Geometric proof | AQA GCSE Maths Notes

Visual model

Geometric proof links facts in a logical chain

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fact first, conclusion second

Gold-standard guide

26 mins

What you will learn

Build logical arguments using geometric facts.

Use a clear step-by-step method for geometric proof.

Check your answer and avoid the most common exam mistake.

Useful before you start

Core number skillsEarlier geometry and measures skillsShowing clear working

Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step

Step 1

Setup: draw radius OA, OB and OC where AB is the diameter

O is the centre

Step 2

Apply the angle at centre theorem

Angle at centre = 2 × angle at circumference on the same arc

Step 3

Substitute and conclude

180° = 2 × angle ACB → angle ACB = 90°

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through

Proof habit

State the angle or shape fact before using it.

Conclusion

Link every statement to the required result.

Worked example

Prove that the angle in a semicircle is always 90°.

Setup: draw radius OA, OB and OC where AB is the diameter: O is the centre. OA = OB = OC (all radii).Angle AOB = 180° (AB is a straight line through the centre, a diameter).

Apply the angle at centre theorem: Angle at centre = 2 × angle at circumference on the same arc.So angle AOB = 2 × angle ACB.

Substitute and conclude: 180° = 2 × angle ACB → angle ACB = 90°. Therefore the angle in a semicircle is always 90°. ∎

Final answer

Angle ACB = 90°

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Prove that the angle in a semicircle is always 90°.

3 marks4 minsgeometric-proof-worked

Show solution

Worked solution

1.Setup: draw radius OA, OB and OC where AB is the diameter: O is the centre. OA = OB = OC (all radii).Angle AOB = 180° (AB is a straight line through the centre, a diameter).
2.Apply the angle at centre theorem: Angle at centre = 2 × angle at circumference on the same arc.So angle AOB = 2 × angle ACB.
3.Substitute and conclude: 180° = 2 × angle ACB → angle ACB = 90°. Therefore the angle in a semicircle is always 90°. ∎

Final answer

Angle ACB = 90°

Mark points

M1: setup: draw radius oa, ob and oc where ab is the diameter
M1: apply the angle at centre theorem
M1: substitute and conclude
A1: Angle ACB = 90°

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Diagnosticrecall

Prove that opposite angles in a cyclic quadrilateral sum to 180°.

1 mark2 minsgeometric-proof-q1

Show solution

Worked solution

1.Spot the skill: Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.
2.Use the setup: draw radius oa, ob and oc where ab is the diameter stage first, then apply the angle at centre theorem.
3.Keep the final answer visible: Each angle pair subtends arcs summing to 360°; using angle at centre = 2 × circumference angle.

Final answer

Each angle pair subtends arcs summing to 360°; using angle at centre = 2 × circumference angle

Mark points

M1: use the correct geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.never assume what you are trying to prove — build from given facts step by step.always end with a clear concluding sentence.
A1: Each angle pair subtends arcs summing to 360°; using angle at centre = 2 × circumference angle

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Easyprocedure

Prove that tangent-radius angle is 90° using a geometric argument.

2 marks3 minsgeometric-proof-q2

Show solution

Worked solution

1.Spot the skill: Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.
2.Use the apply the angle at centre theorem stage first, then substitute and conclude.
3.Keep the final answer visible: Tangent meets circle at exactly one point; the shortest distance from centre to line is perpendicular.

Final answer

Tangent meets circle at exactly one point; the shortest distance from centre to line is perpendicular

Mark points

M1: use the correct geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.never assume what you are trying to prove — build from given facts step by step.always end with a clear concluding sentence.
A1: Tangent meets circle at exactly one point; the shortest distance from centre to line is perpendicular

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Mediumreasoning

Why is SSA not a valid congruence condition?

3 marks4 minsgeometric-proof-q3

Show solution

Worked solution

1.Spot the skill: Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.
2.Use the substitute and conclude stage first, then setup: draw radius oa, ob and oc where ab is the diameter.
3.Keep the final answer visible: Two different triangles can satisfy SSA — the ambiguous case.

Final answer

Two different triangles can satisfy SSA — the ambiguous case

Mark points

M1: use the correct geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.never assume what you are trying to prove — build from given facts step by step.always end with a clear concluding sentence.
A1: Two different triangles can satisfy SSA — the ambiguous case

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Hardproblem solving

Prove that base angles of an isosceles triangle are equal.

3 marks5 minsgeometric-proof-q4

Show solution

Worked solution

1.Spot the skill: Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.
2.Use the setup: draw radius oa, ob and oc where ab is the diameter stage first, then apply the angle at centre theorem.
3.Keep the final answer visible: Draw axis of symmetry; two congruent triangles by SAS; corresponding angles are equal.

Final answer

Draw axis of symmetry; two congruent triangles by SAS; corresponding angles are equal

Mark points

M1: use the correct geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.never assume what you are trying to prove — build from given facts step by step.always end with a clear concluding sentence.
A1: Draw axis of symmetry; two congruent triangles by SAS; corresponding angles are equal

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Exam-stylemulti-step

Prove that alternate angles are equal using parallel lines.

4 marks6 minsgeometric-proof-q5

Show solution

Worked solution

1.Spot the skill: Geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.
2.Use the apply the angle at centre theorem stage first, then substitute and conclude.
3.Keep the final answer visible: Corresponding angles equal (F-angles); co-interior angles sum to 180°; alternate = 180° minus co-interior = corresponding.

Final answer

Corresponding angles equal (F-angles); co-interior angles sum to 180°; alternate = 180° minus co-interior = corresponding

Mark points

M1: use the correct geometric proofs use angle facts, properties of shapes and circle theorems, stated explicitly at each step.never assume what you are trying to prove — build from given facts step by step.always end with a clear concluding sentence.
A1: Corresponding angles equal (F-angles); co-interior angles sum to 180°; alternate = 180° minus co-interior = corresponding

Watch out

Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

Grade 9 stretchproblem solving

Explain why the diagonals of a rhombus meet at right angles.

4 marks7 minsgeo-proof-g9

Show solution

Worked solution

1.Use the equal sides to form congruent triangles.
2.Use congruence to show adjacent angles at the intersection are equal.
3.Angles on a straight line sum to 180 degrees.

Final answer

The equal adjacent angles each equal 90 degrees

Mark points

C1: use congruent triangles
C1: equal adjacent angles
C1: conclude 90 degrees

Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint

16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Geometric proof - 2 marksProve that opposite angles in a cyclic quadrilateral sum to 180°.Mark answer

Answer

Each angle pair subtends arcs summing to 360°; using angle at centre = 2 × circumference angle

2Angles, lines and polygons - 2 marksThe exterior angle of a regular polygon is 24°. How many sides?Mark answer

Answer

3Properties of shapes - 2 marksName all 2D shapes with equal diagonals that bisect each other.Mark answer

Answer

Rectangle, square

4Perimeter, area and volume - 3 marksFind the perimeter of a rectangle with length 13 cm and width 8 cm.Mark answer

Answer

42 cm

Mastery check

I can explain the method for geometric proof.
I can show clear working without skipping key steps.
I can avoid this mistake: Students write 'it is obvious' or use the result they are trying to prove partway through.Every step must come from a named theorem or axiom. Build the proof like a chain of logic, not a shortcut.

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