# Circle Theorems and Angles

## Circle Theorems and Angles Revision

**Circle Theorems**

**Circle theorems** are properties of circles that allow us to consider and work out angles within the geometry of a circle.

You need to be able to **identify**, **utilise**, and **describe** each **theorem**.

**Rule 1 – Angles in a Semicircle**

The angle extended from the **diameter** is **always** a **right angle**.

This may be written as ‘a diameter subtends a right-angle at the circumference’.

**Rule 2 – Tangent/Radius Angle**

The angle where a **tangent** meets a **radius** is **always** a **right angle**.

This may be written as ‘the tangent and radius that meet are perpendicular’.

**Rule 3 – Segment Angles**

Angles drawn from the **same chord** are **equal **when touching the circumference.

The lines may or may not pass through the centre.

**Rule 4 – Centre Angle**

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

**Rule 5 – Cyclic Quadrilateral**

A **cyclic quadrilateral** is a four-sided shape within a circle, with **each corner touching** the circumference.

In a **cyclic quadrilateral**, **opposite** angles add to 180\degree.

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

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

**Rule 6 – Alternate Segment Theory**

The angle between the **tangent** to the circle and the side of the triangle is **equal** to the **opposite interior angle**. This is the same for the angle on the other side of the tangent:

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

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

**Rule 7 – Tangents from the Same Point**

**Tangents** from the same point to the circumference are **equal in length**:

AB = BC

**Rule 8 – Perpendicular Bisector of a Chord**

The **perpendicular bisector** to a **chord** will always pass through the **centre** of the circle.

This can be any length chord anywhere in the circle.

**Rule 9 – Equal Chords are Equidistant from the Centre **

Chords of equal length of a circle are **equidistant** from the centre.

Taking a look at the diagram to the left.

AB=CD and are both chords of the same circle.

This means they are both equal distance from the centre

Therefore EO=OF

**Example 1: Using Circle Theorems**

Use one of the circle theorems to calculate the size of angles x and y.

**[3 marks]**

We can use **rule 5** – this is a cyclic quadrilateral, so the opposite angles add to 180\degree.

So x and 118 add to 180,

180 - 118 = x\\ x = 62\degree\\

And y and 124 add to 180,

180 - 124 = y\\ y = 56\degree\\**Example 2: Using Multiple Theorems**

The line AC passes through the centre of the circle.

Use circle theorems to work out the angles x\degree and y\degree

**[3 marks]**

Firstly, we can use **rule 5**, this is a cyclic quadrilateral, so opposite angles add to 180\degree

We can use this to work out y,

180-105=75\degree\\ y=75

Now, we can use **rule 1** – as AC passes through the centre, this is a diameter of the circle, so the angle that extends from the diameter is a right angle. Therefore, B and D are right angles, so,

## Circle Theorems and Angles Example Questions

**Question 1:** AB is a tangent to the circle at point D. BC is a tangent to the circle at point E.

Given that length BD is 6\text{ cm}, work out the length BE.

**[2 marks]**

We know from rule 7 that tangents from the same point are the same length, so BE is also 6\text{ cm}

**Question 2:** Work out the size of angles x and y.

**[2 marks]**

Using rule 3, we know angles from the same chord are equal when touching the circumference. So:

x=16\degree\\ y=22\degree\\**Question 3: **Given that line BG is the perpendicular bisector of AC and EI is the perpendicular bisector to DF, work out the size of angle x.

**[4 marks]**

We know that perpendicular bisectors to chords pass through the centre (rule 8), so the point where EI and BG cross must be the centre of the circle.

This means we can use rule 4, as we have an angle at the centre and an angle at the circumference. Using this rule, we know x will be half the size of 48, and hence, is 24\degree.