Plane geometry proofs sit at the top of the Sec 3 A-Math geometry strand. They make you combine circle theorems, parallel-line angle facts and triangle similarity or congruence into a single logical chain. The topic tests how well you recall and apply results from earlier topics: angle properties, congruence and similarity, and circle properties.
Exam proofs are rarely longer than 3–5 linked steps, but every step must cite the correct reason. Most marks are lost by mis-stating the reason rather than through wrong logic, because Cambridge mark schemes award marks specifically for the reason.
What these notes cover
- •Circle theorem toolkit
- •Radius–tangent and equal tangents
- •Alternate segment theorem
- •Angles in the same segment and angle at centre
- •Angle in a semicircle
- •Cyclic quadrilaterals
- •Stating the reason for every step
- •Similar triangles in circle proofs (AA similarity)
- •Ratios of corresponding sides
- •Worked examples: tangent-chord, cyclic quadrilateral, touching circles
- •Common mistakes
- •Summary
Key definitions
- Radius–tangent property
- A tangent to a circle is perpendicular to the radius drawn to the point of contact.
- Equal tangents
- Two tangents drawn from the same external point to a circle are equal in length.
- Alternate Segment Theorem
- The angle between a tangent and a chord, at the point of contact, equals the angle in the alternate segment. The alternate segment is the one on the other side of the chord from the angle being compared.
- Angles in the same segment
- Angles in the same segment, subtending the same arc, are equal.
- AA similarity
- Two triangles are similar when two pairs of corresponding angles are equal. The third pair follows automatically from the angle sum of a triangle.
Key formulas
Cyclic quadrilateral
∠ABC + ∠ADC = 180°
ABCD is a cyclic quadrilateral: opposite angles are supplementary. An exterior angle also equals the interior opposite angle.
Angle at centre vs circumference
∠AOB = 2∠ACB
O is the centre. Applies only when both angles subtend the same arc AB.
Angle in a semicircle
∠APB = 90°
AB is a diameter and P is a point on the circle.
Similar triangle ratio
For similar triangles ABC and DEF, corresponding sides are in the same ratio. Cross-multiplying gives results of the form PQ × RS = QR × PS.
Worked example
TA is a tangent to a circle at A. B and C are points on the circle, with C on the opposite side of chord AB from T, such that BC is parallel to TA and ∠TAB = 58°. Prove that triangle ABC is isosceles.
- 1∠ACB = ∠TAB = 58° (alternate segment theorem: the tangent-chord angle equals the angle in the alternate segment).
- 2∠ABC = ∠TAB = 58° (alternate angles, BC ∥ TA, transversal AB).
- 3Since ∠ACB = ∠ABC = 58°, two angles of triangle ABC are equal.
- 4By the converse of the isosceles triangle property (sides opposite equal angles are equal), AB = AC, so triangle ABC is isosceles (shown).
Watch out: Track which segment is the alternate segment. It is the one on the other side of the chord from the angle being compared, which is why C must lie on the opposite side of AB from T.
Common mistakes
Trusting the diagram
Wrong: Assuming lines are parallel or triangles are similar because the diagram looks that way.
Right: Prove it. Every step needs a stated theorem.
Misusing the factor of 2
Wrong: Applying 'angle at centre = 2 × angle at circumference' to any two angles in the figure.
Right: The factor of 2 applies only when both angles subtend the same arc.
Vague reasons
Wrong: Writing '(circle theorem)' as the reason for a step.
Right: Name the specific theorem, e.g. '(alt. segment theorem)' or '(∠s in same segment)'. Mark schemes require the specific theorem name.
Questions students ask
What is the alternate segment theorem?
The angle between a tangent and a chord, at the point of contact, equals the angle in the alternate segment. The alternate segment is the one on the other side of the chord from the angle being compared. It is the theorem students most often forget in O-Level and A-Math papers.
How do you prove two triangles are similar in a circle geometry question?
You usually only need two pairs of equal angles (AA similarity), because the third pair follows from the angle sum of a triangle. Common sources are a shared angle, vertically opposite angles, the alternate segment theorem, angles in the same segment, and alternate or corresponding angles when two lines are stated to be parallel.
Do you need to write reasons in geometry proofs?
Yes. A correct angle equation without a bracketed reason earns partial credit at best, because Cambridge mark schemes award marks specifically for the reason. Name the exact theorem rather than writing 'circle theorem'.
How do I start a plane geometry proof?
Work backwards from the statement to be proved to find which theorem will supply the final missing link. If you need to track a variable angle through several steps, let the unknown angle be x°. Nearly every proof uses 2–3 results from the circle theorem toolkit.
In the full PDF
- •Labelled diagrams for the cyclic quadrilateral, angle at centre and angle in a semicircle results
- •Worked example 2: a cyclic quadrilateral with similar triangles, proving EB × EA = EC × ED
- •Worked example 3: two circles touching externally with a common tangent, proving ∠APB = 90°
- •The standard construction for any question with two circles that touch each other
- •The remaining common mistakes, including how students miss the alternate segment theorem
- •A one-page summary of proof strategy