Classifying Special Quadrilaterals

See how the special quadrilaterals are related in one family tree, give a quadrilateral its most specific name by matching its properties, and decide whether a statement about a shape is always, sometimes or never true.

Before you start

On a drawing, ticks mark equal sides (sides with the same number of ticks are equal), arrows mark parallel sides (sides with the same number of arrows are parallel), and a small square marks a right angle.
The lesson Properties of Special Quadrilaterals set out the properties of the parallelogram, rectangle, rhombus, square, trapezium, isosceles trapezium and kite. They are gathered again in the first section below.
A line of symmetry folds a figure exactly onto itself. The order of rotational symmetry is the number of times a figure fits onto itself in one complete turn, counting the full turn; order 1 means no rotational symmetry (the lesson Line and Rotational Symmetry).
On a grid, from $(1, 1)$ to $(3, 4)$ is 2 right and 3 up. Two sides that go the same steps in the same direction are parallel and equal in length (Lesson 10.1).
If a line crosses two lines and a pair of interior angles add up to 180°, the two lines are parallel; the reason is written (int. ∠s supp.) (Lesson 11.4).

The Special Quadrilaterals and Their Properties

A quadrilateral is a plane figure with four straight sides. Every quadrilateral in this lesson has each of its angles less than 180°. Seven special quadrilaterals have names of their own. Each is fixed by its definition: the properties that make a quadrilateral that shape. Every shape of that kind then has further properties, which follow from its definition (the lesson Properties of Special Quadrilaterals).

The definitions

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

A rectangle is a parallelogram with four right angles.

A rhombus is a parallelogram with four equal sides.

A square has four equal sides and four right angles. Each side makes 90° with the two sides it meets, and $90^{\circ} + 90^{\circ} = 180^{\circ}$ , so its opposite sides are parallel (int. ∠s supp.) and it is a parallelogram too.

A trapezium is a quadrilateral with exactly one pair of parallel sides.

An isosceles trapezium is a trapezium whose two non-parallel sides are equal.

A kite is a quadrilateral with two pairs of equal adjacent sides (two sides that meet at a vertex are adjacent).

Each drawing below shows the marks of its definition.

Parallelogram

Rectangle

Rhombus

Square

Trapezium

Isosceles trapezium

Kite

The table gathers their properties. It has two kinds of row.

Rows (a) are properties that every shape of that kind has, special cases included. A ✓ means every one has it; a – means not every one does (some may, or none).

Rows (b) give the symmetry of the general shape only: a parallelogram that is not a rectangle or a rhombus, a rectangle that is not a square, a rhombus that is not a square, a trapezium that is not isosceles, and a kite that is not a rhombus. A special case may have more symmetry: a square is a rectangle, but it has 4 lines of symmetry, not 2.

Property	Parallelogram	Rectangle	Rhombus	Square	Trapezium	Isosceles trapezium	Kite
(a) Properties every member has
Both pairs of opposite sides parallel	✓	✓	✓	✓	–	–	–
Exactly one pair of opposite sides parallel	–	–	–	–	✓	✓	–
Both pairs of opposite sides equal	✓	✓	✓	✓	–	–	–
All four sides equal	–	–	✓	✓	–	–	–
Two pairs of equal adjacent sides	–	–	✓	✓	–	–	✓
One pair of opposite sides parallel and the other pair equal	✓	✓	✓	✓	–	✓	–
Four right angles	–	✓	–	✓	–	–	–
Both pairs of opposite angles equal	✓	✓	✓	✓	–	–	–
At least one pair of opposite angles equal	✓	✓	✓	✓	–	–	✓
The two angles at the ends of each parallel side equal (base angles)	–	✓	–	✓	–	✓	–
Diagonals bisect each other	✓	✓	✓	✓	–	–	–
Diagonals equal	–	✓	–	✓	–	✓	–
Diagonals perpendicular	–	–	✓	✓	–	–	✓
Each diagonal bisects the angles at its ends	–	–	✓	✓	–	–	–
One diagonal bisects the other diagonal and the angles at its own ends	–	–	✓	✓	–	–	✓
(b) Symmetry of the general shape
Lines of symmetry	0	2	2	4	0	1	1
Where the lines of symmetry are	none	through the midpoints of opposite sides (not the diagonals)	the two diagonals	the two diagonals and the two lines through the midpoints of opposite sides	none	through the midpoints of the two parallel sides	the diagonal joining the two vertices where the equal sides meet
Order of rotational symmetry	2	2	2	4	1	1	1

The Family Tree

Some special quadrilaterals are special cases of others. A square has four equal sides and four right angles, and it is a parallelogram. So it meets the definition of a rectangle (a parallelogram with four right angles) and the definition of a rhombus (a parallelogram with four equal sides). A square is a rectangle, and it is a rhombus: a special one of each.

The family tree shows every such link. A line joins a shape to a special case of it, drawn below it.

The family tree

A square is a special rectangle (one with four equal sides) and a special rhombus (one with four right angles).

A rectangle is a special parallelogram (one with four right angles), and so is a rhombus (one with four equal sides).

A rhombus is a special kite: its four equal sides make two pairs of equal adjacent sides.

The trapezium is on a branch of its own, since it has exactly one pair of parallel sides. The isosceles trapezium is a special trapezium (one whose non-parallel sides are equal).

A special case has every property that every shape above it has (rows (a) of the table), and more. So going up the tree is always true ("every square is a rectangle"), but going down is only sometimes true ("a rectangle may or may not be a square").

Example 1: Reading the family tree

Answer each question with a reason.

(a) Is every square a rhombus?

(b) Is every rhombus a kite?

(d) Name every special quadrilateral that a square is.

Solution

(a) Yes. A square is a parallelogram with four equal sides, which is the definition of a rhombus. In the tree, square is below rhombus.

(b) Yes. A rhombus has four equal sides, so the two sides at one vertex are equal and the two sides at the opposite vertex are equal: two pairs of equal adjacent sides, which is the definition of a kite. In the tree, rhombus is below kite.

(c) No. Rectangle is below parallelogram, and going down is only sometimes true. A parallelogram with angles of 60° and 120° has no right angle, so it is not a rectangle.

(d) Follow every line up from square: to rectangle and on to parallelogram; to rhombus and on to parallelogram and to kite.

(a) Yes (b) Yes (c) No (d) A square is a rectangle, a rhombus, a parallelogram and a kite (and, like every shape here, a quadrilateral).

Classifying by Matching Properties

To classify a quadrilateral is to say which special quadrilaterals it is and to give its most specific name: the lowest name in the family tree that fits it. A square is a rectangle, a rhombus, a parallelogram and a kite, but "square" tells you the most.

You classify by matching: set what you know about the shape beside the definitions and the table. Use only what you are told, in words, in numbers or by marks. A drawing may look like a square without being one.

Classifying by matching

Which special quadrilaterals always have these properties? Find each property's row in the table, rows (a). Keep the shapes with a ✓ in every one of those rows.

Is the shape a …? It is, if it has every property in that shape's definition.

It is not a … if it lacks a property that every shape of that kind has (a ✓ in that column).

Its most specific name: start at the top of the family tree and move down a line whenever the shape meets the definition below. Stop when you can go no further.

Matching works one way

The table tells you what every rhombus has, such as perpendicular diagonals. It does not tell you that a quadrilateral with perpendicular diagonals is a rhombus: a kite that is not a rhombus has them too (Example 6). To show that a shape is a rhombus, use the definition of a rhombus.

Example 2: Which shapes always have these properties?

Use the table. Which special quadrilaterals always have:

(a) equal diagonals;

(b) both pairs of opposite sides equal and perpendicular diagonals;

Solution

(a) The row "Diagonals equal" has a ✓ for the rectangle, the square and the isosceles trapezium.

(b) "Both pairs of opposite sides equal": parallelogram, rectangle, rhombus, square. "Diagonals perpendicular": rhombus, square, kite. The shapes in both lists are the rhombus and the square.

(c) "Two pairs of equal adjacent sides": rhombus, square, kite. "Four right angles": rectangle, square. Only the square is in both lists.

Notice that the square turns up in all three answers. Whenever a shape is on such a list, so is every special case of it below it in the tree.

(a) The rectangle, the square and the isosceles trapezium. (b) The rhombus and the square. (c) The square only.

Example 3: Naming a quadrilateral from words

Give the most specific name of each quadrilateral.

(a) $ABCD$ has $AB \parallel DC$ and $AD \parallel BC$ , four right angles, $AB = 7$ cm and $BC = 4$ cm.

(b) $PQRS$ has $PQ = PS = 5$ cm and $RQ = RS = 8$ cm.

(d) $WXYZ$ has $WX \parallel ZY$ and $WZ \parallel XY$ , all four sides 6 cm, and $\angle W = 75^{\circ}$ .

Solution

(a) Both pairs of opposite sides are parallel: $ABCD$ is a parallelogram. It has four right angles: it is a rectangle. Is it a square? Every square has four equal sides, but $AB = 7$ cm and $BC = 4$ cm. So it is not a square.

(b) $PQ$ and $PS$ meet at $P$ and are equal; $RQ$ and $RS$ meet at $R$ and are equal. Two pairs of equal adjacent sides: $PQRS$ is a kite. Is it a rhombus? Every rhombus has four equal sides, but $PQ = 5$ cm and $RQ = 8$ cm. So it is not a rhombus.

(c) Exactly one pair of sides is parallel: $KLMN$ is a trapezium. Its non-parallel sides $KN$ and $LM$ are equal: it is an isosceles trapezium.

(d) Both pairs of opposite sides are parallel: $WXYZ$ is a parallelogram. All four sides are equal: it is a rhombus. Is it a square? Every square has four right angles, but $\angle W = 75^{\circ}$ . So it is not a square.

(a) A rectangle (b) a kite (c) an isosceles trapezium (d) a rhombus.

Example 4: Naming a quadrilateral from a drawing

Use the marks and the lengths to give the most specific name of each quadrilateral. In (b), $PS$ and $QR$ are not parallel.

(a)

(b)

(c)

Solution

(a) $AB = AD$ (one tick each) and $CB = CD$ (two ticks each): two pairs of equal adjacent sides, so $ABCD$ is a kite. Every rhombus has four equal sides, but $AB = 3$ cm and $BC = 5$ cm, so it is not a rhombus.

(b) The arrows show $PQ \parallel SR$ , and $PS$ and $QR$ are not parallel: exactly one pair of parallel sides, so $PQRS$ is a trapezium. The ticks show $PS = QR$ : its non-parallel sides are equal, so it is an isosceles trapezium.

(c) $WZ$ and $XY$ each make 90° with $WX$ , and $90^{\circ} + 90^{\circ} = 180^{\circ}$ , so $WZ \parallel XY$ (int. ∠s supp.). In the same way $WX \parallel ZY$ . So $WXYZ$ is a parallelogram with four right angles: a rectangle. Every square has four equal sides, but $WX = 6$ cm and $XY = 3.5$ cm, so it is not a square.

(a) A kite (b) an isosceles trapezium (c) a rectangle.

Always, Sometimes or Never

A statement such as "a rhombus is a square" or "the diagonals of a kite are perpendicular" is about every shape of one kind. Decide whether it is always, sometimes or never true, and back your answer up.

Always, sometimes or never

Always true: true for every shape of that kind. Give the reason: the definition, a ✓ in rows (a) of the table, a property that every shape of that kind has, or the family tree.

Sometimes true: true for some shapes of that kind and false for others. Give one shape for which it is true and one for which it is false (a named shape or a drawing).

Never true: true for no shape of that kind. Give the reason.

Remember the special cases

A kite may be a rhombus or a square; a rectangle may be a square; a parallelogram may be a rectangle, a rhombus or a square. A statement about "a kite" is always true only if it is true for these special kites as well.

Example 5: Always, sometimes or never?

Is each statement always, sometimes or never true? Give a reason or a drawing.

(a) A square is a rhombus.

(b) A rhombus is a square.

(d) A kite has both pairs of opposite sides parallel.

(e) A parallelogram has exactly one obtuse angle.

Solution

(a) Always. Every square is a parallelogram with four equal sides, which is the definition of a rhombus.

(b) Sometimes. A rhombus with four right angles is a square. The rhombus $EFGH$ below has angles of 60° and 120°; a square has four right angles, so $EFGH$ is not a square.

(c) Sometimes. A square is a rectangle, and its diagonals are perpendicular. The rectangle $JKLM$ below is 8 cm by 4 cm. Draw it accurately on paper, join its diagonals and measure $\angle KOL$ with a protractor: it is about 53°, not 90°, so these diagonals are not perpendicular.

(d) Sometimes. A rhombus is a kite, and both pairs of its opposite sides are parallel. The kite $ABCD$ of Example 4(a) has $AB = 3$ cm and $DC = 5$ cm. If both pairs of its opposite sides were parallel, it would be a parallelogram, and the opposite sides of a parallelogram are equal; $AB$ and $DC$ are not. So its opposite sides are not both parallel.

(e) Never. The opposite angles of a parallelogram are equal. If one angle is obtuse, the angle opposite it is obtuse too. So a parallelogram has two obtuse angles, or none (a rectangle).

(a) Always (b) sometimes (c) sometimes (d) sometimes (e) never.

Example 6: Three tempting conclusions

Each pupil draws a conclusion. Does it follow from what the pupil knows? Back up your answer with a drawing.

(a) Ali: "The diagonals of $ABCD$ are perpendicular, so $ABCD$ is a rhombus."

(b) Mei Ling: "The diagonals of $PQRS$ are equal, so $PQRS$ is a rectangle."

Solution

(a) Every rhombus has perpendicular diagonals, but so does every kite. The kite $ABCD$ below has $AB = AD = 6.5$ cm and $CB = CD = 10$ cm. Its diagonals cross at right angles at $O$ , but its four sides are not equal, so it is not a rhombus.

(b) Every rectangle has equal diagonals, but so does every isosceles trapezium. The isosceles trapezium $PQRS$ below has $PQ \parallel SR$ and $PS = QR$ , and its diagonals $PR$ and $QS$ are equal. Every rectangle has both pairs of opposite sides parallel, but $PS$ and $QR$ are not parallel, so it is not a rectangle.

(c) The isosceles trapezium $KLMN$ below has $KL \parallel NM$ and $KN = LM$ , but $KN$ and $LM$ are not parallel, so it is not a parallelogram.

None of the three conclusions follows. A kite that is not a rhombus has perpendicular diagonals; an isosceles trapezium has equal diagonals without being a rectangle, and has one pair of sides parallel and the other pair equal without being a parallelogram.

Naming Quadrilaterals on a Grid and in Real Life

On a grid, a side whose ends have the same $y$ -coordinate is horizontal, and one whose ends have the same $x$ -coordinate is vertical. Count squares to find the length of a horizontal or vertical side, compare the steps of two sides to see whether they are parallel, and use a fold along a line of the grid to show that two sides are equal.

Example 7: A quadrilateral on a grid

Plot the points $A(1, 1)$ , $B(7, 1)$ , $C(5, 4)$ and $D(3, 4)$ and join them in order. Give the most specific name of the quadrilateral $ABCD$ .

Solution

$A$ and $B$ both have $y$ -coordinate 1, so $AB$ is horizontal. $D$ and $C$ both have $y$ -coordinate 4, so $DC$ is horizontal. So $AB \parallel DC$ .

From $A$ to $D$ is 2 right and 3 up; from $B$ to $C$ is 2 left and 3 up. $AD$ leans to the right as it rises and $BC$ leans to the left, so they are not parallel.

Exactly one pair of parallel sides: $ABCD$ is a trapezium.

The dashed line $x = 4$ is halfway between $A$ and $B$ (each is 3 units from it) and halfway between $D$ and $C$ (each is 1 unit from it). Folding along it carries $A$ onto $B$ and $D$ onto $C$ , so $AD$ lands exactly on $BC$ : $AD = BC$ .

Its non-parallel sides are equal: $ABCD$ is an isosceles trapezium.

$ABCD$ is an isosceles trapezium.

Example 8: A folding gate

Mr Tan's folding gate is made of straight metal bars. The bars that slope one way are all parallel, and the bars that slope the other way are all parallel. They are pinned together so that each gap between them has four sides of 12 cm. In the drawing of one gap, $ABCD$ , the bars cross at 70°.

(a) Give the most specific name of the gap $ABCD$ .

(b) Mr Tan moves the gate until the bars cross at right angles. Give the most specific name of the gap now.

Solution

(a) $AB \parallel DC$ and $AD \parallel BC$ , since each pair lies along bars that slope the same way: $ABCD$ is a parallelogram. All four sides are 12 cm: it is a rhombus. Every square has four right angles, but $\angle DAB = 70^{\circ}$ , so it is not a square.

(b) The sides are still parallel in pairs and still 12 cm, so the gap is still a rhombus. Each corner of the gap is a place where two bars cross, so now all four angles are 90°. Four equal sides and four right angles: a square.

(c) However far the gate is moved, the gap has both pairs of opposite sides parallel and four sides of 12 cm, so it is always a rhombus (and so also a parallelogram and a kite). It is a square only when the bars cross at right angles: a rhombus is sometimes a square.

(a) A rhombus (b) a square (c) a rhombus, at every position of the gate.

Common Mistakes

"The diagonals are perpendicular, so it is a rhombus." No: a kite that is not a rhombus has perpendicular diagonals too.
"The diagonals are equal, so it is a rectangle." No: an isosceles trapezium has equal diagonals too.
"One pair of sides is parallel and the other pair is equal, so it is a parallelogram." No: an isosceles trapezium is exactly that.
"A square is not a rectangle." It is: a square is a rectangle whose four sides are equal. In the same way every square is a rhombus, and every rhombus is a kite.
Naming a shape by its look. Use only the marks, the numbers and the words: a shape that looks square may not be one, and a square standing on a corner is still a square.
Reading the symmetry rows as "always". "A rectangle has 2 lines of symmetry" is only sometimes true: a square is a rectangle with 4.

Practice Questions

Try each question before you open its solution. Use the table and the family tree as often as you need.

Practice 1: Reading the table

Which special quadrilaterals always have:

(a) four right angles;

(b) two pairs of equal adjacent sides;

(d) diagonals that are equal and bisect each other?

ShowHide solution

(a) The row "Four right angles" has a ✓ for the rectangle and the square.

(b) The row "Two pairs of equal adjacent sides" has a ✓ for the rhombus, the square and the kite.

(d) "Diagonals equal": rectangle, square, isosceles trapezium. "Diagonals bisect each other": parallelogram, rectangle, rhombus, square. In both lists: the rectangle and the square.

(a) Rectangle, square (b) rhombus, square, kite (c) parallelogram, rectangle, rhombus, square (d) rectangle, square.

Practice 2: True or false?

True or false? Give a reason from the definitions or the family tree.

(a) Every rectangle is a parallelogram.

(b) Every kite is a rhombus.

(d) Every parallelogram is a rhombus.

(e) Every trapezium is an isosceles trapezium.

ShowHide solution

(a) True. A rectangle is a parallelogram with four right angles; in the tree, rectangle is below parallelogram.

(b) False. A rhombus is a special kite, not every kite. A kite with sides 3 cm, 3 cm, 5 cm and 5 cm does not have four equal sides, so it is not a rhombus.

(d) False. A parallelogram with sides of 6 cm and 4 cm does not have four equal sides, so it is not a rhombus.

(e) False. A trapezium whose non-parallel sides are 3 cm and 4 cm does not have its non-parallel sides equal, so it is not isosceles.

(a) True (b) false (c) true (d) false (e) false.

Practice 3: Naming from drawings

Give the most specific name of each quadrilateral. In (c), $RU$ and $ST$ are not parallel.

(a)

(b)

(c)

ShowHide solution

(a) The arrows show both pairs of opposite sides parallel: $EFGH$ is a parallelogram. Every rectangle has four right angles, but $\angle E = 65^{\circ}$ , so it is not a rectangle (nor a square). Every rhombus has four equal sides, but $EF = 6$ cm and $EH = 4$ cm, so it is not a rhombus.

(b) Four equal sides and four right angles: $JKLM$ is a square. It stands on a corner, but turning a shape does not change its sides or its angles.

(c) $RS \parallel UT$ , and $RU$ and $ST$ are not parallel: exactly one pair of parallel sides, so $RSTU$ is a trapezium. Its non-parallel sides are 4 cm and 3 cm, which are not equal, so it is not an isosceles trapezium.

(a) A parallelogram (b) a square (c) a trapezium.

Practice 4: Naming from words

(a) $EFGH$ has $EF = FG = 7$ cm and $GH = HE = 4$ cm. Give its most specific name.

(b) $KLMN$ has $KL \parallel NM$ and $KN \parallel LM$ , four right angles, and all four sides 5 cm. Give its most specific name, and every other special name it has.

ShowHide solution

(a) $EF$ and $FG$ meet at $F$ and are equal; $GH$ and $HE$ meet at $H$ and are equal. Two pairs of equal adjacent sides: $EFGH$ is a kite. Every rhombus has four equal sides, but $EF = 7$ cm and $GH = 4$ cm, so it is not a rhombus.

(b) Both pairs of opposite sides parallel: a parallelogram. With four right angles it is a rectangle; with four equal sides it is a rhombus; with both, a square. Going up the family tree from square: rectangle, rhombus, parallelogram and kite.

(a) A kite (b) a square; it is also a rectangle, a rhombus, a parallelogram and a kite.

Practice 5: Always, sometimes or never? (1)

Is each statement always, sometimes or never true? Give a reason or an example.

(a) A parallelogram has four equal sides.

(b) The diagonals of a rhombus are perpendicular.

(d) A rectangle has an angle of 100°.

ShowHide solution

(a) Sometimes. A rhombus is a parallelogram with four equal sides. A parallelogram with sides of 6 cm and 4 cm, like $EFGH$ in Practice 3(a), does not have four equal sides.

(b) Always. The row "Diagonals perpendicular" has a ✓ for the rhombus: every rhombus, a square included, has perpendicular diagonals.

(c) Sometimes. A rhombus is a kite with four equal sides. The kite $ABCD$ of Example 4(a) has sides of 3 cm and 5 cm.

(d) Never. Every angle of a rectangle is a right angle, 90°.

(a) Sometimes (b) always (c) sometimes (d) never.

Practice 6: Always, sometimes or never? (2)

Is each statement always, sometimes or never true? Give a reason or an example.

(a) A rectangle has exactly two lines of symmetry.

(b) The order of rotational symmetry of a parallelogram is 2.

(d) A kite has rotational symmetry of order 2.

ShowHide solution

(a) Sometimes. A rectangle that is not a square has exactly two lines of symmetry, through the midpoints of opposite sides. A square is a rectangle with four lines of symmetry.

(b) Sometimes. A parallelogram that is not a square fits onto itself twice in one complete turn: order 2. A square is a parallelogram that fits onto itself four times: order 4.

(c) Always. The isosceles trapezium has no special case below it in the family tree, so its symmetry row covers every isosceles trapezium: each one has a line of symmetry through the midpoints of its two parallel sides (the lesson Properties of Special Quadrilaterals).

(d) Sometimes. A rhombus that is not a square is a kite with rotational symmetry of order 2. A kite that is not a rhombus has order 1 (no rotational symmetry).

(a) Sometimes (b) sometimes (c) always (d) sometimes.

Practice 7: Mr Tan's claim

Mr Tan says: "A quadrilateral with a line of symmetry is a kite." Is he right? Use a rectangle 6 cm by 3 cm to back up your answer.

ShowHide solution

The rectangle $ABCD$ has two lines of symmetry (dashed), through the midpoints of opposite sides.

A kite has two pairs of equal adjacent sides. At every vertex of this rectangle a 6 cm side meets a 3 cm side, so no two adjacent sides are equal, and it is not a kite.

So a quadrilateral can have a line of symmetry without being a kite.

No. A rectangle 6 cm by 3 cm has two lines of symmetry, but no two of its adjacent sides are equal, so it is not a kite.

Practice 8: A quadrilateral on a grid (1)

Plot $K(4, 1)$ , $L(6, 4)$ , $M(4, 6)$ and $N(2, 4)$ and join them in order. Give the most specific name of $KLMN$ , with reasons.

ShowHide solution

$K$ and $M$ both have $x$ -coordinate 4, so they lie on the line $x = 4$ . $L(6, 4)$ and $N(2, 4)$ are at the same height, each 2 units from $x = 4$ , on opposite sides.

Folding along $x = 4$ carries $L$ onto $N$ and leaves $K$ and $M$ where they are. So $KL$ lands on $KN$ and $ML$ lands on $MN$ : $KL = KN$ and $ML = MN$ .

$KL$ and $KN$ meet at $K$ ; $ML$ and $MN$ meet at $M$ . Two pairs of equal adjacent sides: $KLMN$ is a kite.

Is it a rhombus? Every rhombus has both pairs of opposite sides parallel. From $K$ to $L$ is 2 right and 3 up; from $N$ to $M$ is 2 right and 2 up. For the same 2 across, they rise by different amounts, so $KL$ and $NM$ are not parallel, and $KLMN$ is not a rhombus.

$KLMN$ is a kite.

Practice 9: A flower bed on a plan

Mr Tan plans a flower bed on a grid. Its corners are $W(1, 1)$ , $X(4, 1)$ , $Y(5, 5)$ and $Z(2, 5)$ . Give the most specific name of $WXYZ$ , with reasons.

ShowHide solution

$W$ and $X$ both have $y$ -coordinate 1, and $Z$ and $Y$ both have $y$ -coordinate 5: $WX$ and $ZY$ are horizontal, so $WX \parallel ZY$ .

From $W$ to $Z$ is 1 right and 4 up; from $X$ to $Y$ is also 1 right and 4 up. The same steps in the same direction, so $WZ \parallel XY$ .

Both pairs of opposite sides parallel: $WXYZ$ is a parallelogram.

Is it a rectangle? $WX$ is horizontal, but $WZ$ is not vertical (it moves 1 right as it rises), so $\angle ZWX$ is not 90°. Every rectangle, and every square, has four right angles, so it is neither.

Is it a rhombus? $WZ$ slopes, so it is longer than the straight-up distance of 4 units between the two horizontal sides, while $WX$ is 3 units long. Every rhombus has four equal sides, so it is not a rhombus.

The flower bed $WXYZ$ is a parallelogram.

Practice 10: A window grille

The window grille in Siti's flat is made of metal frames like the one drawn. Each frame has four sides of 20 cm, its opposite sides are parallel, and two of its angles are 60°.

(a) Give the most specific name of the frame.

(b) Mei Ling says the frame is a square turned onto its corner. Is she right?

ShowHide solution

(a) Both pairs of opposite sides are parallel: a parallelogram. All four sides are 20 cm: a rhombus.

(b) No. A square has four right angles, and this frame has angles of 60°.

(c) The frame is a rhombus that is not a square. It has 2 lines of symmetry, along its two diagonals, and rotational symmetry of order 2.

(a) A rhombus (b) no: its angles of 60° are not right angles (c) 2 lines of symmetry (the diagonals); order 2.

Practice 11: A kite for flying

Ali makes a kite for flying from two straight sticks, $AC = 90$ cm and $BD = 60$ cm. They cross at right angles at $O$ , the midpoint of $BD$ , and $AO = 25$ cm. String joins $A$ to $B$ , $B$ to $C$ , $C$ to $D$ and $D$ to $A$ .

(a) Explain why $AB = AD$ and $CB = CD$ .

(b) Give the most specific name of $ABCD$ .

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(a) Fold the kite along $AC$ . $BD$ crosses $AC$ at right angles, and $OB = OD = 30$ cm, so $B$ lands on $D$ , while $A$ and $C$ stay where they are. So $AB$ lands on $AD$ and $CB$ lands on $CD$ : $AB = AD$ and $CB = CD$ . $AC$ is a line of symmetry.

(b) $AB$ and $AD$ meet at $A$ ; $CB$ and $CD$ meet at $C$ . Two pairs of equal adjacent sides: $ABCD$ is a kite.

(c) Every rhombus has diagonals that bisect each other. Here $BD$ cuts $AC$ into $AO = 25$ cm and $OC = 90 - 25 = 65$ cm, which are not equal. So $AC$ is not bisected, and $ABCD$ is not a rhombus.

(a) $AC$ is a line of symmetry, so $AB = AD$ and $CB = CD$ (b) a kite (c) $AO = 25$ cm and $OC = 65$ cm, so the diagonals do not bisect each other, as the diagonals of every rhombus do.

Practice 12: Priya's claim

Priya says: "If the diagonals of a quadrilateral are equal and perpendicular, it is a square."

The points $P(1, 1)$ , $Q(9, 1)$ , $R(6, 6)$ and $S(4, 6)$ are joined in order.

(a) Show that $PQRS$ is an isosceles trapezium.

(b) Show that its diagonals $PR$ and $QS$ are equal and perpendicular.

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(a) $P$ and $Q$ have $y$ -coordinate 1, and $S$ and $R$ have $y$ -coordinate 6, so $PQ$ and $SR$ are horizontal: $PQ \parallel SR$ . From $P$ to $S$ is 3 right and 5 up; from $Q$ to $R$ is 3 left and 5 up. $PS$ and $QR$ lean opposite ways, so they are not parallel. Exactly one pair of parallel sides: a trapezium.

Folding along $x = 5$ carries $P$ onto $Q$ (each 4 units from the line) and $S$ onto $R$ (each 1 unit from it), so $PS$ lands on $QR$ : $PS = QR$ . So $PQRS$ is an isosceles trapezium.

(b) The same fold carries $PR$ onto $QS$ , so $PR = QS$ .

From $P$ to $R$ is 5 right and 5 up: $PR$ goes one square across for every square up, along the diagonals of the grid squares. The diagonal of a square bisects its right angle, so $PR$ makes 45° with $PQ$ . In the same way, $QS$ (5 left and 5 up) makes 45° with $QP$ .

The diagonals cross at $T(5, 5)$ . In $\triangle PTQ$ : $\angle PTQ = 180^{\circ} - 45^{\circ} - 45^{\circ} = 90^{\circ}$ (∠ sum of △: the angles of a triangle add up to 180°). So $PR$ and $QS$ are perpendicular.

(c) $PQRS$ has equal, perpendicular diagonals, but it is not a square: every square has four equal sides, and $PQ = 8$ units while $SR = 2$ units.

(a) $PQ \parallel SR$ , $PS$ is not parallel to $QR$ , and $PS = QR$ (b) $PR = QS$ , and they meet at 90° (c) no: $PQRS$ is an isosceles trapezium with equal, perpendicular diagonals, not a square.

Key Points

A quadrilateral is a special quadrilateral of a kind when it has every property in that kind's definition.
A square is a special rectangle and a special rhombus; rectangles and rhombuses are special parallelograms; a rhombus is a special kite.
The trapezium (exactly one pair of parallel sides) is on a branch of its own, with the isosceles trapezium below it.
A special case has every property that every shape above it in the family tree has (rows (a) of the table), and more.
Rows (a) of the table hold for every member, special cases included; rows (b) give the symmetry of the general shape only.
To find which shapes always have some properties, keep the columns with a ✓ in every one of those rows.
The most specific name is the lowest name in the family tree whose definition the shape meets.
To show a shape is not of some kind, find one property every shape of that kind has, which it lacks.
Always: give a reason. Sometimes: give one shape for which it is true and one for which it is false. Never: give a reason.
Perpendicular diagonals do not make a rhombus, equal diagonals do not make a rectangle, and one pair of sides parallel with the other pair equal does not make a parallelogram.