Circle Theorems Worked ExamplesIGCSE Maths 0580

Check both angles stand on the same arc before doubling. The reason must be stated in full, with the words 'centre' and 'circumference' — 'because it is double' does not earn the mark. The March 2026 examiner report found that "Many candidates did not use the correct geometrical language for 'circumference'" when giving this reason.

Two reasons are needed for full marks. On a question like this, the June 2025 examiner report noted that "many candidates offered just one of the two required reasons", and advised that "Candidates would be well advised to use standard wording directly from the syllabus (such as 'angle in a semicircle')". The theorem only applies when the chord is a diameter — it must pass through the centre.

The reason needs three things: 'opposite', 'angles' and 'cyclic quadrilateral'. The June 2025 examiner report found "the reason was usually not acceptable, either because the word 'cyclic' did not accompany quadrilateral or candidates did not use the phrase 'opposite angles'." The March 2025 report also noted that "Some candidates described opposite sides, rather than opposite angles, as summing to 180°." Pair the angles correctly: $P$ with $R$, and $Q$ with $S$.

Two separate tangent facts are needed here. Part (b) is independent of the $54^{\circ}$ — the tangent-radius angle is always $90^{\circ}$, whatever the geometry.

The two angles must stand on the same chord and lie on the same side of it. The March 2025 examiner report noted a small number of candidates who chose the wrong angle at the right point — an angle there that did not stand on the same chord. If $C$ and $D$ were on opposite sides, $ACBD$ would be a cyclic quadrilateral and the angles would sum to $180^{\circ}$ instead of being equal — a completely different relationship.

When a point sits on the opposite arc, the relevant centre angle is the reflex one. Being able to reach $115^{\circ}$ by two independent routes is a strong check — if the two disagree, you have used the wrong arc.

Spotting that two radii make an isosceles triangle is the step most often missed — the equal sides are not marked, so you have to see them. The reason must say why the sides are equal ("radii"), not just "isosceles". In error-spotting questions, say exactly which angle was used wrongly; "he used the wrong theorem" is too vague for the mark.

For the alternate segment theorem, trace the chord from the point of contact: the tangent–chord angle on one side equals the angle in the segment on the other side of the chord. A common error in part (a) is to pick angle $ACB$ or angle $BAD$ instead. In part (d) the whole angle $ABC$ is needed, not just the $25^{\circ}$ part.

In part (a) the reason 'angle between tangent and radius $= 90^{\circ}$' is needed as well as the angle sum of the quadrilateral; the March 2026 examiner report reminded candidates that "When candidates are required to show a given result, they should show every step in their method". A common error in part (b) is to halve the wrong side, writing $\frac{1}{2}(190 - 2x) = 2(x + 15)$. In part (d), $D$ is on the other arc, so the answer is not $55^{\circ}$: angles in the same segment are only equal when the points are on the same side of the chord.

Each part builds on the one before, so keep going even if you are unsure — a given result like the $59^{\circ}$ in part (a) can be used in the later parts even if you cannot show it. On a question like this, the June 2025 examiner report noted that for the final part "around a fifth of candidates did not attempt the question". The same report found that a common wrong answer for the angle at the circumference came from assuming that two angles of the triangle were equal — here angle $PQR$ is not equal to angle $PRQ$, because $PQ \neq PR$. Use the diagram as a rough check: an answer that clearly disagrees with the size of the angle drawn is probably wrong.

The reason in part (a) must say 'cyclic quadrilateral' — the March 2025 examiner report noted that "A small number omitted the word 'cyclic' when describing the quadrilateral." On a June 2025 question that also combined a tangent, a cyclic quadrilateral and parallel lines, the examiner report found that some candidates could not find the intermediate angles they needed. Write each angle on the diagram as you find it: part (c) needs both earlier answers and the parallel lines.

On a similar question, the June 2025 examiner report found that candidates had more difficulty in finding the last of the three angles — here the cyclic quadrilateral $FHJK$ is not drawn as a separate shape, so you have to look for it. The same report described candidates who assumed an angle marked where two chords cross was at the centre of the circle, and halved it. $M$ is not the centre, so the angle-at-the-centre theorem cannot be used with the $80^{\circ}$.