PAPER · REASONING · PRACTICE

Squares and inclusive quadrilateral classification

Combine rectangle and rhombus conditions, compare diagonals and reject unsupported converse statements.

40 minutes · Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24

Textbook reference: Year 8, lower volume · Chapter 18: Parallelograms · lessons 23–24

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the lower-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

3 min: Predict the strongest conclusion from three cards: parallelogram plus equal diagonals; plus perpendicular diagonals; plus both. 8 min: Sort the cards using a non-square 4-by-3 rectangle, a non-square rhombus with diagonals 6 and 8, and a square as examples. Label which conditions each actually satisfies. Keep a square inside the rectangle, rhombus and parallelogram families. 5 min: Explain why either diagonal condition alone is insufficient for a square and why both suffice for a parallelogram. Give a counterexample to the stronger unsupported claim. 2 min: Individually complete: a rhombus with one right angle is a ____, because ____.

Organise the key idea

Every square is both a rectangle and a rhombus, and therefore a parallelogram. The reverse implications need extra conditions.

Open the original chapter

Use the lesson question here for paper discussion; the original activity opens separately.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: follow the full 18-minute investigation; record predictions before checking.

Worked example1

ABCD is a rectangle with AB=BC. Prove it is a square.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use the rectangle premise for the right angles.
Use equal opposite sides with AB=BC to obtain four equal sides.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

ABCD is a rhombus with one right angle. Explain why it is a square.

Rhombus ⇒ four ____ sides and two pairs of ____ opposite sides.
One angle is $90^{\circ}$ ; its adjacent angle is $180^{\circ}-\square=\square$ .
Four equal sides + four right angles ⇒ ____.

Geometry diagram; not drawn to scale except for construction tasks.

The rhombus premise already gives four equal sides.
A rhombus is a parallelogram: opposite angles are equal and adjacent angles are supplementary.
The one right angle forces all four angles to be right. Combine that with four equal sides.

A rhombus is a parallelogram with equal sides.
One right angle gives the rectangle condition; combine both families.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Does a square count as a rectangle under the inclusive definition? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Four equal sides do not remove its right angles.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

Does a square count as a rhombus? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The extra right angles do not remove the equal-side property.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

A rectangle has perpendicular diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Retain the rectangle premise, then use the rhombus criterion.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

A rhombus has equal diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use the equal-diagonal criterion together with the parallelogram premise inherited from the rhombus.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

A square has side 4 cm. Find its diagonal exactly and name the right triangle used.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Its diagonal forms a right triangle with two 4 cm legs.
$AC^{2}=4^{2}+4^{2}=32$ ; $AC=4\sqrt{2}$ cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A rectangle has adjacent sides 8 cm and 6 cm. A learner calls it a square because the drawing looks nearly equal. Correct the claim.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use the stated lengths, not measured appearance.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Is every square a parallelogram? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Being more specialised does not exclude the broader family.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Must every rectangle be a square? Give a counterexample.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Right angles do not force adjacent sides to be equal.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

A square has $AC=4\sqrt{2}$ cm. Find BD.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

A square is a rectangle, so its diagonals are equal.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

The diagonals of square ABCD meet at O. Find ∠AOB and name the inherited property.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Use the rhombus diagonal property.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

A square has side 4 cm. Find its area and perimeter, keeping the units distinct.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Area=4×4=16 cm²; perimeter=4×4=16 cm.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Four equal sides are given in a simple quadrilateral. Is that sufficient for a square?

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

A square also requires the right-angle condition.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

A general quadrilateral has equal, perpendicular diagonals. Must it be a square? Test the shown kite and identify a missing condition.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

The vertical diagonal is split into lengths 3 and 1.
Equal and perpendicular without bisection do not give a square.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

A parallelogram has equal and perpendicular diagonals. Prove it is a square by using two different criteria.

Geometry diagram; not drawn to scale except for construction tasks.

Separate the given conditions from the conclusion being requested.
Choose the exact property or criterion; a picture alone cannot supply a condition.
Explain the conclusion using that relation and the original givens.

Apply each criterion with the original parallelogram premise.
Combine right angles and four equal sides.

Check your work

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Preparation and assessment notes

Prerequisites

Parallelograms, rectangles, rhombi and their criteria.

Look for in students’ work

Use the two prerequisite checks to choose support. Project the worked and guided tasks, then collect the eight paper core questions. Follow-ups and extensions are optional; accept other complete valid methods. This page does not save learners’ paper answers.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

Original chapter-core practice after the cited lower-volume textbook lessons. References identify skills; textbook exercises are not reproduced.

大湾区初中数学三维课堂（八年级下册）正文.pdf

Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Squares and inclusive quadrilateral classification

Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

ABCD is a rectangle with AB=BC. Prove it is a square.

Geometry diagram; not drawn to scale except for construction tasks.

Guided attempt2

ABCD is a rhombus with one right angle. Explain why it is a square.

Rhombus ⇒ four ____ sides and two pairs of ____ opposite sides.
One angle is $90^{\circ}$ ; its adjacent angle is $180^{\circ}-\square=\square$ .
Four equal sides + four right angles ⇒ ____.

Geometry diagram; not drawn to scale except for construction tasks.

Prerequisite3

Does a square count as a rectangle under the inclusive definition? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Prerequisite4

Does a square count as a rhombus? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Squares and inclusive quadrilateral classification

Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

A rectangle has perpendicular diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice6

A rhombus has equal diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice7

A square has side 4 cm. Find its diagonal exactly and name the right triangle used.

Geometry diagram; not drawn to scale except for construction tasks.

Independent practice8

A rectangle has adjacent sides 8 cm and 6 cm. A learner calls it a square because the drawing looks nearly equal. Correct the claim.

Geometry diagram; not drawn to scale except for construction tasks.

Exit question9

Is every square a parallelogram? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Exit question10

Must every rectangle be a square? Give a counterexample.

Geometry diagram; not drawn to scale except for construction tasks.

Squares and inclusive quadrilateral classification — Further practice

Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24

Further practice11

A square has $AC=4\sqrt{2}$ cm. Find BD.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice12

The diagonals of square ABCD meet at O. Find ∠AOB and name the inherited property.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice13

A square has side 4 cm. Find its area and perimeter, keeping the units distinct.

Geometry diagram; not drawn to scale except for construction tasks.

Further practice14

Four equal sides are given in a simple quadrilateral. Is that sufficient for a square?

Geometry diagram; not drawn to scale except for construction tasks.

Extension15

A general quadrilateral has equal, perpendicular diagonals. Must it be a square? Test the shown kite and identify a missing condition.

Geometry diagram; not drawn to scale except for construction tasks.

Extension16

A parallelogram has equal and perpendicular diagonals. Prove it is a square by using two different criteria.

Geometry diagram; not drawn to scale except for construction tasks.

Squares and inclusive quadrilateral classification — Teacher answers

Year 8, lower volume · text pp. 45–48 · textbook lessons 23–24

Ordinary route: prerequisites 4, short inquiry 5, worked/guided 8, independent 15, correction 5, exit 3 minutes. Advanced route: inquiry 18, organise 4, selected practice 13, exit 5 minutes.

Prerequisites: Parallelograms, rectangles, rhombi and their criteria.

Ordinary route: use the first fixed case for the five-minute demonstration. Advanced route: complete the full investigation.

3′ Predict the strongest conclusion from three cards: parallelogram plus equal diagonals; plus perpendicular diagonals; plus both.
8′ Sort the cards using a non-square 4-by-3 rectangle, a non-square rhombus with diagonals 6 and 8, and a square as examples. Label which conditions each actually satisfies. Keep a square inside the rectangle, rhombus and parallelogram families.
5′ Explain why either diagonal condition alone is insufficient for a square and why both suffice for a parallelogram. Give a counterexample to the stronger unsupported claim.
2′ Individually complete: a rhombus with one right angle is a ____, because ____.

Worked example1

ABCD is a rectangle with AB=BC. Prove it is a square.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

The rectangle gives right angles and equal opposite sides. AB=BC then makes all four sides equal, so it is a square.

Use the rectangle premise for the right angles.
Use equal opposite sides with AB=BC to obtain four equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Guided attempt2

ABCD is a rhombus with one right angle. Explain why it is a square.

Rhombus ⇒ four ____ sides and two pairs of ____ opposite sides.
One angle is $90^{\circ}$ ; its adjacent angle is $180^{\circ}-\square=\square$ .
Four equal sides + four right angles ⇒ ____.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

It has four equal sides and, as a parallelogram with a right angle, four right angles; it is a square.

A rhombus is a parallelogram with equal sides.
One right angle gives the rectangle condition; combine both families.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite3

Does a square count as a rectangle under the inclusive definition? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Yes; it satisfies the rectangle definition.

Four equal sides do not remove its right angles.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Prerequisite4

Does a square count as a rhombus? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Yes; all four sides are equal.

The extra right angles do not remove the equal-side property.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice5

A rectangle has perpendicular diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

A square: perpendicular diagonals make this parallelogram a rhombus; it is already a rectangle.

Retain the rectangle premise, then use the rhombus criterion.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice6

A rhombus has equal diagonals. State and justify its strongest classification.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

A square: equal diagonals make this parallelogram a rectangle; it is already a rhombus.

Use the equal-diagonal criterion together with the parallelogram premise inherited from the rhombus.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice7

A square has side 4 cm. Find its diagonal exactly and name the right triangle used.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

$4\sqrt{2}$ cm.

4\sqrt{2}

Its diagonal forms a right triangle with two 4 cm legs.
$AC^{2}=4^{2}+4^{2}=32$ ; $AC=4\sqrt{2}$ cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Independent practice8

A rectangle has adjacent sides 8 cm and 6 cm. A learner calls it a square because the drawing looks nearly equal. Correct the claim.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

It is a rectangle, not a square, because adjacent side lengths differ.

Use the stated lengths, not measured appearance.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question9

Is every square a parallelogram? Explain.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Yes; both opposite-side pairs are parallel.

Being more specialised does not exclude the broader family.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Exit question10

Must every rectangle be a square? Give a counterexample.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No. A rectangle with sides 8 cm and 6 cm is a counterexample.

Right angles do not force adjacent sides to be equal.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice11

A square has $AC=4\sqrt{2}$ cm. Find BD.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

$4\sqrt{2}$ cm.

4\sqrt{2}

A square is a rectangle, so its diagonals are equal.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice12

The diagonals of square ABCD meet at O. Find ∠AOB and name the inherited property.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

90°; a square is a rhombus, whose diagonals are perpendicular.

Use the rhombus diagonal property.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice13

A square has side 4 cm. Find its area and perimeter, keeping the units distinct.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Area 16 cm²; perimeter 16 cm. The numerical values agree but their units differ.

Area=4×4=16 cm²; perimeter=4×4=16 cm.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Further practice14

Four equal sides are given in a simple quadrilateral. Is that sufficient for a square?

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No; the shown non-square rhombus has four equal sides but no right angles.

A square also requires the right-angle condition.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension15

A general quadrilateral has equal, perpendicular diagonals. Must it be a square? Test the shown kite and identify a missing condition.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

No. Both diagonals are 4 units and perpendicular, but one does not bisect the other. The parallelogram condition is missing.

The vertical diagonal is split into lengths 3 and 1.
Equal and perpendicular without bisection do not give a square.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.

Extension16

A parallelogram has equal and perpendicular diagonals. Prove it is a square by using two different criteria.

Geometry diagram; not drawn to scale except for construction tasks.

Reference answer

Equal diagonals give a rectangle; perpendicular diagonals give a rhombus. The same quadrilateral belongs to both families and is a square.

Apply each criterion with the original parallelogram premise.
Combine right angles and four equal sides.

Use the stated conditions and justify the relation before calculating.
Give an exact answer where possible, with units and any domain restrictions.