# Circle Theorems

This guide has everything you need on using circle theorems. Once you’ve mastered this, you may want to look at how to prove the circle theorems.

The cards below have all the circle theorems you need to know. You need to be able to explain which one you have used so pay attention to the explanations as well the theorems themselves. You can use our quiz to help!

### Theorem by Theorem

Angles in the same segment are equal.

The angle in a semicircle is a right angle

The tangent to a circle is perpendicular to the radius.

Two tangents to a circle from a point are equal.

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

The angle between a tangent and a chord is equal to the angle in the alternate segment. This is called the alternate segment theorem.

The opposite angles of a cyclic quadrilateral add up to 180°.

The perpendicular from the centre to a chord bisects the chord.

The words in bold are the ones you have to include in your reasons to get full marks.  You can download them, for free, right here.

Example
Q, R and S are points on the circumference of a circle with centre O.

OPS = 31°
SP and PR are tangents to the circle.
Find the size of angle SQR.

Give reasons for each stage of your working.

Start by writing down any angles you can spot – each one might open the door for the next one. Don’t worry if you find some angles you don’t need. For each step, start with your statement and then explain why it’s true.

OSP = 90°
The radius OS meets the tangent SP at a right angle.

SOP = 180 – 90 – 31
= 59°
Angles in a triangle add to 180°.

ROP = SOP
The triangles ARP and PSO are congruent.

SOR = 2 × 59
= 118°

SQR = 118 ÷ 2
= 59°

The angle at the centre is double the angle at the circumference.

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