-angle to deduce that \angle BAD = 90\degree.

To find CBA, we just need to subtract from 180\degree.

\angle CBA = 180\degree - 23\degree - 90\degree = 67\degree

**Example 2: Combining Rules**

Below is a circle with centre C.

A, B, and D are points on the circumference.

Angle \angle BCD is 126\degree and angle \angle CDA is 33\degree.

Find angle ABC.

You must show your workings.

**[2 marks]**

The first circle theorem we’re going to use here is: **Rule 3**, the angle at the centre is twice the angle at the circumference.

The angle at the centre is 126\degree, so;

\angle BAD = 126\degree \div 2 = 63\degree.

We now know two out of the four angles inside ABCD.

To find a third, simply observe that angles around a point sum to \textcolor{orange}{360\degree}:

360\degree - 126\degree = 234\degree

Since the angles in a quadrilateral sum to \textcolor{orange}{360\degree}, we can find the angle we’re looking for.

\angle ABC= 360\degree - 33\degree - 63\degree - 234\degree = 30 \degree