Understand the idea.
Make each decision precise
An algorithm must give clear decisions and an outcome for every branch. For triangles described by three side lengths, sort each list from shortest to longest, then compare equal-position entries. Equal lists give SSS congruence; one common ratio gives proportional SSS similarity.
If only some information is supplied, select a test that it supports: SAS, ASA/AAS or RHS for congruence, and AA or a valid proportional-side test for similarity. A failed test does not justify discarding inconvenient information or claiming an unsupported result.
| Step | Action |
|---|---|
| 1 | Order each triangle’s side lengths from smallest to largest. |
| 2 | Compare corresponding lengths. If all are equal, report congruent. |
| 3 | Otherwise compare the three matching ratios. |
| 4 | If all ratios are equal, report similar; otherwise report not similar. |
Explore digitally without mistaking evidence for proof
In dynamic geometry, construct the required constraints before dragging. To investigate a parallelogram, enforce both pairs of opposite sides as parallel. A quadrilateral that merely looks right may stop being a parallelogram when a vertex moves.
Repeated measurements are useful evidence for a conjecture, but rounded readings do not prove an exact universal statement. Use dragging to find possible counterexamples; if a property appears consistent, justify it with exact angle facts, congruence or established theorems.
See the method in action.
Follow the ratio algorithm
Apply the side-based process to triangles with side lists 5, 6, 7 cm and 10, 12, 14 cm.
Worked solution
- 1
Compare the sorted side lists.
The entries are already ordered, so shortest, middle and longest sides correspond.
- 2
Test equality of corresponding lengths.
The side sets differ, so the triangles are not congruent.
- 3
Test all corresponding side ratios.
All three ratios must agree for proportional SSS similarity.
- 4
Report the algorithm's conclusion.
The triangles are similar by proportional SSS and have different sizes, so are not congruent.
Similar, not congruent.
Find a faulty decision
An algorithm says: “If two triangles have the same three angles, report congruent.” Explain the flaw and repair it.
Worked solution
- 1
Identify what the angle check establishes.
Angles determine triangle shape but do not determine its size.
- 2
Give a counterexample to the congruence claim.
An enlargement keeps all angles equal while doubling every side, so these triangles satisfy the rule but are not congruent.
- 3
Repair the branch's conclusion.
The revised conclusion follows from AA; a congruence conclusion would need sufficient equal-side information as well.
The rule confuses AAA similarity with congruence; revise the conclusion or add a valid side check.
Try it yourself.
Check your understanding
A side-ratio algorithm finds factors 2, 2 and 3. Can it conclude similarity?
Show answer & reasoning
No.
The three factors are not one common ratio, so proportional SSS fails. With corresponding sides checked consistently, the triangles cannot be similar because similarity would require all three ratios to agree.
Keep practising
6 worksheet questions · 14 marks
Read the questions online
- Apply the side algorithm to triangles with sides 4, 5, 6 cm and 4, 5, 6 cm. State the result.2 marks
- Apply the side algorithm to side lists 3, 4, 5 cm and 6, 8, 9 cm. State the result.2 marks
- A program declares SAS congruence after checking two sides and any angle. What condition is missing?2 marks
- A digital quadrilateral is intended to be a parallelogram. What construction constraints should be used?2 marks
- A student drags ten parallelograms and observes that each pair of diagonals bisects. Is this a universal proof? Explain.3 marks
- Write a short algorithm to decide whether two given triangles are similar when all side lengths are known.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Apply the side algorithm to triangles with sides 4, 5, 6 cm and 4, 5, 6 cm. State the result.2 marks
The three side pairs are equal, so SSS applies.
Congruent, and therefore also similar. - Apply the side algorithm to side lists 3, 4, 5 cm and 6, 8, 9 cm. State the result.2 marks
The ratios 2, 2 and 1.8 do not all match.
Not similar. - A program declares SAS congruence after checking two sides and any angle. What condition is missing?2 marks
SAS requires the included angle, not an arbitrary angle.
The equal angle must lie between the two checked sides. - A digital quadrilateral is intended to be a parallelogram. What construction constraints should be used?2 marks
Visual appearance alone is not a maintained geometric condition.
Both pairs of opposite sides must be constrained parallel. - A student drags ten parallelograms and observes that each pair of diagonals bisects. Is this a universal proof? Explain.3 marks
Ten examples cannot establish all possible parallelograms.
A congruent-triangle argument can justify the general result.
No. It is useful evidence, but a deductive proof is still needed. - Write a short algorithm to decide whether two given triangles are similar when all side lengths are known.3 marks
Check that each list contains valid positive triangle side lengths.
Match smallest/middle/largest.
Calculate all three new-to-original ratios.
Compare all three factors and give the appropriate conclusion.
Order corresponding sides, calculate three matching ratios, and report similar only if all are equal.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum
Congruent shapes and matching parts; triangle congruence tests; similar shapes and corresponding sides; triangle similarity; properties of quadrilaterals established using triangles; testing congruence/similarity with transformations; decision-tree algorithms and digital geometry.
Official curriculum source ↗