PAPER · REASONING · PRACTICE

Parallelograms: properties, criteria and midsegments

Separate given facts from conclusions, use diagonals carefully, and prove the triangle midsegment theorem.

40 minutes · Year 8, lower volume · text pp. 27–36 · textbook lessons 14–18

Textbook reference: Year 8, lower volume · Chapter 18: Parallelograms · lessons 14–18

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 which of the four half-diagonals in a parallelogram must be equal. Must all four be equal? 8 min: Draw A(0,0), B(4,0), C(6,2), D(2,2); join the diagonals and find their common midpoint. Compare a trapezium with vertices (0,0),(4,0),(3,2),(1,2), whose diagonals are equal but do not bisect each other. In a separate triangle, join two side midpoints and predict the new segment’s direction and length. 5 min: Separate “each diagonal is bisected” from “the diagonals are equal”. Use stated parallelism and triangle congruence or an established criterion to justify the claim; measurement alone is not proof. 2 min: Individually state the two conditions and length relation for a triangle midsegment.

Organise the key idea

A parallelogram has two pairs of parallel opposite sides; its diagonals bisect each other. One pair of equal opposite sides alone is insufficient.

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 parallelogram and AB=8 cm. Find CD and name the 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.

Opposite sides of a parallelogram are equal.
CD=AB=8 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.

Guided attempt2

ABCD is a parallelogram. Its diagonals meet at O and AO=5 cm. Find OC.

The diagonals of a parallelogram ____ each other.
$AO=OC$ , so $OC=\square$ cm.
Check the whole diagonal: $AC=AO+OC=\square$ cm.

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

AO and OC are the two parts of the same diagonal AC.
The parallelogram condition makes O the midpoint of AC, so the two parts are equal.
Copy the given half-length 5 cm to the other half; do not halve 5 again.

The diagonals bisect each other, so AO=OC.
Thus OC=5 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.

Prerequisite3

AB∥CD and AD∥BC. Which quadrilateral family follows directly from the definition?

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.

Both pairs of opposite sides are parallel.

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

D is the midpoint of AB and AB=10 cm. Find AD.

Use the midpoint definition.
AD is half of AB.
Divide 10 by 2.

A midpoint divides the whole into two equal parts: 10/2=5.

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

ABCD is a parallelogram with ∠A=60°. Find ∠B and explain the angle relation.

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+∠B=180°.
∠B=180°−60°=120°.

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

Both pairs of opposite sides of ABCD are equal. State the criterion and conclusion.

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 both equal-side pairs; neither pair is omitted.

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

AB∥CD and AB=CD. Is ABCD a parallelogram? Identify why the same pair matters.

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 parallel and equality conditions concern AB and CD together.

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

D and E are midpoints of AB and AC. Extend DE to F so DE=EF. Prove DE∥BC and DE=BC/2 using congruence and a parallelogram criterion.

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.

AE=EC, DE=EF, and ∠AED=∠CEF are vertical angles; apply SAS.
Corresponding sides and angles give AD=CF and AD∥CF. Since DB=AD, DB and CF are parallel and equal.
Use the parallelogram DBCF to obtain DF∥BC and DF=BC; DE=EF then gives DE=BC/2.

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

The diagonals of ABCD bisect each other at O. What follows?

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 criterion involving both diagonal midpoint conditions.

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

Only AD=BC is given. Must ABCD be a parallelogram? Use the shown isosceles trapezium as 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.

The legs AD and BC are equal, but AB and CD have different lengths.

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

ABCD is a parallelogram with AD=5 cm. Find BC.

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.

BC=AD because they are opposite 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.

Further practice12

A rectangle ABCD has diagonal AC=10 cm. Its diagonals meet at O. Find AO using a parallelogram 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.

A rectangle is a parallelogram, so O bisects AC.
AO=10/2=5 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 practice13

D,E are midpoints of AB,AC and BC=12 cm. Find DE and its direction relative to BC.

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 triangle midsegment is parallel to the third side and half its length.

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

D,E are midpoints of two triangle sides and DE=7 cm. Find the third side BC.

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.

DE=BC/2, so BC=2×7=14 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.

Extension15

A parallelogram has diagonals AC=BD=10 cm, meeting at O, and AB=8 cm. Are the diagonals perpendicular? Test this using Pythagoras. Then state an extra side condition that would give a rhombus.

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 parallelogram gives two diagonal midpoint conditions.
The proposed right triangle would have legs 5 and 5 but hypotenuse 8, contradicting its square relation.

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

AC bisects BD, but AO≠OC. Must ABCD be a parallelogram? Explain using the shown kite.

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.

BO=OD=2, while AO=3 and OC=1.

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

Parallel lines, triangle congruence and midpoints.

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. 27–36 · textbook lessons 14–18

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

Parallelograms: properties, criteria and midsegments

Year 8, lower volume · text pp. 27–36 · textbook lessons 14–18 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

ABCD is a parallelogram and AB=8 cm. Find CD and name the property.

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

Guided attempt2

ABCD is a parallelogram. Its diagonals meet at O and AO=5 cm. Find OC.

The diagonals of a parallelogram ____ each other.
$AO=OC$ , so $OC=\square$ cm.
Check the whole diagonal: $AC=AO+OC=\square$ cm.

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

Prerequisite3

AB∥CD and AD∥BC. Which quadrilateral family follows directly from the definition?

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

Prerequisite4

D is the midpoint of AB and AB=10 cm. Find AD.

Parallelograms: properties, criteria and midsegments

Year 8, lower volume · text pp. 27–36 · textbook lessons 14–18 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

ABCD is a parallelogram with ∠A=60°. Find ∠B and explain the angle relation.

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

Independent practice6

Both pairs of opposite sides of ABCD are equal. State the criterion and conclusion.

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

Independent practice7

AB∥CD and AB=CD. Is ABCD a parallelogram? Identify why the same pair matters.

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

Independent practice8

D and E are midpoints of AB and AC. Extend DE to F so DE=EF. Prove DE∥BC and DE=BC/2 using congruence and a parallelogram criterion.

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

Exit question9

The diagonals of ABCD bisect each other at O. What follows?

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

Exit question10

Only AD=BC is given. Must ABCD be a parallelogram? Use the shown isosceles trapezium as a counterexample.

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

Parallelograms: properties, criteria and midsegments — Further practice

Year 8, lower volume · text pp. 27–36 · textbook lessons 14–18

Further practice11

ABCD is a parallelogram with AD=5 cm. Find BC.

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

Further practice12

A rectangle ABCD has diagonal AC=10 cm. Its diagonals meet at O. Find AO using a parallelogram property.

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

Further practice13

D,E are midpoints of AB,AC and BC=12 cm. Find DE and its direction relative to BC.

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

Further practice14

D,E are midpoints of two triangle sides and DE=7 cm. Find the third side BC.

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

Extension15

A parallelogram has diagonals AC=BD=10 cm, meeting at O, and AB=8 cm. Are the diagonals perpendicular? Test this using Pythagoras. Then state an extra side condition that would give a rhombus.

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

Extension16

AC bisects BD, but AO≠OC. Must ABCD be a parallelogram? Explain using the shown kite.

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

Parallelograms: properties, criteria and midsegments — Teacher answers

Year 8, lower volume · text pp. 27–36 · textbook lessons 14–18

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: Parallel lines, triangle congruence and midpoints.

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

3′ Predict which of the four half-diagonals in a parallelogram must be equal. Must all four be equal?
8′ Draw A(0,0), B(4,0), C(6,2), D(2,2); join the diagonals and find their common midpoint. Compare a trapezium with vertices (0,0),(4,0),(3,2),(1,2), whose diagonals are equal but do not bisect each other. In a separate triangle, join two side midpoints and predict the new segment’s direction and length.
5′ Separate “each diagonal is bisected” from “the diagonals are equal”. Use stated parallelism and triangle congruence or an established criterion to justify the claim; measurement alone is not proof.
2′ Individually state the two conditions and length relation for a triangle midsegment.

Worked example1

ABCD is a parallelogram and AB=8 cm. Find CD and name the property.

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

Reference answer

$CD=8$ cm.

Opposite sides of a parallelogram are equal.
CD=AB=8 cm.

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 parallelogram. Its diagonals meet at O and AO=5 cm. Find OC.

The diagonals of a parallelogram ____ each other.
$AO=OC$ , so $OC=\square$ cm.
Check the whole diagonal: $AC=AO+OC=\square$ cm.

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

Reference answer

$OC=5$ cm.

OC=5

The diagonals bisect each other, so AO=OC.
Thus OC=5 cm.

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

Prerequisite3

AB∥CD and AD∥BC. Which quadrilateral family follows directly from the definition?

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

Reference answer

A parallelogram.

Both pairs of opposite sides are parallel.

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

Prerequisite4

D is the midpoint of AB and AB=10 cm. Find AD.

Reference answer

5 cm.

5

A midpoint divides the whole into two equal parts: 10/2=5.

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

Independent practice5

ABCD is a parallelogram with ∠A=60°. Find ∠B and explain the angle relation.

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

Reference answer

120°; adjacent angles between parallel sides are supplementary.

∠A+∠B=180°.
∠B=180°−60°=120°.

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

Independent practice6

Both pairs of opposite sides of ABCD are equal. State the criterion and conclusion.

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

Reference answer

ABCD is a parallelogram by the two-pairs-of-equal-opposite-sides criterion.

Use both equal-side pairs; neither pair is omitted.

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

Independent practice7

AB∥CD and AB=CD. Is ABCD a parallelogram? Identify why the same pair matters.

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

Reference answer

Yes. One pair of opposite sides is both parallel and equal.

The parallel and equality conditions concern AB and CD together.

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

Independent practice8

D and E are midpoints of AB and AC. Extend DE to F so DE=EF. Prove DE∥BC and DE=BC/2 using congruence and a parallelogram criterion.

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

Reference answer

△ADE≅△CFE by SAS. Thus AD=CF and AD∥CF; DB=AD gives DB=CF. Hence DBCF is a parallelogram, so DF∥BC and DF=BC. Since DF=2DE, the claims follow.

AE=EC, DE=EF, and ∠AED=∠CEF are vertical angles; apply SAS.
Corresponding sides and angles give AD=CF and AD∥CF. Since DB=AD, DB and CF are parallel and equal.
Use the parallelogram DBCF to obtain DF∥BC and DF=BC; DE=EF then gives DE=BC/2.

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

Exit question9

The diagonals of ABCD bisect each other at O. What follows?

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

Reference answer

ABCD is a parallelogram.

Use the criterion involving both diagonal midpoint conditions.

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

Exit question10

Only AD=BC is given. Must ABCD be a parallelogram? Use the shown isosceles trapezium as a counterexample.

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

Reference answer

No. One equal opposite-side pair does not ensure both opposite-side pairs are parallel.

The legs AD and BC are equal, but AB and CD have different lengths.

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

Further practice11

ABCD is a parallelogram with AD=5 cm. Find BC.

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

Reference answer

$BC=5$ cm.

BC=AD because they are opposite sides.

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

Further practice12

A rectangle ABCD has diagonal AC=10 cm. Its diagonals meet at O. Find AO using a parallelogram property.

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

Reference answer

$AO=5$ cm.

A rectangle is a parallelogram, so O bisects AC.
AO=10/2=5 cm.

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

Further practice13

D,E are midpoints of AB,AC and BC=12 cm. Find DE and its direction relative to BC.

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

Reference answer

DE=6 cm and DE∥BC.

The triangle midsegment is parallel to the third side and half its length.

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

Further practice14

D,E are midpoints of two triangle sides and DE=7 cm. Find the third side BC.

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

Reference answer

BC=14 cm.

DE=BC/2, so BC=2×7=14 cm.

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

Extension15

A parallelogram has diagonals AC=BD=10 cm, meeting at O, and AB=8 cm. Are the diagonals perpendicular? Test this using Pythagoras. Then state an extra side condition that would give a rhombus.

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

Reference answer

No. $AO=BO=5$ cm; if the diagonals were perpendicular, $AB^{2}=5^{2}+5^{2}=50$ , but $AB^{2}=64$ . Equal adjacent sides would give a rhombus.

The parallelogram gives two diagonal midpoint conditions.
The proposed right triangle would have legs 5 and 5 but hypotenuse 8, contradicting its square relation.

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

Extension16

AC bisects BD, but AO≠OC. Must ABCD be a parallelogram? Explain using the shown kite.

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

Reference answer

No. Both diagonals must bisect each other; one bisection condition is insufficient.

BO=OD=2, while AO=3 and OC=1.

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