Algebraic & Geometric Proof – Grade 10 IGCSE Maths

Welcome to Proof!

Mathematical proof is the art of showing something is always true using logic and algebra — not just for a few examples. At IGCSE Grade 10, you will prove algebraic identities, divisibility results, and geometric facts using rigorous reasoning.

Proof = General argument | Examples only DISPROVE | One counterexample is enough to disprove

Three pillars of proof at IGCSE:
1. Algebraic proof — manipulate expressions to prove an identity or divisibility
2. Counterexample — find one specific case that makes a statement false
3. Geometric proof — use congruence, coordinate geometry, or circle theorems

Algebraic Proof

Number representations, divisibility, identities

Counterexample

Disproving statements with one example

Geometric Proof

Congruence, coordinate, circle, vectors

Worked Examples

6 fully worked proof questions

Proof Builder

Interactive algebraic proof visualiser

Practice (25q)

Self-marking practice questions

Learn 1 — Algebraic Proof

Algebraic proof uses variables to represent any integer, then manipulates expressions to show a property is universally true.

Number Representations

Even number2n (n is any integer)

Odd number2n + 1

Consecutive integersn, n+1, n+2

Consecutive even2n, 2n+2, 2n+4

Consecutive odd2n+1, 2n+3, 2n+5

Multiple of kkn

Proving Divisibility

To prove an expression is divisible by k, show it can be written as k × (some integer).

Example: Prove that the sum of 3 consecutive integers is divisible by 3.
Let the integers be n, n+1, n+2.
Sum = n + (n+1) + (n+2) = 3n + 3 = 3(n + 1)
Since 3(n+1) = 3 × (integer), the sum is always divisible by 3. ✓

Example: Prove the difference of squares of consecutive odd numbers is always divisible by 8.
Let the odds be (2n−1) and (2n+1).
(2n+1)² − (2n−1)² = (4n²+4n+1) − (4n²−4n+1) = 8n
Since 8n = 8 × n, the result is always divisible by 8. ✓

Proving Identities

An identity is true for all values. Expand the LHS (or both sides) until they match.

Example: Prove that (n+1)² − (n−1)² = 4n for all integers n.
LHS = (n²+2n+1) − (n²−2n+1)
= n²+2n+1 − n²+2n−1
= 4n = RHS ✓

"Show that" questions: You must show every algebraic step. Writing "it is obvious" or checking with numbers is NOT acceptable as proof. Write LHS = ... = ... = RHS, giving reasons at each stage.

Key Strategy

Always start by expressing numbers algebraically using the representations above
Expand and simplify — collect like terms
Factorise to show the required divisibility or identity
Write a conclusion sentence: "Since ... = k×(integer), the result is divisible by k"

Learn 2 — Proof by Counterexample

A counterexample is a single specific case that makes a universal statement false. You only need one to disprove — but you cannot prove a statement true using examples.

Critical rule: Examples can only DISPROVE statements, never PROVE them.
"I checked it for n = 1, 2, 3, 4, 5 and it works" is NOT proof that it works for all n.

Classic Counterexamples

Statement: "n² + n + 41 is prime for all positive integers n"
Try n = 41: 41² + 41 + 41 = 1681 + 41 + 41 = 1763 = 41 × 43 → NOT prime
Counterexample: n = 41 gives a composite number, so the statement is FALSE.

Statement: "All square numbers are even"
Counterexample: 9 = 3², and 9 is odd → statement is FALSE.

Statement: "If n > 0 and m > 0 then n/m > 1"
Counterexample: n = 1, m = 3 → 1/3 < 1 → statement is FALSE.

Statement: "n² − n is always divisible by 4"
Try n = 3: 9 − 3 = 6. Is 6 divisible by 4? No → statement is FALSE.

How to Write a Counterexample

State clearly: "Let n = ___ (or x = ___, etc.)"
Calculate: "Then [expression] = ___"
Conclude: "This is not [prime/even/greater than 1/etc.], so the statement is FALSE."

Strategy for finding counterexamples:
• Try small numbers first: 0, 1, 2, 3
• Try negative numbers if the domain allows
• For prime number statements, try multiples of the constant
• For inequality statements, try fractions

Common trap: Students find a value that WORKS and stop — you need one that FAILS. Keep looking until the statement breaks down.

Learn 3 — Geometric Proof

Congruence Criteria (Triangle Proof)

Two triangles are congruent if they satisfy one of these criteria:

SSS Three sides equal SAS Two sides + included angle ASA Two angles + included side AAS Two angles + any corresponding side RHS Right angle, hypotenuse, side

Proof structure:
State matching sides/angles with reasons (e.g. "AB = CD given", "angle ACB = angle ACD — alternate angles"), then name the criterion used.

Coordinate Geometry Proof

Prove a quadrilateral is a parallelogram:
Method 1: Show midpoints of both diagonals are the same point (diagonals bisect each other)
Method 2: Show two pairs of opposite sides have equal gradients (parallel) AND equal lengths

Gradient formula: m = (y₂−y₁)/(x₂−x₁)
Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²]

Prove a triangle is right-angled:
Show that two sides have gradients m₁ and m₂ where m₁ × m₂ = −1 (perpendicular).
Or use the converse of Pythagoras: verify a² + b² = c².

Circle Theorem Proof

Proving angle at centre = 2 × angle at circumference:
Draw OC where O is centre. Triangles OAC and OBC are isosceles (equal radii).
Let ∠OAC = ∠OCA = α and ∠OBC = ∠OCB = β.
Exterior angle of OAC: ∠AOC = 2α. Exterior angle of OBC: ∠BOC = 2β.
Total ∠AOB = 2α + 2β = 2(α + β) = 2 × ∠ACB. ✓

Vector Proof

Prove lines are parallel: Show vector AB = k × vector CD for some scalar k.
Prove collinearity: Show AC = k × AB (same direction, share point A).

Golden rule for geometric proof: Every statement must have a reason in brackets. "AB = DC (opposite sides of parallelogram)", "∠BAC = ∠DCA (alternate angles, AB∥CD)"

Example 1 — Prove n³ − n is divisible by 6

Factorise: n³ − n = n(n² − 1) = n(n−1)(n+1)

Rewrite: n(n−1)(n+1) = (n−1) · n · (n+1) — product of 3 consecutive integers

Among any 3 consecutive integers, one is divisible by 2 and one is divisible by 3

Therefore their product is divisible by 2 × 3 = 6. ✓

Example 2 — Prove (n+2)² − n² = 4(n+1)

Expand LHS: (n+2)² − n² = n² + 4n + 4 − n²

Simplify: = 4n + 4

Factorise: = 4(n + 1) = RHS ✓

Example 3 — Find counterexample to "n² − n + 1 is always prime"

Test n = 1: 1 − 1 + 1 = 1 (not prime, but let's check more)

n = 3: 9 − 3 + 1 = 7 ✓ prime; n = 7: 49 − 7 + 1 = 43 ✓ prime

n = 8: 64 − 8 + 1 = 57 = 3 × 19 → NOT prime

Counterexample: n = 8 gives 57 = 3 × 19, which is composite. Statement is FALSE.

Example 4 — Prove triangle A(0,0), B(4,0), C(2, 2√3) is equilateral

AB = √[(4−0)² + (0−0)²] = √16 = 4

BC = √[(2−4)² + (2√3−0)²] = √[4 + 12] = √16 = 4

CA = √[(0−2)² + (0−2√3)²] = √[4 + 12] = √16 = 4

Since AB = BC = CA = 4, the triangle is equilateral. ✓

Example 5 — Prove ABCD is a parallelogram given A(1,2), B(5,4), C(7,8), D(3,6)

Midpoint of AC = ((1+7)/2, (2+8)/2) = (4, 5)

Midpoint of BD = ((5+3)/2, (4+6)/2) = (4, 5)

Both diagonals share the same midpoint (4,5), so the diagonals bisect each other.

Therefore ABCD is a parallelogram. ✓

Example 6 — Prove triangles ABC and PQR congruent: AB=PQ=5cm, ∠A=∠P=60°, ∠B=∠Q=75°

In △ABC: ∠A = 60°, ∠B = 75°, so ∠C = 180°−60°−75° = 45°

In △PQR: ∠P = 60°, ∠Q = 75°, so ∠R = 45°

AB = PQ = 5 cm (given)

∠A = ∠P = 60° and ∠B = ∠Q = 75° (given)

By ASA (two angles and the included side), △ABC ≡ △PQR. ✓

Common Mistakes in Proof

Mistake 1: Using examples to PROVE
Checking that a statement works for n = 1, 2, 3, 4, 5 does NOT prove it for all integers. You must use a general algebraic argument. Examples only help you DISPROVE (counterexample).

Mistake 2: Skipping algebraic steps
"I can see that both sides are equal" is not acceptable. Show every expansion, every collection of like terms, every factorisation. Examiners award marks for each step shown.

Mistake 3: Wrong counterexample direction
You need a value where the statement FAILS, not one where it holds. If the statement says "always prime" and you find n = 5 gives 29 (prime), you haven't disproven it — keep looking.

Mistake 4: Geometric proof without reasons
"Side AB = side CD" is not enough — you must state WHY: "(given)", "(CPCTC)", "(opposite sides of a parallelogram)", "(radii of circle)", "(alternate angles)". Every assertion needs a justification.

Mistake 5: Proof by exhaustion confusion
Proof by exhaustion checks ALL possible cases (valid for finite sets). Counterexample disproves by finding ONE case that fails. Don't confuse these — for infinite sets you cannot use exhaustion.

Key Formulas & Results

Even number2n

Odd number2n + 1

n consecutive from nn, n+1, …, n+k

Multiple of kk·n

Difference of squaresa² − b² = (a−b)(a+b)

Perfect square(a+b)² = a²+2ab+b²

Distance (coords)√[(x₂−x₁)²+(y₂−y₁)²]

Midpoint((x₁+x₂)/2, (y₁+y₂)/2)

Gradient(y₂−y₁)/(x₂−x₁)

Perpendicular gradientsm₁ × m₂ = −1

Congruence Criteria

Criterion	What you need	Watch out
SSS	All 3 pairs of sides equal	No angle needed
SAS	2 sides + INCLUDED angle	Angle must be between the two sides
ASA	2 angles + INCLUDED side	Side between the two angles
AAS	2 angles + any corresponding side	Any pair of corresponding sides
RHS	Right angle + hypotenuse + one side	Only for right-angled triangles

Proof checklist:
✓ Start from a general expression (not a specific number)
✓ Show every algebraic manipulation step
✓ Factorise to show divisibility
✓ Write a clear conclusion sentence
✓ For geometric: give a reason for every statement

Proof Builder — Algebraic Divisibility

Select a proof type and watch the algebra build step by step.

Proof type:

Select a proof type and click Build Proof to see the step-by-step algebra.

Test a value: n =

Exercise 1 — Divisibility Proofs

Enter the correct algebraic expression at each step. Use the format shown (e.g. 3(2n+1) means 3 times (2n+1)).

Q1. The sum of 4 consecutive integers starting from n is n + (n+1) + (n+2) + (n+3). Simplify this. Enter the factorised form (e.g. 2(2n+3)).

Hint: add them up first, then factorise

Q2. Prove that (2n+1)² − (2n−1)² is always divisible by 8. What is the simplified result?

Hint: expand both squares then subtract

Q3. Show that the product of two consecutive even numbers 2n and 2n+2 is divisible by 4. Write the product in factorised form.

Q4. Prove that n² + n is always even. Write n² + n in fully factorised form.

Hint: factorise, then consider what must always be even

Q5. The sum of 5 consecutive integers n, n+1, …, n+4. What is the simplified sum?

Hint: 5n + ...

Q6. Prove n(n+1)(n+2) is divisible by 6. What are the three consecutive integers (using n)?

Exercise 2 — Prove Identities

Q1. Prove (a+b)² − (a−b)² = 4ab. What does the LHS simplify to?

Q2. Prove (n+3)² − (n+1)² = 4(n+2). Expand (n+3)² and enter the result.

Format: n²+6n+9

Q3. Prove that (2n+3)² − (2n−1)² = 8(n+1). What is the simplified LHS?

Q4. Show that n² − (n−2)² = 4(n−1). What is the LHS when expanded and simplified?

Q5. Prove (a+b+c)² − (a−b−c)² = 4a(b+c). What is the factorised form of the LHS?

Hint: use difference of squares: X²−Y² = (X+Y)(X−Y)

Q6. Prove that 3n² + 6n + 3 = 3(n+1)². What is 3n² + 6n + 3 factorised completely?

Exercise 3 — Find Counterexamples

For each statement, enter a specific integer value of n that disproves it.

Q1. "n² − n is always divisible by 4." Find a counterexample (enter a value of n).

Hint: try small values like 2, 3, 5...

Q2. "2n + 1 is always prime." Find a counterexample value of n (n ≥ 1).

Q3. "n² + 1 is never divisible by 5." Find a counterexample.

Hint: try n = 2, 3, 4, 5, 7...

Q4. "For all integers n, n² ≥ n." Find a counterexample. Enter the value of n.

Hint: think about what happens between 0 and 1... or try a fraction

Q5. "n³ > n² for all positive integers n." Find a counterexample (positive integer only).

Q6. "The sum of two prime numbers is always even." Find a counterexample. Enter the smaller prime.

Hint: 2 is the only even prime...

Exercise 4 — Coordinate Geometry Proofs

Give numerical answers where requested. Round to 2 decimal places where needed.

Q1. A(0,0), B(3,4). What is the length AB?

Q2. A(1,3), B(5,3), C(5,7), D(1,7). What is the midpoint of diagonal AC? Enter as (x,y).

Q3. A(2,1), B(6,3). What is the gradient of AB?

Q4. Line AB has gradient 3/4. What is the gradient of a line perpendicular to AB?

Hint: m₁ × m₂ = −1

Q5. P(0,0), Q(4,2), R(6,6), S(2,4). To prove PQRS is a parallelogram, find the midpoint of PR. Enter as (x,y).

Q6. A(0,0), B(4,0), C(4,3). Is this triangle right-angled? Enter 1 for yes, 0 for no.

Hint: check if any two sides have gradients where m₁ × m₂ = −1, or use Pythagoras

Exercise 5 — Combined & Geometric Proof

Q1. Prove that the sum of an even number and an odd number is always odd. Is the sum even (0) or odd (1)?

Q2. Two triangles share side BC. In △ABС: AB=7cm, BC=9cm, ∠B=52°. In △DBC: DB=7cm, BC=9cm, ∠B=52°. Which congruence criterion applies? Enter: SSS, SAS, ASA, AAS, or RHS.

Q3. Prove n² + 2n is always divisible by 2 if n is even. Write n = 2k and simplify n² + 2n. What is it divisible by? Enter the integer.

Q4. Statement: "For all real x, x² + 1 ≥ 1". Is this true or false? Enter 1 for true, 0 for false.

Q5. A(−1,2), B(3,4), C(5,0), D(1,−2). The midpoint of AC is (2,1). What is the midpoint of BD? Enter as (x,y).

Q6. Using vectors: OA = (2,3) and OB = (6,9). Express OB in terms of OA. OB = k × OA. What is k?

Practice — 25 Questions

Enter 1 for True, 0 for False on T/F questions. Give simplified algebraic or numerical answers elsewhere.

Challenge — 12 Questions

Harder proof reasoning questions.

Exam Style — 5 Questions

Multi-step proof questions in IGCSE exam format.