**Revision Notes**

What are circle theorems?

You would have learned a lot of angle facts for your GCSE – angles in polygons, angles with parallel lines. **Circle Theorems** deal with angle facts that occur with shapes and lines drawn within and connected to a circle. There are quite a few of them!

What do I need to know?

Firstly, it helps to be familiar with the names of parts of a circle and lines within and outside of a circle as per the diagram below:

You need to know the following circle theorems:

1.The angle at the circumference in a semicircle is a right angle – see below

2. Angles at the circumference subtended by the same arc are equal – see below

3. Angle subtended by an arc at the centre is twice the angle at the circumference – see below

4. A radius and a tangent are perpendicular

5. Two tangents that meet are equal in length

6. Opposite angles in a cyclic quadrilateral add up to 180°

7. The perpendicular bisector of a chord is a radius

8. Alternate Segment Theorem

These are not in any particular order but generally get more difficult to work with!To solve harder problems you may need to use the angle facts you are already familiar with from triangles, polygons, parallel lines and you may have to use **circumference** and area so ensure you’re familiar with the two formulas: *C=πd and A=πr ^{2}*.

The first three angles theorems …

**1.The angle at the circumference in a semicircle is a right angle**

This theorem should be self-explanatory form its name/title. The semicircle arises if you ignore the right-hand side of the diameter in the diagram above. Look out for triangles hidden amongst other lines/shapes within the circle

**2. Angles at the circumference subtended by the same arc are equal**

This is easier to see that its name! **Subtended** means the equal angles are created by drawing chords from the ends of the arc PQ. Theses chords may or may not pass through the centre.

**3. Angle subtended by an arc at the centre is twice the angle at the circumference**

Similar to above the chords (radii) to the centre and the chords to the circumference are both drawn from (subtended by) the ends of the arc PQ.

This theorem can also happen when the ‘triangle parts’ overlap:

**Example**

**Question**

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