Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 3 - Congruence, similarity and quadrilateral properties

Congruence, similarity and quadrilateral properties

Use congruence and similarity conditions and angle or side properties to reason about triangles and quadrilaterals.

Updated 2026-07-26 - 12 min read

Congruence, similarity and quadrilateral properties is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Distinguish congruence and similarity

Congruent figures have the same shape and size; similar figures have equal corresponding angles and proportional corresponding side lengths. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by distinguish congruence and similarity and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Same shape with a different scale is similarity; congruence requires scale factor 1.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Two triangles have corresponding sides 3, 4, 5 and 6, 8, 10. How are they related?

Step 1 - identify the governing idea: Congruent figures have the same shape and size; similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: All side lengths use scale factor 2, so shape is preserved but size differs.

Result: Similar, not congruent

The relationship is visible in the working: All side lengths use scale factor 2, so shape is preserved but size differs. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Congruent — It does not agree with the required relationship: All side lengths use scale factor 2, so shape is preserved but size differs.
  • Neither similar nor congruent — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Equal in area — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Two squares both have side length 7 cm. How are they related?

Step 1 - identify the governing idea: Congruent figures have the same shape and size; similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: They have identical side lengths and angles, so the similarity scale factor is 1.

Result: Congruent and similar

The relationship is visible in the working: They have identical side lengths and angles, so the similarity scale factor is 1. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Similar only — It does not agree with the required relationship: They have identical side lengths and angles, so the similarity scale factor is 1.
  • Congruent only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Neither — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A shape is enlarged by scale factor 1.5. Is the image congruent to the original?

Step 1 - identify the governing idea: Congruent figures have the same shape and size; similar figures have equal corresponding angles and proportional corresponding side lengths.

Step 2 - apply it to this evidence: Lengths change while corresponding angles and proportions remain.

Result: No, but it is similar

The relationship is visible in the working: Lengths change while corresponding angles and proportions remain. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • Yes, because the shape is unchanged — It does not agree with the required relationship: Lengths change while corresponding angles and proportions remain.
  • No, and it is not similar — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only if it is a triangle — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Apply triangle congruence and similarity conditions

Use sufficient corresponding information such as SSS, SAS, ASA/AAS or RHS for congruence and AA or proportional-side conditions for similarity. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by apply triangle congruence and similarity conditions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: AAA fixes shape but not size, so it proves similarity rather than congruence.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Two triangles have all three corresponding sides equal. Which condition proves congruence?

Step 1 - identify the governing idea: Use sufficient corresponding information such as SSS, SAS, ASA/AAS or RHS for congruence and AA or proportional-side conditions for similarity.

Step 2 - apply it to this evidence: Three matching side lengths determine a unique triangle shape and size.

Result: SSS

The relationship is visible in the working: Three matching side lengths determine a unique triangle shape and size. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • AAA — It does not agree with the required relationship: Three matching side lengths determine a unique triangle shape and size.
  • AA — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • SSA always — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Two triangles have angles 40° and 65° in corresponding positions. What can be concluded?

Step 1 - identify the governing idea: Use sufficient corresponding information such as SSS, SAS, ASA/AAS or RHS for congruence and AA or proportional-side conditions for similarity.

Step 2 - apply it to this evidence: Two equal corresponding angles force the third angle and the same shape.

Result: They are similar by AA

The relationship is visible in the working: Two equal corresponding angles force the third angle and the same shape. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • They are congruent by AA — It does not agree with the required relationship: Two equal corresponding angles force the third angle and the same shape.
  • They have equal area — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Nothing can be concluded — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Two right triangles have equal hypotenuse and one equal corresponding leg. Which condition applies?

Step 1 - identify the governing idea: Use sufficient corresponding information such as SSS, SAS, ASA/AAS or RHS for congruence and AA or proportional-side conditions for similarity.

Step 2 - apply it to this evidence: Right angle, equal hypotenuse and equal side determine congruence.

Result: RHS congruence

The relationship is visible in the working: Right angle, equal hypotenuse and equal side determine congruence. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • AAA congruence — It does not agree with the required relationship: Right angle, equal hypotenuse and equal side determine congruence.
  • SSS similarity only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • No condition — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Reason with quadrilateral properties

Classify quadrilaterals using defining side, angle and diagonal properties and use those properties rather than visual appearance. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by reason with quadrilateral properties and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Orientation does not change properties; four right angles define a rectangle regardless of how it is drawn.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Which property is sufficient to identify a parallelogram?

Step 1 - identify the governing idea: Classify quadrilaterals using defining side, angle and diagonal properties and use those properties rather than visual appearance.

Step 2 - apply it to this evidence: This is a defining property of a parallelogram.

Result: Both pairs of opposite sides are parallel

The relationship is visible in the working: This is a defining property of a parallelogram. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • One pair of sides is equal — It does not agree with the required relationship: This is a defining property of a parallelogram.
  • One angle is acute — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The diagonals are different lengths — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A quadrilateral has four equal sides and four right angles. What is it?

Step 1 - identify the governing idea: Classify quadrilaterals using defining side, angle and diagonal properties and use those properties rather than visual appearance.

Step 2 - apply it to this evidence: It satisfies the defining properties of both a rhombus and rectangle, specifically a square.

Result: A square

The relationship is visible in the working: It satisfies the defining properties of both a rhombus and rectangle, specifically a square. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • A kite only — It does not agree with the required relationship: It satisfies the defining properties of both a rhombus and rectangle, specifically a square.
  • A trapezium only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A general parallelogram only — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

In a parallelogram, one interior angle is 68°. Find an adjacent angle.

Step 1 - identify the governing idea: Classify quadrilaterals using defining side, angle and diagonal properties and use those properties rather than visual appearance.

Step 2 - apply it to this evidence: Adjacent angles in a parallelogram are supplementary: 180° − 68° = 112°.

Result: 112°

The relationship is visible in the working: Adjacent angles in a parallelogram are supplementary: 180° − 68° = 112°. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 68° — It does not agree with the required relationship: Adjacent angles in a parallelogram are supplementary: 180° − 68° = 112°.
  • 22° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 292° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Two squares both have side length 7 cm. How are they related?
  2. Two triangles have angles 40° and 65° in corresponding positions. What can be concluded?
  3. A quadrilateral has four equal sides and four right angles. What is it?

Answers

  1. Congruent and similar — They have identical side lengths and angles, so the similarity scale factor is 1.
  2. They are similar by AA — Two equal corresponding angles force the third angle and the same shape.
  3. A square — It satisfies the defining properties of both a rhombus and rectangle, specifically a square.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources