Circle theorems
All eight with a preferred reason, accepted equivalent wording and an example that is too vague. Mark schemes accept equivalent statements that identify the whole theorem; you do not need one sentence word-for-word. Flip on Test yourself to drill them.
Use the preferred wording or an equivalent that identifies the full theorem. Vague labels do not earn the reason mark.
1. Angle at the centre
Preferred: The angle at the centre is twice the angle at the circumference (subtended by the same arc).
Accepted equivalent: The centre angle is double the circumference angle standing on the same arc.
Too vague: “Angle at the centre theorem” — it does not state the relationship.
2. Angle in a semicircle
Preferred: The angle in a semicircle is 90°.
Accepted equivalent: The angle subtended by a diameter at the circumference is a right angle.
Too vague: “Semicircle theorem” — name the right angle.
3. Angles in the same segment
Preferred: Angles in the same segment are equal.
Accepted equivalent: Angles subtended by the same arc at the circumference are equal — or “subtended by the same chord, on the same side of it”.
Too vague: “Angles in a circle are equal” — this names no segment, arc or chord. Beware also “angles subtended by the same chord are equal”: that is not vague but wrong, since a point on the other arc gives the supplementary angle.
4. Cyclic quadrilateral
Preferred: Opposite angles of a cyclic quadrilateral add up to 180°.
Accepted equivalent: Opposite angles in a cyclic quadrilateral are supplementary.
Too vague: “Angles add to 180°” — identify the opposite angles and cyclic quadrilateral.
5. Tangent and radius
Preferred: A tangent is perpendicular to the radius at the point of contact.
Accepted equivalent: The radius meets the tangent at 90° at the point of contact.
Too vague: “Radius and tangent” — it does not state the perpendicular relationship.
6. Two tangents
Preferred: Two tangents drawn from an external point are equal in length.
Accepted equivalent: Tangents from the same external point to a circle have equal lengths.
Too vague: “Tangents are equal” — say they come from the same external point.
7. Alternate segment
Preferred: The angle between a tangent and a chord equals the angle in the alternate segment.
Accepted equivalent: The tangent–chord angle equals the angle in the opposite segment.
Too vague: “Alternate angles are equal” — that names a different parallel-line fact.
8. Perpendicular to a chord
Preferred: The perpendicular from the centre to a chord bisects the chord.
Accepted equivalent: A line from the centre perpendicular to a chord cuts the chord in half.
Too vague: “A perpendicular bisects the chord” — identify that it comes from the centre.
Learn the language of proofs
“State”, “prove”, “show that” — each wants a different kind of answer.
Command words guideFree revision tool by Oscar Song Tutoring
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