The idea, step by step
Make each decision precise
An algorithm is a sequence of clear instructions and decisions. To compare triangles, begin by matching vertices or sorting corresponding sides consistently. Then test the information actually supplied: SSS, SAS, ASA/AAS or RHS for congruence; AA or common corresponding-side ratios for similarity.
A flowchart branch must specify what happens when a condition fails. If the side ratios are not all equal, the algorithm must reject similarity by proportional SSS instead of ignoring the inconvenient ratio.
| 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
A dynamic geometry tool can help test an idea by constructing a shape with constraints and dragging it. To investigate parallelograms, construct both pairs of opposite sides parallel, then draw and measure diagonals. Merely drawing something that looks parallel does not preserve the required conditions when dragged.
Repeated measurements suggest whether a claim may hold, but rounding and selected examples cannot prove a universal theorem. After testing, justify the claim using angle facts, congruent triangles or another valid chain of reasoning.
A common trap
Treating a diagram measurement rounded to the nearest degree as exact proof.
Instead: Measurements test examples; a proof must use exact givens and valid mathematical deductions.
See the thinking
Follow the ratio algorithm
Apply the side-based process to triangles with side lists 5, 6, 7 cm and 10, 12, 14 cm.
- The ordered lists are already matched. They are not equal, so congruence by SSS does not apply.
- The ratios are , , .
- All ratios match. Report similar with scale factor 2, but 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.
- Equal angles fix shape but not size.
- A 3–4–5 triangle and its 6–8–10 enlargement have equal angles and different lengths.
- Replace “congruent” with “similar”, or require sufficient equal side information before reporting congruent.
Check your understanding
A side-ratio algorithm finds factors 2, 2 and 3. Can it conclude similarity?
Show answer
No.
All three corresponding-side factors must be equal for proportional SSS.
Your turn
6 questions · 14 marks · Work at your own pace.
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 ↗