The idea, step by step
A test can disprove a claim
A universal statement must hold for every allowed case. One valid counterexample disproves it. Choose an example satisfying the hypotheses, then show that the proposed conclusion fails.
A degenerate triangle or a point not on the stated circle is not a valid counterexample when the original claim excludes it. Check the conditions before rejecting the theorem.
Validate the chain and the answer
Check that units, vertex correspondence and angle regions remain consistent. Recalculate numerical results, but also check the logical theorem conditions. A correct number can arise from an invalid argument.
For an algorithm, test different orientations and sizes, including boundary cases. Explain why the exact method works; measurements and software rounding are secondary checks.
A common trap
Treating many successful numerical tests as a general proof.
Instead: Use tests to locate errors; use a connected theorem-based argument for a universal conclusion.
See the thinking
Disprove a similarity claim
A student claims any two triangles containing a 60° angle are similar. Give a counterexample.
- Use angle sets 60°, 60°, 60° and 60°, 50°, 70°.
- Both contain 60°, but their remaining angles differ.
- They satisfy the premise and fail the similarity conclusion.
Check a chord result
A radius 5 cm circle has an 8 cm chord. A student reports a centre-to-chord distance 4 cm. Check the answer.
- Half-chord is 4 cm.
- A centre-to-chord distance 4 cm would require radius √(4²+4²)=√32, not 5.
- The correct perpendicular distance is √(25−16)=3 cm.
Check your understanding
Does one valid counterexample disprove an “always” statement?
Show answer
Yes.
An always statement must hold for every allowed case.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- Disprove: all quadrilaterals with four equal sides have four right angles.3 marks
- A claimed triangle has sides 3, 4, 8. Explain why it cannot serve as a triangle example.2 marks
- A cyclic quadrilateral has opposite angles 85° and 95°. Is this information consistent?2 marks
- A radius 10 circle has a chord 12 long. A student finds its centre distance 8. Check it.2 marks
- A square-construction algorithm is tested only on horizontal sides. Give another important test and explain.3 marks
- Why must corresponding vertices be named consistently when using triangle similarity?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Disprove: all quadrilaterals with four equal sides have four right angles.3 marks
Its sides are equal but its angles need not be 90°.
A non-square rhombus. - A claimed triangle has sides 3, 4, 8. Explain why it cannot serve as a triangle example.2 marks
3+4=7<8.
It violates the triangle inequality. - A cyclic quadrilateral has opposite angles 85° and 95°. Is this information consistent?2 marks
85+95=180°.
Yes. - A radius 10 circle has a chord 12 long. A student finds its centre distance 8. Check it.2 marks
Half-chord 6.
6²+8²=100=10².
Correct. - A square-construction algorithm is tested only on horizontal sides. Give another important test and explain.3 marks
A correct geometric construction should not depend on page orientation.
This can reveal hidden horizontal/vertical assumptions.
Use a sloping side. - Why must corresponding vertices be named consistently when using triangle similarity?3 marks
A wrong ordering can produce ratios between non-corresponding sides and an invalid answer.
They determine which angles and sides match.
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 ↗