Angles in a semicircle - Higher

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

Triangle (ABP) within circle, passing through centre, O

Angle APB = 90°

Example

Calculate the angle z.

Triangle (STU) within circle, passing through centre, O

The angle at the circumference in a semicircle is 90°.

Angle STU = 90°

Angles in a triangle add up to 180°.

z = 180 - 90 - 31 = 59^\circ

Proof

The angle on a straight line is 180°. The angle VOY = 180°.

Triangle (VWY) within circle, passing through centre, O

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

Angle VWY = \frac{1}{2} of angle VOY = \frac{1}{2} \times 180.

Angle VWY = 90°