## Circle Theorems 1

### Circle Theorems

#### Investigation

### Angles in the same segment are equal

The dotted line divides the circle into two segments. This line is called a chord.

Angles in the same chord as their start and finish and that are in the same segment are equal.

Examples

Find the values of x and y

Angle has the same start and finish as

Angle has the same start and finish as

Find the values x,y and zAngle has the same start and finish as Angle Angle has the same start and finish as

### Angle at the centre is twice the angle at the circumference

Like the theorem above both of these angles need to be in the same segment

#### Proof

Examples

Find the value of x

Since the angle at the centre is twice the angle at the circumference

Find the value of x

Since the angle at the centre is twice the angle at the circumference

### Angle in a semi-circle

The angle in a semi-circle is always

Opposite angles in a cyclic quadrilateral

The opposite angles in a cyclic quadrilateral are supplementary (add up to )

Example

Find the value of

Since PQRS is a cyclic quadrilateral angles in a cyclic quad are supplementary.

angles on a line add up to