Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 2 - Deductive geometric proof

Deductive geometric proof

Construct deductive proofs using definitions, established angle and congruence properties and clearly linked statements.

Updated 2026-07-26 - 12 min read

Deductive geometric proof 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.

Separate givens, definitions and diagram assumptions

Use only stated givens, marked properties, definitions and previously established results; appearance is not evidence. 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 separate givens, definitions and diagram assumptions 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: Parallelism must be stated, marked or proven before parallel-line angle facts apply.

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

A diagram shows two equal-looking sides with no marks. What can be concluded?

Step 1 - identify the governing idea: Use only stated givens, marked properties, definitions and previously established results; appearance is not evidence.

Step 2 - apply it to this evidence: Scale drawings do not establish exact equality.

Result: Nothing about equality from appearance alone

The relationship is visible in the working: Scale drawings do not establish exact equality. 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:

  • The sides are equal — It does not agree with the required relationship: Scale drawings do not establish exact equality.
  • The triangle is isosceles — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The sides are parallel — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Which statement is a definition?

Step 1 - identify the governing idea: Use only stated givens, marked properties, definitions and previously established results; appearance is not evidence.

Step 2 - apply it to this evidence: Equality of the two segments is built into midpoint.

Result: A midpoint divides a segment into two equal lengths

The relationship is visible in the working: Equality of the two segments is built into midpoint. 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:

  • Opposite angles look equal — It does not agree with the required relationship: Equality of the two segments is built into midpoint.
  • All triangles are isosceles — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Parallel lines meet — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

If AB ∥ CD, which information is available?

Step 1 - identify the governing idea: Use only stated givens, marked properties, definitions and previously established results; appearance is not evidence.

Step 2 - apply it to this evidence: Parallel lines permit established transversal angle facts.

Result: Corresponding and alternate angle relationships

The relationship is visible in the working: Parallel lines permit established transversal angle facts. 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:

  • All segment lengths are equal — It does not agree with the required relationship: Parallel lines permit established transversal angle facts.
  • Every angle is 90° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The lines intersect — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Write linked deductive proofs

Each proof step states a conclusion and a valid reason that follows from earlier givens or established facts. 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 write linked deductive proofs 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: The chain must show how each fact advances the required conclusion.

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

Angles on a straight line are 3x and x. Find x.

Step 1 - identify the governing idea: Each proof step states a conclusion and a valid reason that follows from earlier givens or established facts.

Step 2 - apply it to this evidence: 3x+x=180°, so 4x=180° and x=45°.

Result: 45°

The relationship is visible in the working: 3x+x=180°, so 4x=180° and x=45°. 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:

  • 60° — It does not agree with the required relationship: 3x+x=180°, so 4x=180° and x=45°.
  • 90° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 135° — 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 two equal sides and included equal angles. Which congruence reason applies?

Step 1 - identify the governing idea: Each proof step states a conclusion and a valid reason that follows from earlier givens or established facts.

Step 2 - apply it to this evidence: The equal angle lies between the two equal corresponding sides.

Result: SAS

The relationship is visible in the working: The equal angle lies between the two equal corresponding sides. 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:

  • SSA — It does not agree with the required relationship: The equal angle lies between the two equal corresponding sides.
  • AAA — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • RHS — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

If a triangle is isosceles with AB=AC, what angle conclusion follows?

Step 1 - identify the governing idea: Each proof step states a conclusion and a valid reason that follows from earlier givens or established facts.

Step 2 - apply it to this evidence: Base angles opposite equal sides are equal.

Result: ∠B = ∠C

The relationship is visible in the working: Base angles opposite equal sides are equal. 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 = 90° — It does not agree with the required relationship: Base angles opposite equal sides are equal.
  • All angles are equal — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • ∠B + ∠C = 90° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Use converses and counterexamples carefully

A converse reverses implication and requires its own justification; one valid counterexample disproves a universal geometric claim. 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 use converses and counterexamples carefully 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: Reversing a statement can change its truth conditions and must be tested independently.

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

The theorem says 'square ⇒ rectangle'. Is 'rectangle ⇒ square' true?

Step 1 - identify the governing idea: A converse reverses implication and requires its own justification; one valid counterexample disproves a universal geometric claim.

Step 2 - apply it to this evidence: A non-square rectangle is a counterexample.

Result: No

The relationship is visible in the working: A non-square rectangle is a counterexample. 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 — It does not agree with the required relationship: A non-square rectangle is a counterexample.
  • Only for large rectangles — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The statements are identical — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Which condition proves a parallelogram is a rectangle?

Step 1 - identify the governing idea: A converse reverses implication and requires its own justification; one valid counterexample disproves a universal geometric claim.

Step 2 - apply it to this evidence: In a parallelogram, adjacent angles are supplementary and opposite angles equal, so one right angle forces four.

Result: One angle is 90°

The relationship is visible in the working: In a parallelogram, adjacent angles are supplementary and opposite angles equal, so one right angle forces four. 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 equal — It does not agree with the required relationship: In a parallelogram, adjacent angles are supplementary and opposite angles equal, so one right angle forces four.
  • Diagonals intersect — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • One angle is acute — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

How can 'all rhombuses are squares' be disproved?

Step 1 - identify the governing idea: A converse reverses implication and requires its own justification; one valid counterexample disproves a universal geometric claim.

Step 2 - apply it to this evidence: It has four equal sides but not four right angles.

Result: Give a rhombus with a non-right angle

The relationship is visible in the working: It has four equal sides but not four right angles. 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:

  • Draw a square — It does not agree with the required relationship: It has four equal sides but not four right angles.
  • Name a rectangle — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Measure one square — 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. Which statement is a definition?
  2. Two triangles have two equal sides and included equal angles. Which congruence reason applies?
  3. Which condition proves a parallelogram is a rectangle?

Answers

  1. A midpoint divides a segment into two equal lengths — Equality of the two segments is built into midpoint.
  2. SAS — The equal angle lies between the two equal corresponding sides.
  3. One angle is 90° — In a parallelogram, adjacent angles are supplementary and opposite angles equal, so one right angle forces four.

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