Writing a geometric proof
State a reason for each angle or length equality, then connect them to the required result.
A diagram suggests relationships but does not establish them. Start with the stated facts: equal radii, a tangent, parallel lines, a cyclic quadrilateral or a midpoint. Use a theorem only after checking its conditions. Keep angle labels unambiguous by giving all three letters, with the vertex in the middle.
Congruence proves corresponding lengths and angles equal. Similarity gives equal angles and proportional corresponding lengths. When proving a circle theorem, do not use that same theorem as a step in the proof. A proof applies to every configuration satisfying its stated assumptions, so a numerical angle measurement is only a check.
Worked example
A and B lie on a circle with centre O. Prove that OA and OB make equal angles with chord AB.
- OA = OB because both are radii of the same circle.
- Triangle OAB is therefore isosceles.
- The base angles of an isosceles triangle are equal.
Answer: Angle OAB = angle ABO
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Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 8365 · core · 6.2: Geometric proof
- 7M20 · core · G4.1: Geometric proof
Next practice: Circle theorems and angle reasoning, Sine rule, cosine rule and triangle area.