The idea, step by step
Plan backwards, write forwards
Identify exactly what must be proved. Ask which theorem would establish it and what facts that theorem needs. Then write the argument starting from the given information, not the desired result.
A proof should name points and corresponding vertices clearly. Statements such as “these are equal” become ambiguous when several angles or sides appear.
Every link needs support
Useful support includes a given condition, a definition, a shared side, an established equality or a named theorem. A diagram alone is not evidence that lengths, parallel lines or right angles are exact.
One unexplained leap can invalidate an otherwise correct result. A counterexample is enough to refute a claim, but several confirming examples do not prove it universally.
A common trap
Using the conclusion as a reason halfway through the proof.
Instead: Mark every fact as given or already established; do not assume the target equality.
See the thinking
A chord midpoint proof
O is the centre of a circle. M is the midpoint of a chord AB that is not a diameter. Prove OM⊥AB.
- OA=OB because they are radii.
- AM=BM because M is the midpoint, and OM is common.
- Triangles OMA and OMB are congruent by SSS.
- Angles OMA and OMB are equal and form a straight angle, so each is 90°.
Cyclic angle proof
Explain why opposite interior angles of a cyclic quadrilateral sum to 180°.
- Each interior angle equals half its opposite intercepted arc.
- For opposite vertices, the two intercepted arcs together make the complete 360° circle.
- Their half-angles sum to 180°.
Check your understanding
Why does one shared side supply equal lengths in a congruence proof?
Show answer
A segment has the same length as itself.
This is a common-side fact, not an additional measurement.
Your turn
6 questions · 16 marks · Work at your own pace.
Read the questions online
- In triangle ABC, AB=AC. D is the midpoint of BC. Prove triangles ABD and ACD congruent.3 marks
- Use the previous conditions to prove AD⊥BC.3 marks
- A proof says “AB=CD because the picture looks symmetric.” Explain why this is insufficient.2 marks
- Prove that an angle in a semicircle is 90° using the centre-angle theorem.3 marks
- Triangles ABC and DEF are similar with correspondence A↔D, B↔E, C↔F. AB=DE. Prove they are congruent.3 marks
- A student verifies a circle claim for three drawings and calls it proved. What is still needed?2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- In triangle ABC, AB=AC. D is the midpoint of BC. Prove triangles ABD and ACD congruent.3 marks
AB=AC; BD=CD; AD is shared.
Apply SSS.
SSS congruence. - Use the previous conditions to prove AD⊥BC.3 marks
SSS gives ∠ADB=∠ADC.
They total 180°, so each is 90°.
The two adjacent angles at D are 90°. - A proof says “AB=CD because the picture looks symmetric.” Explain why this is insufficient.2 marks
A sketch may not be to scale.
Prove symmetry or use a given equality.
Visual appearance is not a stated mathematical condition. - Prove that an angle in a semicircle is 90° using the centre-angle theorem.3 marks
A diameter subtends a 180° central angle.
The circumference angle on the same arc is 90°.
The inscribed angle is half 180°. - Triangles ABC and DEF are similar with correspondence A↔D, B↔E, C↔F. AB=DE. Prove they are congruent.3 marks
AB/DE=1.
All corresponding lengths therefore match, so SSS congruence follows.
Their similarity scale factor is 1. - A student verifies a circle claim for three drawings and calls it proved. What is still needed?2 marks
Three cases do not cover every possible allowed configuration.
A general argument from valid definitions and theorems.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum
Angle and similarity reasoning; congruent/similar triangles in proofs; chords, radii and tangents; angles at the centre and circumference; circle theorems and their applications; connected deductive arguments; construction/spatial algorithms; testing and justifying solutions.
Official curriculum source ↗