Algorithms for Congruence and Similarity — AC9M8SP04

• Translate geometric criteria into ordered decisions.
• Create a flowchart/decision tree for congruence and similarity.
• Trace algorithms using examples, near-matches and insufficient-information cases.
• Debug a geometric algorithm and explain why each branch is necessary.

Recall congruent/similar shapes, corresponding parts, triangle congruence tests, scale factor and the meaning of sequence, decision, condition, branch, output and test case.

An algorithm is an explicit sequence of steps and decisions. For geometric classification, every decision must ask about measurable or stated criteria, not appearance.

A good flowchart separates outcomes such as congruent, similar but not congruent, not established and insufficient information. It should not force a classification when the data are inadequate.

Testing means more than confirming easy examples. Include rotated/reflected shapes, scale factor 1, non-unit scale factors, mismatched correspondence, AAA cases, SSA traps and near-matches that should be rejected.

Preserve the full AC9 elaboration sequence: list criteria, convert triangle congruence/similarity conditions into a sorting algorithm such as a flow chart, then evaluate the algorithm for accuracy and explain how it works.

• An algorithm should always return yes/no. Sometimes the correct result is insufficient information.
• Passing one test case proves correctness. Test contrasting and boundary cases designed to expose faults.
• Correspondence can be ignored. Wrong vertex matching can produce false ratios or angle comparisons.
• A flowchart is only presentation. Branch order and conditions change the mathematical output.

1. List the criteria your algorithm needs for SSS, SAS and AAA.
2. Create a branch distinguishing congruent from similar-but-not-congruent triangles.
3. Trace a 3-4-5 triangle against a 6-8-10 triangle.
4. Test a case with insufficient information and choose an appropriate output.

1. Write pseudocode for an SSS congruence test.
2. Create a flowchart that uses scale factor to identify similarity.
3. Trace your algorithm with two congruent triangles in different orientations.
4. Test a similar pair with scale factor 2.
5. Test an AAA pair and explain the correct classification.
6. Find a bug in an algorithm that accepts SSA as congruence.
7. Design a minimum test suite containing success, failure, boundary and insufficient-information cases.

Two algorithms classify the same pair differently. Algorithm A checks correspondence before side ratios; Algorithm B compares the sides in the order they are listed. Explain which design is more reliable, construct a test case that exposes the weaker algorithm and describe the repair.

1. What makes a geometric algorithm reliable? Explicit valid criteria, correct correspondence and tests covering success and failure cases.
2. Why include insufficient information? Some data do not justify either conclusion.
3. What is debugging? Finding which step/condition causes an incorrect output and repairing it.
4. Why test near-matches? They reveal overly broad criteria that accept shapes they should reject.

1. Design a flowchart to classify triangle pairs as congruent, similar-but-not-congruent, or not established.
   Include correspondence, valid conditions and an insufficient-information route.
2. A proposed algorithm accepts any pair with two equal sides and one equal angle as congruent. Evaluate it and repair the faulty condition.
   Construct an SSA counterexample and replace with SAS/RHS as appropriate.
3. Create four test cases that give strong evidence your algorithm works and explain why each is included.
   Use positive, negative, boundary and insufficient-information cases.

• I translate geometry into decisions.
• I trace every branch.
• I test edge/counterexample cases.
• I can identify and repair a faulty condition.

Exit ticket: Why is “insufficient information” sometimes a more mathematically correct output than “not congruent”?

Australian Curriculum v9.0 — AC9M8SP04: design, create and test algorithms involving sequences of steps and decisions that identify congruency or similarity, and describe how the algorithm works.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8SP04: Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-GEO-C-01; MAO-WM-01: Partial. NSW Stage 4 geometry and Working mathematically supply classification/reasoning content, but there is no single Stage 4 outcome mirroring the national geometric-algorithm requirement.

• AC9M8SP01 — congruence and similarity conditions
• AC9M8SP02 — quadrilateral reasoning

• Australian Curriculum v9.0
• Victorian Curriculum F–10 Version 2.0 Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Design, create and test algorithms involving a sequence of steps...

Mapped skill: design, create and test algorithms involving a sequence of steps and decisions that identify congruency or similarity of shapes, and describe how the algorithm works

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.