- 1. The Language of the Circle and the Subtended Angle
Identify and name the chord, arc, segment, and sector in a circle figure, distinguish the minor arc from the major arc, and identify and name the angle an arc or chord subtends at the centre and at the circumference.
- 2. The Isosceles Triangle Hidden in Every Circle
Recognise that two radii drawn to the ends of a chord form an isosceles triangle with equal base angles, and calculate an unknown base angle from the apex angle, or the apex angle from a base angle, giving a reason for each step — including short WAEC essay-style questions that combine the radius triangle with angles on a straight line.
- 3. The Angle at the Centre is Twice the Angle at the Circumference
State and apply the theorem that the angle an arc subtends at the centre is twice the angle it subtends at the circumference: halve to go from centre to circumference, double to go the other way, and combine the result with the isosceles radius triangle in short essay-style questions, giving a reason for each step.
- 4. Reflex Angles at the Centre
Decide whether the angle at the centre is the ordinary angle or the reflex angle from where the circumference point sits, switch between them using the 360-degrees-at-a-point rule, and apply the centre theorem correctly when the point lies on the minor arc, giving a reason for each step.
- 5. Angles in the Same Segment are Equal
State the same-segment theorem, recognise two angles standing on the same chord from the same side, transfer a known angle to its equal elsewhere in a figure, and combine the result with the exterior-angle rule and the triangle angle-sum in essay-style questions, giving a reason for each step.
- 6. The Tangent and the Radius: The Right Angle at the Point of Contact
Define a tangent and its point of contact, state that the radius drawn to the point of contact is perpendicular to the tangent, and use that right angle with the triangle angle-sum and the isosceles radius triangle to solve essay-style problems, giving a reason for each step.
- 7. Two Tangents from an External Point are Equal
State that two tangents from the same external point are equal, recognise the symmetry of the kite figure, use the perpendicular radii and the quadrilateral angle-sum to relate the angle at the external point and the angle at the centre, and chain the result with the centre theorem in essay-style problems, giving reasons.
- 8. The Tangent–Chord Angle (Alternate Segment Theorem)
Identify the tangent–chord angle and its alternate segment, state and apply the alternate segment theorem, handle the two-chord version where each tangent–chord angle matches its own alternate angle, and chain the theorem with isosceles triangles and cyclic quadrilaterals in essay-style problems, giving reasons.
- 9. Combining Tangent Theorems with Cyclic Quadrilaterals
Recognise figures that combine a tangent with a cyclic quadrilateral, sequence the alternate-segment theorem, isosceles triangles, and the supplementary-opposite-angles rule in the right order, and write complete, justified solutions to composite essay questions.
- 10. Reading and Annotating the Exam Diagram
Annotate an unfamiliar circle figure systematically, identify the givens and the required angle, select a starting theorem, and write a complete two-column (statement and reason) solution that earns full method marks in the WAEC style.
