Angles at the Centre and Circumference

Angles at the Centre and Circumference

The angle at the centre is twice the angle at the circumference subtended by the same arc. Angles in the same segment are equal. Match the endpoints of the chord/arc before applying either theorem; geometric diagrams can look similar while subtending different arcs.

Worked reasoning

If angle $AOB$ at the centre is $120^\circ$, any angle $ACB$ on the circumference subtending chord AB is $60^\circ$.

Exam method

1. Highlight the two endpoints of the subtended arc. 2. Locate vertex at centre or circumference. 3. Apply twice/half or equal-same-segment with a stated reason.

Trace the arc, not the picture

When several arcs appear, lightly mark the two endpoints of the chord each angle subtends. An angle $ACB$ stands on arc AB, regardless of where C sits. This is the safest way to decide whether two angles are in the same segment or whether one is twice another. Write the chord/arc letters in your reason if the diagram is crowded. It prevents a correct theorem from being applied to the wrong pair of angles.

One-minute retrieval: Angles at the Centre and Circumference

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.