A **tangent** to a circle is a line that just touches the circle without crossing it. If we draw a line from the point where the tangent touches the circle to the centre of the circle, this line will make a right angle with the tangent.

From a **diameter** of a circle, if we draw a triangle such that the third point of the triangle is on the circle, this will always be a right-angled triangle.

Use this information to answer questions 2 to 13 below, and then we will look at some more advanced properties of the circle which are part of the extended mathematics syllabus below.

**Exercise**: Find the size of the angles marked by letters.

The answers are below:

(2.) 90º, (3.) 65º, (4.) 45º, (5.) 90º, (6.) e = 40º, f = 50º, (7.) g = 30º, (8.) h = 90º, i = 60º, (9.) j = 49º, (10.) 45º, (11.) l = 60º, m = 50º, (12.) n = 40º, (13.) 50º

**Extended Work: Circle Theory**

**Circle Theorem 1**: If we draw any **chord** in a circle, the angle that it **subtends** at the centre of the circle will be twice as large as the angle it subtends on the circle.

**Circle Theorem 2**: If we draw any chord in a circle, the angles that it subtends on the circle are all equal to each-other.

**Exercise**

Let’s use this information to find the size of the angles marked with letters in exercise 11 on pages 159 and 160 of the extended textbook:

The answers are below:

**Circle Theorem 3**: In a cyclic quadrilateral, each pair of opposite angles sums to 180º

⍺+ɣ=180º and β+𝜹 = 180º

**Circle Theorem 4**: The angle in a semicircle is a right-angle (we already know this from above)

**Exercise**

Let’s apply this knowledge to calculate the size of the angles marked with letters in exercise 12 on pages 161 to 162 of the extended textbook:

The answers are below:

**Circle Theory 5**

(Already considered in core work above):

**Circle Theory 6**

**Exercise**

**Answers**

**Circle Theory 7**

**Exercise**

**Answers**

Linked here are a selection of 12 circle theory exam question with mark scheme detailed below.