044i - Reasoning About Shape Properties

Lesson 44 — Reasoning About Relationships Between Shape Properties

Strand: Space | Descriptor: AC9M7SP02 | Duration: 45 minutes

Learning Intentions

• To reason about relationships between the properties of shapes.
• To use counterexamples to test and disprove general claims about shapes.

Success Criteria

I can:

1. Decide whether a general statement about shapes is always, sometimes or never true.
2. Produce a counterexample to disprove a false claim.
3. Explain why one property forces, or fails to force, another.
4. Use a Venn diagram to represent overlapping shape families.

Warmup

(6 minutes — always, sometimes, never; pairs)

Decide for each, and be ready to justify.

1. A rhombus has four right angles.
2. A square has four equal sides.
3. A parallelogram has equal diagonals.
4. A triangle has three lines of symmetry.

Answers: 1. Sometimes — only when it is a square. 2. Always — by definition. 3. Sometimes — only when it is a rectangle. 4. Sometimes — only when it is equilateral.

The key idea to name: to show a claim is not always true, you need only one counterexample. To show it is always true, you must give a reason covering every case.

Activities

Activity 1 — Explicit Instruction: Counterexamples (12 min)

The principle:

A single counterexample destroys a general claim. No number of supporting examples proves one.

This is the same logic as Lesson 30’s algebraic proofs, now in a geometric setting.

I do. Claim: “If a quadrilateral has four equal sides, it must be a square.”

Counterexample: a rhombus with angles . All four sides are equal, yet it is not a square. Claim disproved.

I do. Claim: “If a quadrilateral has four right angles, it must be a square.”

Counterexample: a rectangle cm by cm. Four right angles, but not a square. Claim disproved.

I do — a true claim needs an argument, not an example. Claim: “Every square is a parallelogram.”

Reason: a parallelogram is defined by having two pairs of parallel sides. A square has two pairs of parallel sides. Therefore every square satisfies the definition. Claim proved — and note that no drawing was needed.

We do — always, sometimes or never? Justify each:

1. A kite has two pairs of parallel sides.
2. A rectangle has two lines of symmetry.
3. An isosceles triangle has a right angle.
4. A regular polygon is convex.
5. A trapezium has equal diagonals.

(Answers: 1. Never (except in the special case where the kite is a rhombus — worth debating). 2. Always, unless it is a square, which has four. 3. Sometimes — the triangle. 4. Always. 5. Sometimes — only when it is isosceles.)

Discussion of Q1. A rhombus is a kite (two pairs of adjacent equal sides) and does have parallel sides. So the honest answer is “sometimes”. This kind of edge case is exactly why precise definitions matter.

Activity 2 — Venn Diagrams of Shape Families (12 min)

Making the hierarchy from Lesson 42 visual.

I do. Draw two overlapping circles labelled Rectangles and Rhombuses, inside a larger region labelled Parallelograms.

• What sits in the overlap? (Squares — they are both.)
• What sits in Rectangles only? (Non-square rectangles.)
• What sits in Rhombuses only? (Non-square rhombuses.)
• What sits in Parallelograms but neither circle? (Parallelograms with no right angles and unequal sides.)

You do — place each shape in the correct region:

square, rectangle , rhombus with angle, parallelogram with sides and and no right angles, rhombus with angles.

(Answers: square → overlap; rectangle → Rectangles only; rhombus → Rhombuses only; the parallelogram → outside both circles; rhombus with angles → this is a square, so the overlap.)

Second Venn — triangles. Draw circles for Isosceles and Right-angled triangles.

• Overlap: the triangle.
• Isosceles only: e.g. .
• Right-angled only: e.g. .
• Outside both: e.g. .

Where do equilateral triangles sit? (Entirely inside Isosceles, under the “at least two equal sides” convention from Lesson 41, and never in the overlap.)

Activity 3 — Inquiry: Property Implications (10 min)

Pairs. The reasoning task of the lesson.

For each pair of properties, decide whether the first forces the second. If it does, explain why. If not, give a counterexample.

1. All sides equal all angles equal
2. All angles equal all sides equal
3. Four right angles opposite sides parallel
4. Diagonals bisect at all sides equal
5. Two pairs of parallel sides opposite angles equal
6. Equilateral acute-angled (triangles)

Socratic scaffolding for Q4:

Prompt	Purpose
Understand: what does “forces” mean?	Whenever the first is true, the second must also be true — with no exceptions.
Which shapes have diagonals bisecting at ?	Check the Lesson 42 table: rhombus and square.
What about a kite?	A kite’s diagonals cross at , but only one bisects the other.
So does the property as stated include kites?	No — “bisect” means each cuts the other in half, which excludes the general kite.
Do all rhombuses have equal sides?	Yes, by definition. And squares are rhombuses.
So is the implication true?	Yes — the property picks out exactly the rhombus family.
Looking back	Precise wording changed the answer. “Cross at ” would have admitted kites and made it false.

Answers:

1. No. A rhombus has equal sides but unequal angles.
2. No. A rectangle has equal angles but unequal sides.
3. Yes. Four right angles force both pairs of opposite sides to be parallel — the shape is a rectangle.
4. Yes. Diagonals that bisect each other at right angles characterise the rhombus family, all of which have equal sides.
5. Yes. In a parallelogram, opposite angles are always equal.
6. Yes. Equal angles totalling must each be , which is acute.

Checks for Understanding

(5 minutes — exit ticket)

1. Always, sometimes or never: “A rhombus has four right angles.” Justify.
2. Give a counterexample to: “Every quadrilateral with equal diagonals is a rectangle.”
3. Reasoning. Explain why “all sides equal” does not force “all angles equal” for quadrilaterals, but does for triangles.
4. In a Venn diagram of Rectangles and Rhombuses, what sits in the overlap?
5. True or false, with a reason: “Some kites are rhombuses.”

Answers: 1. Sometimes — only when the rhombus is a square; 2. An isosceles trapezium has equal diagonals but is not a rectangle; 3. A triangle is rigid — fixing three side lengths fixes the angles completely. A quadrilateral is not: a square can be “pushed over” into a rhombus, keeping all sides equal while the angles change; 4. Squares; 5. True — a rhombus has two pairs of adjacent equal sides, satisfying the kite definition.

Common Misconceptions

Misconception	How to pre-empt it
Believing several supporting examples prove a claim.	Repeat the principle: examples can only disprove. Proof needs an argument.
Giving an example instead of a counterexample.	A counterexample must satisfy the first condition and fail the second. Check both.
Assuming implications work both ways.	”Every square is a rhombus” is true; the reverse is not. Test each direction separately.
Overlooking edge cases such as “a rhombus is a kite”.	Return to definitions rather than mental pictures.
Treating “sometimes” as a way of avoiding commitment.	A “sometimes” answer must name the case where it holds and the case where it fails.
Drawing Venn circles as disjoint when families nest.	Model the nesting explicitly with the parallelogram diagram.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of these is always true?

(A) Every rhombus is a square. (B) Every square is a rhombus. (C) Every rectangle is a rhombus. (D) Every parallelogram is a rectangle.

Answer

(B) — a square has four equal sides and two pairs of parallel sides, which is exactly the rhombus definition. The others each fail on a counterexample.

E2 (AMC Junior style). A quadrilateral has two pairs of parallel sides and diagonals of equal length. What must it be?

Answer

Two pairs of parallel sides make it a parallelogram; equal diagonals make it a rectangle (possibly a square).

E3 (Challenge). Give a counterexample to each false claim:

(a) Every triangle with two equal angles is equilateral.

(b) Every polygon with equal sides is regular.

(c) Every quadrilateral with perpendicular diagonals is a rhombus.

Answer

(a) A triangle with angles is isosceles but not equilateral.

(b) A rhombus with a angle has equal sides but unequal angles.

(c) A kite with unequal side pairs has perpendicular diagonals but is not a rhombus.

E4 (Challenge). A shape belongs to all three families: parallelogram, kite and rectangle. What must it be? Justify.

Answer

Rectangle gives four right angles; kite gives two pairs of adjacent equal sides, which combined with the parallelogram property forces all four sides equal. A rectangle with four equal sides is a square.

E5 (Reasoning challenge). Explain why a triangle is “rigid” but a quadrilateral is not, and why this matters in construction.

Answer

Three side lengths determine a triangle’s shape completely — the angles cannot change without changing a side. A quadrilateral with fixed side lengths can still flex, as a square deforms into a rhombus. This is why bridges, roof trusses and scaffolding are braced with triangles: the triangle holds its shape under load.

Homework

1. Always, sometimes or never? Justify each: (a) a square is a rhombus (b) a rhombus is a square (c) a parallelogram has a line of symmetry (d) a regular polygon is convex (e) an isosceles triangle is obtuse.
2. Give a counterexample to each false claim: (a) Every quadrilateral with four equal sides is a square. (b) Every triangle with a right angle is isosceles. (c) Every polygon with all angles equal is regular.
3. Explain why each of these is always true: (a) Every square is a parallelogram. (b) Every equilateral triangle is acute-angled. (c) Every rectangle has two lines of symmetry or more.
4. Draw a Venn diagram for Rectangles and Rhombuses within Parallelograms, and place: square, rectangle , rhombus with angle, parallelogram with sides and and no right angles.
5. Draw a Venn diagram for Isosceles and Right-angled triangles, and give an example for each region, including outside both.
6. Reasoning. Does “diagonals cross at right angles” force “all sides equal”? Explain.
7. Reasoning. Explain why one counterexample is enough to disprove a claim, but a hundred examples are not enough to prove one.
8. Challenge. A quadrilateral has exactly one line of symmetry and no parallel sides. Name it and describe its properties.

Answers: Q1 — (a) always (b) sometimes (c) sometimes; only if it is a rectangle or rhombus (d) always (e) sometimes. Q2 — (a) a rhombus with a angle (b) a –– triangle (c) a rectangle. Q3 — (a) it has two pairs of parallel sides (b) equal angles totalling are each (c) four right angles and equal opposite sides give two lines of symmetry, and four if it is a square. Q6 — no; a kite has perpendicular diagonals but generally unequal side pairs. Q7 — a general claim asserts something about every case, so one failure refutes it; but examples can never exhaust infinitely many cases, so proof requires an argument. Q8 — a kite: two pairs of adjacent equal sides, one pair of equal opposite angles, and diagonals crossing at with one bisecting the other.