Circle Theorems: O-Level / IGCSE Maths (0580)

This topic looks at the special angle relationships that arise when straight lines, chords and tangents meet a circle, covering syllabus objectives E4.7 and E4.8. The central idea is a small collection of circle theorems that link angles drawn inside or around a circle to one another, allowing you to find unknown angles by reasoning rather than measurement. The key results are: the angle subtended at the centre is twice the angle subtended at the circumference from the same arc ($\text{angle at centre} = 2 \times \text{angle at circumference}$); the angle in a semicircle is a right angle ($90^{\circ}$); angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to $180^{\circ}$; and the two tangents drawn from an external point are equal in length, each meeting the radius at $90^{\circ}$.

The usual method is to identify which theorem applies to the figure, write down the angle relationship it gives, then combine it with familiar facts such as angles in a triangle summing to $180^{\circ}$, the base angles of an isosceles triangle (often formed by two radii) being equal, or angles on a straight line summing to $180^{\circ}$. Many exam-style questions chain two or three of these steps together, so it helps to label each angle and state the reason at every stage. Always check whether two angles stand on the same arc before equating them, and watch for radii, diameters and tangents, since these unlock the semicircle and tangent results.

The worked examples below are original, written from the syllabus objective to show how each theorem is selected and applied, with every step of the worked solution justified.