Geometry · GCSE Maths
Circle theorems: writing reasons
GCSE Maths circle-theorem reasons: write the official phrases on every line — angle in a semicircle, centre twice circumference, alternate segment, cyclic quadrilateral — as a drill beside the theorems hub.
A correct angle without a reason is a guess. Write the theorem in brackets on the same line: “56°, angle at the centre is twice the angle at the circumference.”
The important bits
What you need to know
- 1
Angle in a semicircle: an angle standing on a diameter, at the circumference, is 90°. Phrase: “angle in a semicircle”.
- 2
Angle at the centre is twice the angle at the circumference when both stand on the same arc. Phrase: “angle at centre is twice angle at circumference”.
- 3
Angles in the same segment, standing on the same chord, are equal. Phrase: “angles in the same segment”.
- 4
Opposite angles of a cyclic quadrilateral sum to 180°. Phrase: “opposite angles of a cyclic quadrilateral sum to 180°”.
- 5
A tangent is perpendicular to the radius at the point of contact. Two tangents from an external point are equal. Phrases: “tangent perpendicular to radius”; “tangents from a point to a circle are equal”.
- 6
Alternate segment theorem: the angle between tangent and chord equals the angle in the alternate segment. Phrase: “alternate segment theorem”.
- 7
Two radii make an isosceles triangle: “base angles of an isosceles triangle (radii)”. Ordinary facts still count: “angles in a triangle sum to 180°”, “angles on a straight line”.
- 8
Chain the reasons. Each new angle gets its own line and its own phrase. Three correct numbers with one reason at the bottom usually lose two reason marks.
Quotations worth analysing
Short evidence. Real method.
“Angle in a semicircle”
Use this exact idea when the angle sits on a diameter. “It looks like 90°” scores nothing. Mark the diameter first so the examiner sees why the phrase applies.
“Angle at the centre is twice the angle at the circumference”
The same-arc condition is part of the theorem. If you cannot point to the shared arc, do not double. A reflex centre angle still follows the rule on the major arc.
“Alternate segment theorem”
The angle in the other segment, not the one you are standing in. Shade the two segments in revision so the “alternate” is visible.
Go deeper
Reasons are method marks with official wording
Circle-theorem questions are marked like short proofs. A lonely 74° on the answer line, even if correct, often scores zero on the reason line. Keep a list of phrases and use them almost verbatim. Mix in school geometry where needed: vertically opposite angles, isosceles base angles from two radii. Write each on the line where you use it. Example: AB diameter, C on the circle, angle BAC = 35°. Angle ACB = 90° (angle in a semicircle). Angle ABC = 55° (angles in a triangle). Angle ABC stands on arc AC, so angle AOC = 110° (angle at centre is twice angle at circumference). The caution is the point of this drill page: match the arc before you double.
Go deeper
Same arc, cyclic quads and tangents
Same-segment angles are equal because they look at the same chord from the same side. Opposite angles in a cyclic quadrilateral add to 180°, which is the other side of the same-arc idea: they look at opposite arcs. If you have just found 56° at the circumference, the opposite angle in the cyclic quad is 124°, reason written in full. Tangents: mark 90° where the radius meets the tangent immediately, then look for two equal tangents from a point, hence an isosceles triangle, hence equal base angles. Alternate segment: from the tangent, along a chord; the angle between tangent and chord equals the angle in the far segment on the other side of that chord. It does not equal the angle in the near segment. A wrong but confident “alternate segment” on the near angle is a common full-method, wrong-theorem line. Shade, then quote.
Go deeper
A three-mark chain, written as the exam wants it
O is the centre, A, B, C on the circumference, angle BAC = 31°, and you need angle BOC. Angle BOC and angle BAC stand on arc BC, so angle BOC = 62° (angle at centre is twice angle at circumference). If ACBO is cyclic? A, C, B already on the circle; if a fourth point D is given, use cyclic quad next. Another chain: tangent at A, chord AB, angle between tangent and AB is 40°, angle in the alternate segment (angle ACB) is 40° (alternate segment theorem), then angle at the centre on the same arc is 80°. Each clause is a mark. Do not batch three theorems into one sentence without labelling which angle used which theorem. Number the angles on the diagram (1), (2), (3) and refer to those numbers in the working if the picture is crowded. This page does not replace the theorems hub; it is the sentence-writing drill that hub assumes.
See the idea in action
AB is a diameter, C is on the circumference, angle BAC = 35°. Find angle ABC and, if O is the centre, find angle AOC standing on the same arc as angle ABC. Step 1: Angle ACB = 90° (angle in a semicircle). Step 2: Angle ABC = 180° − 90° − 35° = 55° (angles in a triangle sum to 180°). Step 3: Angle ABC stands on arc AC, so the angle at the centre on arc AC is angle AOC. Step 4: Angle AOC = 2 × 55° = 110° (angle at the centre is twice the angle at the circumference). Note: doubling 35° to get 70° would be the centre angle on arc BC, which is angle BOC, a different arc. Match the arc before you double.
Exam technique
Turn knowledge into marks
Annotate the diagram and write the theorem in brackets beside every angle you find. The answer line is for the number; the working is for the reason. Use the standard phrases, not nicknames.
Common mistakes
Do not give these marks away
- 01
Doubling or halving an angle that does not stand on the same arc as the other angle.
- 02
Quoting the alternate segment theorem for the angle in the same segment rather than the alternate one.
- 03
Giving a correct angle with no reason, which often scores zero on the reason mark even when the number is right.
AB is a diameter and C is on the circumference. The reason that angle ACB = 90° is
Aangle in a semicircle
Bangle at the centre is twice the angle at the circumference
Calternate segment theorem
Dopposite angles of a cyclic quadrilateral sum to 180°
Show the answer
angle in a semicircle. The angle standing on a diameter at the circumference is 90°. Twice-at-the-centre needs a centre angle. Alternate segment needs a tangent. Cyclic quad needs four vertices on the circle and talks about 180°, not 90° automatically.
Quick questions
If this is the bit you searched
Do I have to write the theorem name?
Write a standard reason, not a private nickname. “Angle in a semicircle” or “angle at centre is twice angle at circumference” is what the mark scheme lists.
What is a cyclic quadrilateral?
A four-sided shape whose four vertices all lie on the same circle. Opposite angles sum to 180°. That phrase is the reason you write.
How do I know I am on the same arc?
Both angles must stand on the same chord, looking at the same piece of circumference. If you cannot point to that arc, do not claim the centre-twice or same-segment theorem.
When do I use the alternate segment theorem?
When there is a tangent and a chord from the point of contact. The angle between tangent and chord equals the angle in the other segment of the circle.