G6: Geometric Proof
Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to conjecture and derive results
📋 Key Concepts
Definition: A geometric proof uses logical reasoning and known facts to prove a geometric statement is true.
Methods of Proof:
- Deductive reasoning from known facts
- Congruence proofs (SSS, SAS, ASA, RHS)
- Similarity proofs
- Algebraic methods
📝 Structure of a Geometric Proof
Steps:
- State what you need to prove
- Mark given information on a diagram
- Use known facts to find new information
- State reasons for each step
- Write a clear conclusion
Example 1
Prove that the base angles of an isosceles triangle are equal.
Proof:
Given: Triangle ABC with AB = AC
Construction: Draw the angle bisector of angle A, meeting BC at D
In triangles ABD and ACD:
- AB = AC (given)
- Angle BAD = Angle CAD (AD is angle bisector)
- AD is common
Therefore triangles ABD and ACD are congruent (SAS)
Therefore Angle B = Angle C (corresponding angles in congruent triangles)
📝 Proving Properties of Quadrilaterals
Example 2
Prove that opposite sides of a parallelogram are equal.
Proof:
Given: Parallelogram ABCD with AB ∥ DC and AD ∥ BC
Draw diagonal AC
In triangles ABC and CDA:
- Angle BAC = Angle DCA (alternate angles, AB ∥ DC)
- Angle BCA = Angle DAC (alternate angles, AD ∥ BC)
- AC is common
Therefore triangles ABC and CDA are congruent (ASA)
Therefore AB = CD and BC = AD (corresponding sides in congruent triangles)
📝 Circle Theorem Proofs
Example 3
Prove that the angle at the centre is twice the angle at the circumference subtended by the same arc.
Proof:
Given: Points A, B, C on a circle, with O at the centre
Case: When one side of the angle at the circumference passes through the centre
Join OA and OB
OA = OB (radii), so triangle OAB is isosceles
Let angle OAB = angle OBA = x
Angle AOB = 180° - 2x
Angle at circumference = x
Angle at centre = 180° - 2x = 2(90° - x)
Using similar reasoning for other cases, the angle at the centre = 2 × angle at circumference
📝 Using Similarity in Proofs
Example 4
Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
Proof:
Given: Triangle ABC, D and E are midpoints of AB and AC
In triangles ADE and ABC:
- Angle ADE = Angle ABC (corresponding angles)
- Angle AED = Angle ACB (corresponding angles)
- Angle A is common
Therefore triangles ADE and ABC are similar (AAA)
Since D and E are midpoints, AD/AB = AE/AC = 1/2
Scale factor = 1/2, so DE = (1/2)BC
DE ∥ BC (corresponding angles are equal)
🧮 Worked Examples — Exam Style
Geometric proof = a numbered chain, each line justified by a named fact. The marks are for the REASONS as much as the steps.
Prove that the base angles of an isosceles triangle are equal.
Draw the angle bisector from the apex. The two triangles share: the bisected angle (construction), two equal sides (definition of isosceles), and the common base line.
By SAS the triangles are congruent → corresponding angles equal → base angles equal ∎
M1 for a valid construction, A1 for the congruence, A1 for the conclusion. Naming the criterion (SAS) is the reason mark.
Exam wording to expect: "Prove that …", "give reasons for each stage".
❓ Practice Questions
Q1: What are the four criteria for congruent triangles?
Q2: Prove that the diagonals of a parallelogram bisect each other.
Q3: Prove that vertically opposite angles are equal.
Q4: What does AAA prove instead of congruence?
Q5: State the key steps in writing a geometric proof.
Q6: Prove that angles in the same segment are equal (outline the steps)
Q7: ABCD is a parallelogram. Prove triangle ABC is congruent to triangle CDA.
✅ Answers
- SSS, SAS, ASA, RHS
- Use congruent triangles formed by diagonals to show they bisect each other
- Use angles on a straight line: both pairs of adjacent angles sum to 180°
- AAA proves similarity (same shape, possibly different sizes)
- 1) State what to prove, 2) Mark given info, 3) Use known facts, 4) Give reasons, 5) Write conclusion
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. SSS, SAS, ASA, RHS
Q2. Diagonals AC and BD meet at M. ABM ≅ CDM: AB = CD (opposite sides), alternate angles ABM = CDM and BAM = DCM (parallel lines). ASA → congruent → AM = CM and BM = DM → diagonals bisect ∎
Q3. Angles a + b = 180 (straight line) and b + c = 180 (straight line) → a = c (subtract b) → vertically opposite angles equal ∎
Q4. AAA proves similarity — same angles, any size
Q5. State the claim → mark given information → apply known facts → give a reason per line → conclude
Q6. Join A and B to the centre O. Angle AOB (centre) = 2 × angle at C and 2 × angle at D (angle at centre theorem) → angles at C and D are equal ∎
Q7. AB = CD (opposite sides of parallelogram); angle BAC = angle DCA (alternate, AB ∥ CD); AC is common → SAS → congruent ∎
📌 Key Facts to Memorise
- Congruence criteria: SSS, SAS, ASA, RHS (AAS counts as ASA)
- Proof structure: claim → given → facts → reasons → ∎ (or QED)
- Common reasons: angles on a line (180°), vertically opposite, parallel-line rules, isosceles base angles
- Circle-theorem proofs: angle at centre = 2 × angle at circumference is the root theorem
- AAA = similarity NOT congruence — the classic distractor
🗺️ In the Exam
Geometric proof: 3–5 marks on Paper 1 (Higher-leaning but examined at Foundation grade 5). Every line needs its named reason — bare assertions score zero.
Links to: congruence criteria (G5), circle theorems (G10), angle rules (G3).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Always write "Given:" and "To prove:" at the start
- Use correct notation (AB for a side, ∠ABC for an angle)
- Justify every statement with a reason
- End with a clear conclusion statement
- Draw a clear, labelled diagram
- Practise writing proofs step by step
🧠 Problem-Solving Strategies
Problem-Solving
Structure geometric proofs logically: start with given information, state each step with a reason, and conclude clearly. Use key phrases: "given", "since", "therefore", "hence". Common reasons: angles in a triangle sum to 180°, alternate angles are equal, vertically opposite angles are equal, angles on a straight line sum to 180°.
Multi-Step Problem
Prove that the angle sum of a quadrilateral is 360° by splitting it into two triangles.
Solution: Draw diagonal AC in quadrilateral ABCD. This creates two triangles: ABC and ACD. The angle sum of △ABC = 180° (angles in a triangle). The angle sum of △ACD = 180°. The angles of both triangles together = 360°. The angles of the two triangles make up all four angles of the quadrilateral. Therefore, the angle sum of ABCD = 360°.
⚠️ Common Errors
Watch Out!
1. Wrong: Making a statement without giving a reason (e.g. "∠A = ∠C because they look equal") Correct: Every statement must have a geometric reason, e.g. "∠A = ∠C (base angles of an isosceles triangle)" or "(alternate angles, AB ∥ CD)".
2. Wrong: Using circular reasoning — assuming what you need to prove Correct: Never use the conclusion as a step in the proof. Start only from given facts and known theorems.
3. Wrong: Proving similarity when the question asks for congruence (or vice versa) Correct: Congruence means identical shape AND size. Similarity means same shape but possibly different sizes. Read the question carefully.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: ABCD is a parallelogram. E is the midpoint of AB and F is the midpoint of DC. Prove that (a) △ADE ≡ △BCF, and (b) DE is parallel to BF.
(a) In parallelogram ABCD: AD = BC (opposite sides of parallelogram), ∠DAE = ∠BCF (opposite angles of parallelogram), AE = EB and DF = FC (midpoints), so AE = FC (each is half of AB = half of DC since AB = DC). Therefore △ADE ≡ △BCF by SAS.
(b) Since △ADE ≡ △BCF, ∠ADE = ∠CBF (corresponding angles in congruent triangles). Since ∠ADE and ∠CBF are alternate angles formed by transversal DB crossing DE and BF, and they are equal, DE is parallel to BF (alternate angles equal → lines parallel).
Mark scheme: M1 identifying equal sides/angles, A1 parallelogram properties stated, M1 SAS congruence, A1 proven, M1 corresponding angles, A1 parallel lines conclusion
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student tries to prove that the base angles of an isosceles triangle are equal by drawing the triangle and measuring the angles.
(a) Why is measurement not a valid method of proof?
(b) Outline a proper geometric proof that the base angles of an isosceles triangle are equal.
(c) Another student says "I've shown it works for three different isosceles triangles, so it must be true for all." What is wrong with this argument?
Answers: (a) Measurement has limited precision and only checks specific cases — a proof must work for ALL cases using logical reasoning, not empirical observation. (b) Draw the angle bisector of the apex angle to meet the base at D. Then △ABD ≡ △ACD by SAS (AB = AC given, AD common, ∠BAD = ∠CAD by construction). Therefore ∠ABD = ∠ACD (corresponding angles in congruent triangles). (c) Checking a finite number of examples is not a proof — it provides evidence but doesn't rule out a counterexample. A proof must use logic that applies universally.
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