Geometry · GCSE Maths
Circle theorems
Use the GCSE circle theorems with reasons: angle in a semicircle, centre and circumference, cyclic quadrilaterals, tangents and the alternate segment.
Name the theorem as you use it. A diagram without a reason is a guess; a diagram with “angle in a semicircle” is a proof.
The important bits
What you need to know
- 1
The angle in a semicircle, standing on the diameter, is 90°. If AB is a diameter and C is on the circle, angle ACB = 90°.
- 2
The angle at the centre is twice the angle at the circumference when both stand on the same arc.
- 3
Angles in the same segment, standing on the same chord, are equal.
- 4
Opposite angles in a cyclic quadrilateral add to 180°.
- 5
A tangent is perpendicular to the radius at the point of contact. Two tangents from an external point are equal in length.
- 6
Alternate segment theorem: the angle between a tangent and a chord equals the angle in the alternate segment.
- 7
Isosceles triangles appear whenever two radii are drawn: those two sides are equal, so the base angles are equal.
- 8
Write a reason on every line. Marks are for “angle at centre is twice angle at circumference”, not for a lonely 74°.
Go deeper
Same arc, different viewpoints
Several theorems are the same idea in different clothes: they compare angles that stand on the same arc. The angle at the centre is the full turn of that arc as seen from the middle; the angle at the circumference is the same arc as seen from the rim, so it is half. Angles in the same segment are equal because they both stand on the same chord, looking at the same arc. If you cannot point to the arc, you cannot yet claim the theorem. Draw the arc with a highlighter in revision, even if you cannot do that in the exam: it trains the eye. A reflex angle at the centre still follows the rule, but you must be looking at the major arc if the circumference angle sits on that major arc. State which arc you mean.
Go deeper
Tangents create right angles and isosceles pairs
Where a tangent meets a radius, mark 90° immediately. That right angle often unlocks Pythagoras or a trigonometric ratio in a later part of a multi-mark question. Two tangents from a point T to points of contact A and B give TA = TB, so triangle TAB is isosceles, and the line from the centre to T is an axis of symmetry. The alternate segment theorem is the one students can quote and still mis-apply. From the tangent, look along a chord; the angle between tangent and chord equals the angle in the other piece of the circle, the segment on the far side of that chord. It does not equal the angle in the segment you are already standing in. A quick sketch of the two segments, shaded differently, prevents the swap.
Go deeper
Reasons are the method marks
Circle-theorem questions are closer to geometry proofs than to calculator work. A correct angle with no reason often scores zero on the reason line. Keep a short list of official phrases and use them verbatim: “opposite angles of a cyclic quadrilateral sum to 180°”; “tangent is perpendicular to radius”; “base angles of an isosceles triangle are equal” (radii). You may need ordinary facts too: angles on a straight line, vertically opposite angles, angles in a triangle. Chain them. If the centre angle is 112° on arc AB, the circumference angle on the same arc is 56°, reason: angle at the centre is twice the angle at the circumference. Then a cyclic quadrilateral might turn 56° into 124° opposite. Each sentence is a mark.
See the idea in action
AB is a diameter, C is a point on the circle, and angle BAC = 35°. Angle ACB = 90° (angle in a semicircle). In triangle ABC, angle ABC = 180° − 90° − 35° = 55° (angles in a triangle). If O is the centre, angle AOC = 2 × 35° = 70° only if that centre angle stands on the same arc as angle ABC — so check the arc before you double.
Exam technique
Turn knowledge into marks
Annotate the diagram with every angle you find, and write the theorem in brackets beside it. The answer line is for the number; the working is for the reason.
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
Using the alternate segment theorem on the angle in the same segment rather than the alternate one.
- 03
Forgetting that two radii make an isosceles triangle, and treating all three angles as unknown.
AB is a diameter of a circle and C is a point on the circumference. Angle ACB is
A45°
B60°
C90°
D180°
Show the answer
90°. The angle in a semicircle is a right angle. This does not depend on where C sits on the remaining circumference, provided it is not at A or B.
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. Its opposite angles sum to 180°.