Why prove?
Measuring a few triangles might suggest the angles add to 180°, but measurements are never perfect and can’t cover every triangle. A proof shows that something is true for every case, using deductive reasoning.
The building blocks
| Term | Meaning | Example |
|---|---|---|
| Definition | precise meaning of a word | A right angle measures 90°. |
| Postulate (axiom) | accepted without proof | Through two points there is exactly one line. |
| Theorem | proven statement | The angles of a triangle add to 180°. |
| Given | what you’re told is true | AB ∥ CD |
| To prove | the conclusion you need | ∠1 = ∠2 |
Useful facts for geometry proofs
- Vertically opposite angles are equal.
- Angles on a straight line add to 180°.
- Parallel lines: corresponding and alternate angles are equal; co-interior angles add to 180°.
- Triangle angle sum: 180°.
- Isosceles triangle: base angles are equal.
- Congruence tests: SSS, SAS, ASA, AAS, RHS.
A two-column proof
Prove: Vertically opposite angles are equal.
Given: Lines AB and CD intersect, forming ∠1, ∠2 and ∠3 in order around the point, where ∠1 and ∠3 are vertically opposite.
| Statement | Reason |
|---|---|
| ∠1 + ∠2 = 180° | angles on a straight line |
| ∠2 + ∠3 = 180° | angles on a straight line |
| ∠1 + ∠2 = ∠2 + ∠3 | both equal 180° |
| ∠1 = ∠3 | subtract ∠2 from both sides |
This is a two-column proof: statements on the left, reasons on the right.
A paragraph proof
Prove: The angles of a triangle add to 180°.
Draw triangle ABC and a line through A parallel to BC. The angles on the line at A are ∠x, ∠A and ∠y, which add to 180° (straight line). Since the line is parallel to BC, ∠x = ∠B and ∠y = ∠C (alternate angles). So ∠B + ∠A + ∠C = 180°. ∎
Congruence proofs
Given: AB = AD and BC = DC. Prove: ∠B = ∠D.
| Statement | Reason |
|---|---|
| AB = AD | given |
| BC = DC | given |
| AC = AC | common side |
| △ABC ≅ △ADC | SSS |
| ∠B = ∠D | corresponding angles of congruent triangles |
Coordinate proofs
Place figures on a grid and use slope, distance and midpoint formulas.
Prove the diagonals of rectangle O(0, 0), A(a, 0), B(a, b), C(0, b) are equal:
- OB = √(a² + b²); AC = √(a² + b²) ✓
Tips
- Draw and label a clear diagram.
- Mark given information on it.
- Work backwards from what you need to prove.
- Every statement needs a reason.
Worked examples
Example 1: Reason for "∠A + ∠B + ∠C = 180°" in △ABC? Triangle angle sum theorem.
Example 2: Two triangles share a side and have two other pairs of equal sides. Which test? SSS.
Example 3: Why isn’t measuring a proof? Measurements are approximate and only test specific cases.
Common mistakes
- Assuming what you’re trying to prove.
- Statements without reasons.
- Using a diagram’s appearance ("it looks equal") as a reason.
Try it yourself
- Give the reason: "∠1 + ∠2 = 180° because ∠1 and ∠2 form a straight line."
- Which congruence test uses two angles and the side between them?
- In a coordinate proof, which formula shows two segments are equal?
- What’s the difference between a postulate and a theorem?
Answers: 1) angles on a straight line (supplementary) 2) ASA 3) the distance formula 4) A postulate is accepted without proof; a theorem is proven.
Quick recap
- A proof uses deductive reasoning from given facts, definitions, postulates and theorems.
- Formats: two-column proof, paragraph proof, coordinate proof.
- Every statement needs a reason.