Properties of quadrilaterals: a week of ready-to-teach maths

Five lessons for Year 8 Space. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus quadrilaterals drawn with their true angles, a parallelogram to measure, six shapes to cut along a diagonal, twelve property cards, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets, one copy per pair: the quadrilaterals to cut (Day 3) and the property cards (Day 4).
4. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
5. Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.

The idea in plain words

A diagonal cuts any quadrilateral into two triangles, so facts about triangles become facts about quadrilaterals. The two triangles hold 180° each, so the four angles of a quadrilateral add to 360°. In a parallelogram a diagonal makes two congruent triangles (the diagonal is shared and the parallel sides give equal alternate angles, so ASA applies), and that one argument shows opposite sides are equal and opposite angles are equal. Similar arguments give the properties of the rectangle, rhombus, square and kite. A property is established when there is a reason that works for every shape of that kind, not just the one we measured.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

A note on the trapezium

This pack, like the interactive unit, takes a trapezium to have exactly one pair of parallel sides. Some books say at least one pair, which makes every parallelogram a trapezium too. Either is fine if the class agrees on one and says which it is using.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 turns every quadrilateral into two triangles; Day 2 collects the parallelogram’s properties by measuring; Day 3 proves them with congruent triangles; Day 4 places each special quadrilateral in the family; and Day 5 uses the properties to solve problems with reasons. It builds on angle facts for parallel lines and the classification of quadrilaterals in Year 7, and on the congruence tests from the previous unit.

Materials for the week (one trip)

• From the classroom: pencils, rulers, protractors, scissors, this pack printed.
• Cut-out sheet 1, one per pair: six quadrilaterals to cut along a diagonal (Day 3).
• Cut-out sheet 2, one per pair: twelve property cards (Day 4).
• Optional: tracing paper, to compare the two triangles without cutting.

Words for the word wall

quadrilateral, diagonal, angle sum, reflex angle, parallel, alternate angles, co-interior angles, parallelogram, rectangle, rhombus, square, kite, trapezium, bisect, perpendicular, congruent, property, reason, prove

Dear families

This week in maths, Year 8 is finding out why four-sided shapes behave as they do: why their angles always add to 360°, and why a parallelogram’s opposite sides are equal. The tool is a diagonal that cuts the shape into two triangles.

Try this at home

• Tear the four corners off a paper quadrilateral and fit them together at a point: they make a full turn, 360°.
• Find parallelograms, rhombuses and kites around the house or street, such as tiles, fences and signs.
• Ask: is a square a rectangle? (Yes: it has four right angles. It is a rhombus too.)
• Fold a paper kite along its long diagonal: the two halves match exactly.

My quadrilaterals this week

Fill one row a day. Name a shape, one property, and the reason it is true.

Printed from the free seegongsik Properties of quadrilaterals teaching pack · seegongsik.com/au/y8/space/AC9M8SP02/pack

The angle sum

A diagonal cuts every quadrilateral into two triangles, so its angles add to 2 × 180° = 360°. Students show this on four quadrilaterals, one of them an arrowhead with a reflex angle, and use it to find missing angles.

We are learning to

• explain why the angles of any quadrilateral add to 360°, using a diagonal,
• find a missing angle, including a reflex angle,
• write an equation from the angle sum and solve it.

Success criteria

• I can show the 360° by drawing a diagonal and naming the two triangles.
• I can find the reflex angle of an arrowhead from its other three angles.

You need

The worksheet, a ruler and a protractor each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Thinking a bigger quadrilateral has a bigger angle sum.
• Measuring the small angle outside a reflex angle instead of the reflex angle itself.
• Drawing a diagonal that falls outside an arrowhead.

Answers

• Part A: 1. x = 360 − 255 = 105°; 2. x = 360 − 290 = 70°; 3. x = 360 − 120 = 240° (reflex); 4. x = 360 − 300 = 60°.
• Part B: each triangle holds 180°, and together their angles make up the four corners, so 360°. In shape 3, draw CA, from the reflex corner; the other diagonal lies outside the shape.
• Part C: x + 2x + 3x + 90 = 360, so 6x = 270 and x = 45: the angles are 45°, 90°, 135° and 90°.
• Part D: a pentagon makes 3 triangles, 540°; a hexagon 4 triangles, 720°; n sides make n − 2 triangles, (n − 2) × 180°.

The angle sum

A. Find the missing angle

Each shape is drawn with its true angles. Find x, then check one answer with a protractor.

B. Why 360°?

Draw a diagonal in shape 1 and explain why its angles must add to 360°. In shape 3, which diagonal must you draw, and why not the other one?

C. Angles with letters

A quadrilateral has angles of x, 2x and 3x, and a right angle. Write an equation, find x, and give all four angles.

D. Stretch: more sides

Split a pentagon and a hexagon into triangles from one corner. What do their angles add to? What about a shape with n sides?

Parallelogram properties

From one fact, two pairs of parallel sides, a parallelogram gets a set of properties. Students measure a parallelogram to find them, explain its angles with parallel-line angle facts, and use the properties to find missing lengths and angles.

We are learning to

• find the properties of a parallelogram by measuring,
• use co-interior and alternate angles to explain its angles,
• use the properties to find missing sides, angles and diagonal lengths.

Success criteria

• I can name three properties of every parallelogram.
• I can find all four angles of a parallelogram from one of them, giving reasons.

You need

The worksheet, a ruler and a protractor each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B.

Watch for these ideas

• Thinking the diagonals of every parallelogram are equal: that is only true of a rectangle.
• Thinking the diagonals cross at right angles: that is only true of a rhombus.
• Mixing up alternate angles (equal) and co-interior angles (adding to 180°).

Answers

• Part A: AB = DC = 7 cm and AD = BC = 4 cm; ∠A = ∠C = 65° and ∠B = ∠D = 115°; AM = MC ≈ 4.7 cm and BM = MD ≈ 3.2 cm. The diagonals (9.4 cm and 6.4 cm) are not equal. A smaller print shrinks every length, but the pairs stay equal.
• Part B: AB ∥ DC with AD crossing them, so ∠A + ∠D = 180° (co-interior). In the same way ∠D + ∠C = 180° (AD ∥ BC), so ∠A = ∠C.
• Part C: 1. 108°, 72° and 108°; 2. MC = 4.6 cm and AC = 9.2 cm; 3. 9 cm, 6 cm and 6 cm.
• Part D: always a parallelogram. The two triangles on opposite sides of the crossing are congruent (SAS, with vertically opposite angles), so alternate angles are equal and each pair of opposite sides is parallel.

Parallelogram properties

A. Measure and notice

ABCD is a parallelogram and M is where its diagonals cross. Measure, then say what you notice.

What do you notice about the sides, the angles and the diagonals?

B. Angles from parallel lines

AB ∥ DC and AD ∥ BC. Explain why ∠A + ∠D = 180°. Then explain why ∠A = ∠C.

C. Missing parts

1. One angle of a parallelogram is 72°. Find the other three angles.
2. The diagonals of parallelogram ABCD cross at M, and AM = 4.6 cm. Find MC and AC.
3. A parallelogram has a perimeter of 30 cm, and one side is 9 cm. Find the other three sides.

D. Stretch: the other way round

Draw two line segments that cross at their midpoints, then join their ends. What shape do you always get? Why?

Proving with congruent triangles

A proof gives a reason that works for every shape of a kind. Students complete the parallelogram proof, cut six quadrilaterals along a diagonal to test the halves, and prove a fact about kites.

We are learning to

• follow and complete a proof that uses congruent triangles,
• name the congruence test and the angle facts a proof relies on,
• prove a property of another quadrilateral in the same way.

Success criteria

• I can give a reason for every line of the parallelogram proof.
• I can prove that a kite has one pair of equal opposite angles.

You need

Cut-out sheet 1 and scissors, one per pair; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the cutting.

Watch for these ideas

• Giving a reason that only holds for the drawing, such as “it looks equal”.
• Naming SSA, or a test the facts do not fit.
• Thinking any diagonal of any quadrilateral makes two congruent triangles.

Answers

• Part A, the reasons in order: alternate angles, AB ∥ DC; alternate angles, AD ∥ BC; the same side (common); ASA; matching sides of congruent triangles.
• Part B: P congruent, by a half turn; Q congruent, after a flip; R not congruent; S congruent, by a half turn; T not congruent; U not congruent.
• Part C, the reasons in order: given (the short sides); given (the long sides); common side; SSS; matching angles of congruent triangles.
• Part D: a rhombus has four equal sides, so it is a kite both ways. The four small triangles at the centre are congruent (SSS: equal sides and equal half-diagonals), so the four angles at the centre are equal, and as they make 360° each is 90°.

Proving with congruent triangles

A. Opposite sides of a parallelogram are equal

B. Cut and test

Cut out the six shapes on cut-out sheet 1 and cut each along its dashed diagonal. Do the two halves match exactly?

C. A kite has a pair of equal angles

D. Stretch: a rhombus

Use congruent triangles to explain why the diagonals of a rhombus cross at right angles.

The quadrilateral family

The special quadrilaterals form a family, and each shape keeps the properties of the shapes it belongs to. Students sort property cards for parallelograms, fill a property table, and decide which family statements are true.

We are learning to

• sort properties into true for every parallelogram, for some, or for none,
• record the properties of each special quadrilateral in a table,
• explain family statements such as “every square is a rhombus”.

Success criteria

• I can say which property makes a parallelogram a rectangle, and which makes it a rhombus.
• I can explain why a square is both a rectangle and a rhombus.

You need

Cut-out sheet 2 (the property cards) and scissors, one set per pair; the worksheet each.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the table.

Watch for these ideas

• Thinking a square is not a rectangle because its sides are equal.
• Giving a kite properties it does not have, such as equal diagonals or parallel sides.
• Treating “some” as “none”: some parallelograms (rectangles) do have equal diagonals.

Answers

• Part A: every parallelogram 1, 3, 4, 7, 10; some 2, 5, 9, 12; none 6, 8, 11.
• Part B: both pairs of opposite sides parallel: parallelogram, rectangle, rhombus, square; all sides equal: rhombus, square; all angles 90°: rectangle, square; diagonals bisect each other: parallelogram, rectangle, rhombus, square; diagonals equal: rectangle, square; diagonals at right angles: rhombus, square, kite.
• Part C: 1 true, 2 true, 3 false, 4 false, 5 true, 6 true, 7 false, 8 true.
• Part D: quadrilateral at the top; below it parallelogram, kite and trapezium; below the parallelogram, rectangle and rhombus (the rhombus is also a kite); the square at the bottom, below both the rectangle and the rhombus.

The quadrilateral family

A. Every, some or none?

Sort the twelve property cards into three piles: true for every parallelogram, for some, or for none. Record the card numbers.

B. The property table

Tick each shape that always has the property.

para. = parallelogram, trap. = trapezium

C. True or false?

D. Stretch: a family tree

Draw a family tree with “quadrilateral” at the top and an arrow from each shape to the shapes that are special cases of it. Where does the square go? Does any shape have two parents?

Problems with reasons

A property is only useful if we can name it as a reason. Students find angles and lengths in a rhombus, a kite, a trapezium and a rectangle, giving a reason for every step, and play the self-check quiz as a class game.

We are learning to

• choose the property of a quadrilateral that unlocks a problem,
• find missing angles and lengths in steps,
• write a reason beside every step.

Success criteria

• I can find every angle of a rhombus from one of them, with reasons.
• I can use equal diagonals in a rectangle to find a length and an angle.

You need

The worksheet each; the self-check quiz at the bottom of the unit page for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B and the quiz.

Watch for these ideas

• Writing an answer with no reason, or a reason that is a measurement.
• Using the kite’s equal angles at the wrong corners: they are between the unequal sides.
• Taking the diagonals of a rectangle to meet at right angles.

Answers

• Part A: rhombus b = 116° (co-interior), c = 64° (opposite angles), d = 116°; AC bisects ∠A (SSS), so 32°. Kite ∠D = 95° (equal angles), ∠C = 60° (angle sum).
• Part B: trapezium d = 122° (co-interior), b = 58°, c = 122° (symmetric). Rectangle BD = AC = 13.0 cm (equal diagonals bisect each other); MA = MB, so ∠MBA = 34° and ∠AMB = 112°.
• Part C: 3x + 10 = 2x + 20 (opposite angles), so x = 10; those two angles are 40° and the other two 140°.
• Part D: a rectangle. Diagonals that bisect each other make a parallelogram; with equal diagonals, triangles ABC and BAD are congruent (SSS), so ∠B = ∠A, and as they add to 180° each is 90°.

Problems with reasons

Give a reason for every step: name the property or the angle fact you use.

A. A rhombus and a kite

B. A trapezium and a rectangle

C. Opposite angles with x

Two opposite angles of a parallelogram are (3x + 10)° and (2x + 20)°. Find x and all four angles.

D. Stretch: equal diagonals

The diagonals of a quadrilateral are equal and bisect each other. What must the quadrilateral be? Explain why.

Six quadrilaterals to cut

For Day 3, one sheet per pair

Cut out each shape along its outline, then cut it in two along the dashed diagonal. Try to lay one half exactly on the other, turning it, or flipping it over if you need to.

Write the shape’s letter on both halves before you cut, so the pieces stay together.

Twelve property cards

For Day 4, one set per pair

Cut out the cards. Is each statement true for every parallelogram, for some parallelograms, or for none?

Teacher note: every 1, 3, 4, 7, 10; some 2, 5, 9, 12; none 6, 8, 11. No row or column of cards gives a pile away.

What we know: Properties of quadrilaterals

Work on your own. Give a reason for each step.

1. Three angles of a quadrilateral are 95°, 80°, 110°. Find the fourth angle. ______________________
2. An arrowhead (a quadrilateral with a reflex angle) has angles of 30°, 40°, 40°. Find its reflex angle. ______________________
3. One angle of a parallelogram is 48°. Find the other three angles. ______________________
4. The diagonals of parallelogram PQRS cross at M. PM = 5.5 cm and QM = 3 cm. Find PR and QS. ______________________
5. A diagonal splits a parallelogram into two triangles. Which test shows they are congruent, and which angle facts does it use? ______________________
6. True or false: (a) every rectangle is a parallelogram; (b) the diagonals of every rhombus are equal. ______________________
7. A kite has an angle of 128° between its two short sides, and its two equal angles are 68° each. Find the fourth angle. ______________________
8. Explain, using congruent triangles, why the two diagonals of a rectangle are equal. ______________________

Answers and marking guide

Answers with working

1. 360 − 285 = 75° (angle sum of a quadrilateral).
2. 360 − 110 = 250°.
3. 132° (co-interior), 48° (opposite) and 132°.
4. The diagonals bisect each other: PR = 11 cm and QS = 6 cm.
5. ASA: the diagonal is a common side, and the parallel sides give two pairs of equal alternate angles.
6. (a) true: it has two pairs of parallel sides; (b) false: only a square’s are; a rhombus’s diagonals cross at right angles.
7. 360 − 128 − 2 × 68 = 96°.
8. In rectangle ABCD, triangles ABC and BAD share AB, have BC = AD (opposite sides) and right angles at B and A, so they are congruent (SAS); the matching sides AC and BD are equal.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions and gives a property or angle fact as each reason.

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: the angle sum, parallelogram properties, proof, the family, and problems with reasons.