### 1) Angles at the centre and circumference

The angle **subtended** by an arc at the centre is twice the angle subtended at the **circumference**. More simply, the angle at the centre is double the angle at the circumference.

**Example**

Calculate the missing angles x and y.

*x* = 50° x 2 = 100°, and *y* = 40° x 2 = 80°

**Proof**

On the diagram angle OGK = *x* and angle OGH = *y*.

Angle OGK (*x*) = angle OKG because triangle GOK is **isosceles**. Lengths OK and OG are both radii.

Angle OGH (*y*) = angle OHG because triangle GOH is also isosceles. Lengths OH and OG are also both radii.

Angle GOH = 180°-2*y* (because angles in a triangle add up to 180°).

Angle JOK = 2*x *(because angles on a straight line add up to 180°).

Angle JOH = 2*y* (because angles on a straight line add up to 180°).

The angle at the centre KOH (2*x* + 2*y*) is double the angle at the circumference KGH (*x* + *y*).