Circle Properties

1. Angle in a semicircle. Any triangle inscribed in a semicircle (with one side a diameter) has a RIGHT ANGLE at the opposite vertex.

This means: if you draw a diameter and pick any other point on the circle, the angle there is $90°$.

2. Angle at centre vs angle at circumference. The angle subtended at the centre by an arc is TWICE the angle subtended at the circumference by the same arc: $\angle_{\text{centre}} = 2 \times \angle_{\text{circumference}}.$

3. Angles in the same segment. Two angles subtended at the circumference by the SAME ARC (in the same segment of the circle) are EQUAL.

4. Cyclic quadrilateral. If all four vertices of a quadrilateral lie on a circle, OPPOSITE ANGLES sum to $180°$: $\angle A + \angle C = 180°, \quad \angle B + \angle D = 180°.$

5. Radius and tangent. A radius is PERPENDICULAR to the tangent at the point of contact.

6. Two tangents from a point. Tangents drawn from an external point to a circle are EQUAL IN LENGTH.

Worked example. Triangle inscribed in a circle with AB as diameter. $\angle ACB$ is the third angle. What is it?

• AB is the diameter → C is on the semicircle → $\angle ACB = 90°$ (angle in semicircle).

Worked example. $\angle$ at centre = $80°$. Find the angle at circumference subtended by the same arc.

• Half: $80 / 2 = 40°$.

Worked example. Cyclic quadrilateral with three angles $80°, 95°, 110°$. Find the fourth.

• Use opposite angles sum to $180°$.
• Identify pairs of opposite angles from the quadrilateral.
• For a quadrilateral $ABCD$ in order: $\angle A + \angle C = 180°$, $\angle B + \angle D = 180°$.
• If $\angle A = 80°$ and $\angle C = 110°$, then $80 + 110 = 190 \ne 180$ — so these aren't opposite. Try $\angle A = 80°$, $\angle C = ?$: need $100°$.
• So the fourth angle is the missing $100°$ (opposite the $80°$).

For criterion C, your working must state which theorem you used at each step:

• 'By the angle in a semicircle theorem, $\angle ACB = 90°$.'
• 'By cyclic quadrilateral, opposite angles sum to $180°$.'

This is exactly the style expected at MYP Extended Level.