Grade 5 · 5.G.B.3

Parallelograms, rectangles, rhombuses and squares: what each one guarantees

A parallelogram, a rectangle, a rhombus and a square all have four sides. What does each name promise, and why does a square keep every promise the others make?

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Shapes in more than one category, and quadrilaterals in noneGrade 3 Parallel linesGrade 4

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Classifying shapes in a hierarchyGrade 5

Where you are 7 in Quadrilaterals

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Two-dimensional shapes15

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▴ 2 above Rectangles, rhombuses and squares Rhombuses, rectangles and squares Shapes in more than one category, and quadrilaterals in none The quadrilateral family tree

Parallelograms, rectangles, rhombuses and squares: what each one guarantees

Why this exists

At the kite festival, every toy that flies is called a kite. In this lesson the toys are always called toy kites, and the word kite on its own means a shape, which comes in Part 4. A toy kite’s sail can be almost any shape: a parallelogram, a rectangle, a rhombus, a square, a trapezoid, or a kite shape.

Golnaz sews edge tape around the sails, and Dumisani judges the sail contest. Neither of them wants to measure every side and every corner of every sail. A shape’s name can save that work, because each name is a promise. Once you know a sail is a parallelogram, you know things about it that nobody measured.

The second saving is bigger. These names fit inside one another, as in Shapes in more than one category, and quadrilaterals in none: every square is also a rectangle, and every rectangle is also a parallelogram. So a promise made to every parallelogram is kept by every rectangle and every square too. You learn a fact once, at the top, and it holds all the way down.

Definition

Two words from earlier lessons do most of the work. Two sides are parallel when the lines they lie on never meet, however far they go, like the two rails of a track (Parallel lines). Two sides are perpendicular when they meet in a right angle, a square corner of 90^\circ (Perpendicular lines). We write AB \parallel CD for “side AB is parallel to side CD”.

A quadrilateral is a flat shape with four straight sides. Six special quadrilaterals have names:

A parallelogram is a quadrilateral with two pairs of parallel sides: each side is parallel to the side across from it.
A rectangle is a parallelogram with four right angles.
A rhombus is a parallelogram with four sides of equal length.
A square is a parallelogram with four right angles and four equal sides. So a square is a rectangle and a rhombus at the same time.
A trapezoid is a quadrilateral with at least one pair of parallel sides.
A kite is a quadrilateral with two pairs of equal sides, where the two sides in each pair are next to each other (they meet at a corner).

About the trapezoid. This lesson uses the “at least one pair” definition. Under it, a parallelogram, with its two pairs, counts as a trapezoid. Some books use “exactly one pair” instead, and under that definition no parallelogram is a trapezoid. Both are in use, so when you read about trapezoids, check which one the book means.

In Figure 1, sides with the same number of ticks are equal in length, sides with the same number of arrows are parallel, and a small square in a corner marks a right angle.

Figure 1The six special four-sided shapes. Matching ticks mean equal lengths, matching arrows mean parallel sides, and a small square marks a right angle.

Intuition

Part 1: what a parallelogram guarantees

A parallelogram is defined by its parallel sides only. Nothing in the definition mentions lengths or angles. Yet if Golnaz cuts out any parallelogram and measures it with a ruler and a protractor, she always finds three more things:

Opposite sides are equal in length. The top is as long as the bottom, and the left side is as long as the right.
Opposite angles are equal. The angle at one corner matches the angle at the corner across from it.
Two angles next to each other add to 180^\circ. Walk along any one side: the angles at its two ends together make a straight angle.

Here is a way to see the first two without a ruler. Trace a parallelogram on tracing paper, put a pencil point at its center, and turn the paper half a turn. The tracing lands exactly on the shape again. The bottom side lands on the top side, so they are the same length, and each corner lands on the corner across from it, so those angles are the same size.

Part 2: a rectangle is a parallelogram with square corners

A rectangle is not a different kind of thing from a parallelogram. It is a parallelogram that happens to have four right angles. So everything a parallelogram promises, a rectangle keeps: its opposite sides are parallel and equal, its opposite angles are equal (all are 90^\circ), and neighboring angles add to 90^\circ + 90^\circ = 180^\circ.

Some books define a rectangle as “a quadrilateral with four right angles” and do not mention parallel sides. Nothing is lost. The left and right sides of such a shape both make a right angle with the bottom side, and two sides that are both perpendicular to the same side are parallel. The same goes for the top and bottom. So a quadrilateral with four right angles is always a parallelogram.

Part 3: a rhombus is a parallelogram with equal sides

A rhombus is a parallelogram whose four sides all have the same length. It may lean, the way a diamond on a playing card leans, or it may stand upright. Being a parallelogram, it keeps all the parallelogram promises: opposite sides parallel, opposite angles equal, neighboring angles adding to 180^\circ. Its own promise is the equal sides.

As with the rectangle, the order can be turned around. A quadrilateral with four equal sides is always a rhombus: its opposite sides always come out parallel. Push on a frame of four equal sticks and it changes its angles, but its opposite sticks stay parallel.

Part 4: the square, the trapezoid and the kite

A square is a rectangle and a rhombus at once, so it collects the promises of both, and of the parallelogram above them: four right angles, four equal sides, two pairs of parallel sides.

A trapezoid promises one thing only: at least one pair of parallel sides. Every parallelogram keeps that promise, so (under the “at least one pair” definition) every parallelogram, rectangle, rhombus and square is also a trapezoid.

A kite promises two pairs of equal sides that meet at corners. Fold a kite along the line through the two corners where equal sides meet, as in Figure 2. The two halves land exactly on each other. So the kite has one more promise: the two angles at the other two corners are equal.

Figure 2Fold a kite along the line through the corners where equal sides meet, and the two halves match.

A rhombus is a kite too: all four of its sides are equal, so any two sides that meet at a corner are equal. That makes every square a kite as well. A kite need not be a parallelogram, though; the kite in Figure 2 has no parallel sides at all.

Results

What each shape guarantees.

Each shape keeps its own promise from the definition, and also every promise of the shapes listed beside it.

Shape	Also keeps the promises of
parallelogram	trapezoid
rectangle	parallelogram, trapezoid
rhombus	parallelogram, trapezoid, kite
square	rectangle, rhombus, parallelogram, trapezoid, kite
trapezoid	quadrilateral only
kite	quadrilateral only

The parallelogram promises. In every parallelogram, opposite sides are equal in length, opposite angles are equal, and two angles next to each other add to 180^\circ.

The kite promise. In every kite, the two angles at the corners where a side from one equal pair meets a side from the other pair are equal.

Inheritance. If every shape of one kind is also a shape of a second kind, then everything promised for the second kind is true for the first. Promises pass down, from the bigger group to the smaller group inside it. They never pass up.

Why it is true

The parallelogram promises can be checked by measuring and by the tracing paper half-turn of Part 1. A full proof, using the angles that parallel lines make with a line crossing them, comes in Properties of a parallelogram.

Inheritance needs no measuring at all. A rectangle is a parallelogram. If a statement is true of every parallelogram, and this rectangle is one of the parallelograms, then the statement is true of this rectangle. The same reasoning works for every row of the table.

That reasoning only runs one way. A statement true of every rectangle, such as “it has four right angles”, need not be true of every parallelogram, because most parallelograms are not rectangles.

The kite promise comes from the fold. The fold line passes through the corner where the two short sides meet and the corner where the two long sides meet. Folding lays each short side on the other short side, since they are the same length, and each long side on the other long side. So the left corner, where a short side meets a long side, lands on the right corner, where the other short side meets the other long side, and the angles there match.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.

Example 1

Measuring Pono’s sail

Pono’s toy kite has a sail shaped like parallelogram ABCD (Figure 3). Golnaz measures AB = 48 cm, AD = 30 cm and the angle at A, which is 65^\circ. Without measuring again, find the other two sides, the other three angles, and the length of edge tape that goes once around the sail.

Figure 3Pono’s sail. Two sides and one angle are measured; the rest are promised.

Solution

The sail is a parallelogram, so its opposite sides are equal. Side CD is across from AB, so CD = 48 cm. Side BC is across from AD, so BC = 30 cm.

Opposite angles are equal, and C is across from A, so the angle at C is 65^\circ.

The angles at A and B sit at the two ends of side AB, so they add to 180^\circ. The angle at B is 180^\circ - 65^\circ = 115^\circ. The angle at D is across from B, so it is also 115^\circ.

The edge tape runs along all four sides:

48 + 30 + 48 + 30 = 156

Golnaz needs 156 cm of tape. As a check, she sets her protractor on corner B and reads about 115^\circ.

Example 2

A sail that breaks a promise

Dumisani judges the parallelogram class. One entry has sides measuring 40 cm, 25 cm, 40 cm and 31 cm, going around the sail in order. Can it be a parallelogram?

Solution

Going around in order, the first and third sides are across from each other, and so are the second and fourth. Every parallelogram has equal opposite sides. The first pair passes: 40 = 40. The second pair fails: 25 cm and 31 cm are 6 cm apart. A ruler can be off by a few millimeters, but not by 6 cm.

One broken promise is enough. Whatever the sail is, it is not a parallelogram, and Dumisani moves it out of the class. Notice what the test does: a promise that fails rules a shape out. A promise that holds is good news but, on its own, is only a measurement. That equal opposite sides are in fact enough to make a parallelogram is proved in later geometry.

Example 3

Two measurements for a whole banner

The festival banner is a rectangle. Golnaz measures one long side, 90 cm, and one short side, 35 cm, and stops. Explain why that is enough to know all four sides and all four angles, and find the length of ribbon to go once around the banner.

Solution

A rectangle is a parallelogram, so it keeps the parallelogram promise of equal opposite sides. The long side across from the measured one is also 90 cm, and the short side across from the measured one is also 35 cm. All four sides are known: 90, 35, 90 and 35 cm.

The angles need no measuring at all: a rectangle has four right angles by definition, so each is 90^\circ.

The ribbon runs around all four sides:

90 + 35 + 90 + 35 = 250

Golnaz needs 250 cm of ribbon. She measured two lengths, and the name “rectangle” handed her the other six facts.

Example 4

One square corner is enough

Pono builds a sail frame whose opposite sticks are parallel, so the frame is a parallelogram. He checks one corner with a square corner of paper and it fits exactly: that angle is 90^\circ. Is the frame a rectangle?

Solution

The other three angles follow from the parallelogram promises.

The corner next to the right angle shares a side with it, so the two add to 180^\circ. That corner is 180^\circ - 90^\circ = 90^\circ.

The corner across from the right angle is equal to it: 90^\circ. The fourth corner is across from the second one, so it is 90^\circ too.

All four angles are right angles, so the frame is a parallelogram with four right angles: a rectangle. Pono only had to check one corner. This works because the frame was a parallelogram to begin with. A kite shape, for instance, can have one right angle at its top corner and no other right angle.

Example 5

A rhombus sail

Golnaz cuts a sail shaped like rhombus EFGH, with EF = 35 cm and the angle at E equal to 50^\circ. Find every side, every angle, and the length of tape around the edge.

Solution

A rhombus has four equal sides by definition, so EF, FG, GH and HE are each 35 cm.

The angles come from the parallelogram promises, since a rhombus is a parallelogram. The angle at G is across from E, so it is 50^\circ. The angle at F is next to E, along side EF, so it is 180^\circ - 50^\circ = 130^\circ. The angle at H is across from F, so it is 130^\circ.

The tape goes around four sides of 35 cm:

4 \times 35 = 140

Golnaz needs 140 cm of tape. The equal sides came from the rhombus definition, and the angles came from the parallelogram above it.

Example 6

The frame that leans

Pono joins four sticks, each 60 cm long, with a loose pin at each corner. Standing upright, the frame makes the left shape in Figure 4. Pushed over, it makes the right shape, with a 60^\circ corner. Which facts stay true as the frame leans, and which change?

Figure 4Four equal sticks make a rhombus in every position, and a square in only one.

Solution

Four equal sides make a rhombus in every position, so the frame is always a rhombus, and therefore always a parallelogram. Whatever its lean, it keeps every promise of both:

all four sides are 60 cm, because the sticks do not stretch;
opposite sides are parallel;
opposite angles are equal, and neighboring angles add to 180^\circ.

In the leaning position the angles are 60^\circ, 180^\circ - 60^\circ = 120^\circ, 60^\circ and 120^\circ, going around.

What changes is the size of the angles. Upright, all four are 90^\circ, so the frame is a rhombus with four right angles: a square. Leaning, it has no right angle, so it is not a rectangle and not a square. Every square is a rhombus, but a rhombus is a square only in the one position where its corners are right angles.

Example 7

Everything a square floor tile promises

The judges’ tent has a floor of square tiles, each 25 cm on a side. Name every shape the tile is, say what each name promises, and find the distance around one tile.

Solution

Start at the square and climb up through every name it carries.

Square: four right angles and four equal sides.
Rectangle: it is a parallelogram with four right angles.
Rhombus: it is a parallelogram with four equal sides.
Parallelogram: two pairs of parallel sides, opposite sides equal, opposite angles equal, neighboring angles adding to 180^\circ.
Trapezoid: at least one pair of parallel sides; it has two.
Kite: two pairs of equal sides meeting at corners. Every side is 25 cm, so any two sides that meet are equal.
Quadrilateral: four straight sides.

So the tile carries seven names, and every promise on the list holds for it. The distance around it is

4 \times 25 = 100

that is, 100 cm.

Example 8

Two ways to define a trapezoid

Dumisani runs a trapezoid class at the contest. Three sails are entered. Sail (a) has exactly one pair of parallel sides. Sail (b) is a rectangle, 80 cm by 50 cm. Sail (c) is a kite shape with sides 40, 40, 70 and 70 cm in order around it, and no parallel sides. Which sails are trapezoids under the “at least one pair” definition, and which under “exactly one pair”? Which are kites?

Solution

Count the pairs of parallel sides. Sail (a) has one pair. Sail (b), a rectangle, is a parallelogram, so it has two pairs. Sail (c) has none.

Under “at least one pair”, sails (a) and (b) are trapezoids, since one pair and two pairs are both at least one. Sail (c) is not.

Under “exactly one pair”, only sail (a) is a trapezoid. Sail (b) has too many pairs, and sail (c) has none.

Now the kites. Sail (c) has two 40 cm sides meeting at a corner and two 70 cm sides meeting at a corner, so it is a kite shape. Sail (b) is not: its sides that meet are 80 cm and 50 cm, never equal. Sail (a) is not known to be a kite, since nothing is said about its sides. All three are sails of toy kites, but only one is a kite shape. The definition decides the class, so Dumisani announces which definition he is using before he judges.

Boundaries

Four straight sides, on a flat surface. Every name here is for flat shapes with four straight sides. A sail with a curved edge is none of them.
Promises pass down, never up. Every rectangle keeps the parallelogram promises, but a parallelogram need not keep the rectangle’s: most parallelograms have no right angle. The same holds between rhombus and parallelogram, and between square and everything above it.
A name gives a guarantee; a measurement gives an estimate. A protractor reading of 89^\circ may be a right angle measured a little roughly. A shape is a rectangle because it is known to have four right angles, not because its corners look square.
Trapezoid has two definitions. Under “at least one pair of parallel sides”, every parallelogram is a trapezoid. Under “exactly one pair”, none is. Say which one is in use.
In a kite the equal sides must meet. Equal sides that sit across from each other, as in a parallelogram that is not a rhombus, do not make a kite.
Many quadrilaterals have no special name. Four sides of different lengths with no parallel pair make a quadrilateral and nothing more.

Where this goes next

The next lesson, Classifying shapes in a hierarchy, draws all of these names as one family tree and sorts triangles the same way, by which group sits inside which.

Much later, the same family returns with new tools. The diagonals of each shape (the lines joining opposite corners) turn out to make promises of their own, and each promise in this lesson gets a proof. That is the quadrilateral family tree of geometry, and the sibling lessons on rectangles, rhombuses and squares and on trapezoids and kites.

Exercises

Part 1: what a parallelogram guarantees

In parallelogram PQRS, PQ = 52 cm and QR = 27 cm. Find RS, SP, and the distance around the parallelogram.
One angle of a parallelogram measures 72^\circ. Find the other three angles, going around in order.
Golnaz measures a sail’s sides in order around it: 45 cm, 28 cm, 45 cm and 30 cm. Can the sail be a parallelogram? Explain.
Dumisani measures a sail’s angles in order around it: 75^\circ, 105^\circ, 75^\circ and 115^\circ. Can it be a parallelogram? Explain.
The distance around a parallelogram is 140 cm, and one of its sides is 44 cm. Find the lengths of the other three sides.
In parallelogram WXYZ, the angle at W measures 58^\circ. Golnaz says the angle at X is also 58^\circ. Pono says the angle at Y is. Who is right? Find the angle at X.

Part 2: rectangles

A rectangular banner is 120 cm long and 45 cm wide. Give all four side lengths and the distance around it.
List four promises a rectangle keeps because it is a parallelogram.
The distance around a rectangle is 96 cm, and one side is 30 cm. Find the other three sides.
Pono measures a parallelogram’s angles as 88^\circ, 92^\circ, 88^\circ and 92^\circ. Is it a rectangle? Explain.
Two rectangular floor tiles, each 60 cm by 30 cm, are laid with their long sides touching. What shape do they make together, and what is the distance around it? Give its most exact name.

Part 3: rhombuses

A rhombus has a side of 28 cm. Find the distance around it.
One angle of a rhombus measures 40^\circ. Find the other three angles, going around in order.
The distance around a rhombus is 180 cm. How long is each side?
Golnaz joins four sticks, each 55 cm long, at loose corners. Which names is the frame guaranteed to have in every position? Which names does it have in only one position?
A rectangle has two sides of 30 cm that meet at a corner. Use the promises of this lesson to show it is a square.

Part 4: squares, trapezoids and kites

The distance around a square tile is 132 cm. How long is each side?
A kite shape has two short sides of 30 cm and two long sides of 50 cm. Find the distance around it, and say which sides meet at a corner.
In a kite shape, the angle where one short side meets one long side measures 115^\circ. Which other angle must also measure 115^\circ? Explain with folding.
For each shape, say whether it is a trapezoid under “at least one pair” and under “exactly one pair”: (a) a rectangle 70 cm by 40 cm; (b) a kite shape with no parallel sides; (c) a quadrilateral with exactly one pair of parallel sides.
A sail has four sides of 35 cm, two pairs of parallel sides, and angles of 70^\circ, 110^\circ, 70^\circ and 110^\circ. List every name it has, and the names it does not have from the six in this lesson.

Reasoning

Explain why every promise made to all parallelograms is also true of every square.
Dumisani says, “A rectangle cannot be a parallelogram, because parallelograms lean.” Explain what is wrong.
Is every rhombus a kite? Is every kite a rhombus? Explain using the sides.
Explain why a quadrilateral with four right angles must have both pairs of opposite sides parallel.

Answers

RS = 52 cm and SP = 27 cm, because opposite sides of a parallelogram are equal. The distance around is 52 + 27 + 52 + 27 = 158 cm.
108^\circ, 72^\circ, 108^\circ. The neighbor of the 72^\circ angle is 180^\circ - 72^\circ = 108^\circ, and each angle across matches.
No. The second and fourth sides are across from each other, and 28 cm and 30 cm are not equal, so the promise of equal opposite sides is broken.
No. The second and fourth angles are across from each other, and 105^\circ and 115^\circ are not equal.
44 cm, 26 cm and 26 cm. The side across from the 44 cm side is also 44 cm, which uses 88 cm. The other two sides share 140 - 88 = 52 cm equally, so each is 26 cm.
Pono is right: Y is across from W, so it measures 58^\circ. The angle at X is next to W, so it is 180^\circ - 58^\circ = 122^\circ.
120, 45, 120 and 45 cm. The distance around is 120 + 45 + 120 + 45 = 330 cm.
Opposite sides are parallel; opposite sides are equal in length; opposite angles are equal; neighboring angles add to 180^\circ. (It is also a trapezoid, with at least one pair of parallel sides.)
30 cm, 18 cm and 18 cm. The two 30 cm sides use 60 cm, and 96 - 60 = 36 cm is shared by the other two, 18 cm each.
No. A rectangle needs four angles of exactly 90^\circ, and none of these is. It is still a parallelogram: opposite angles match and 88^\circ + 92^\circ = 180^\circ.
A square with sides of 60 cm; the distance around is 240 cm. Each long side is 60 cm and the two widths together make 30 + 30 = 60 cm, so all four sides are 60 cm, and every corner is a tile corner, a right angle. It is a rectangle with four equal sides, so its most exact name is square.
4 \times 28 = 112 cm.
140^\circ, 40^\circ, 140^\circ. The neighbor is 180^\circ - 40^\circ = 140^\circ, and angles across match.
180 \div 4 = 45 cm, since all four sides are equal.
In every position: rhombus, parallelogram, trapezoid, kite and quadrilateral, since four equal sides always make a rhombus. Only in the upright position, with right angles: rectangle and square.
A rectangle is a parallelogram, so opposite sides are equal. The side across from each 30 cm side is also 30 cm, so all four sides are 30 cm. A parallelogram with four equal sides is a rhombus, and a shape that is a rectangle and a rhombus at once is a square.
132 \div 4 = 33 cm.
30 + 30 + 50 + 50 = 160 cm. The two 30 cm sides meet at one corner, and the two 50 cm sides meet at the opposite corner.
The angle at the other corner where a short side meets a long side. Folding along the line through the corner of the two short sides and the corner of the two long sides lays one half on the other, so those two corners land on each other, and both angles are 115^\circ.
(a) “At least one pair”: yes, it has two pairs. “Exactly one pair”: no. (b) No under both, since it has no parallel sides. (c) Yes under both.
Rhombus, parallelogram, trapezoid, kite and quadrilateral. Not a rectangle and not a square, because its angles are 70^\circ and 110^\circ, not 90^\circ.
A square has two pairs of parallel sides, so it is one of the parallelograms. A statement true of every parallelogram is therefore true of this one.
A parallelogram only needs two pairs of parallel sides; it is allowed to lean but does not have to. A rectangle has two pairs of parallel sides, so it is a parallelogram that happens not to lean.
Every rhombus is a kite: all four sides are equal, so the two sides at each corner are equal. Not every kite is a rhombus: a kite with two 30 cm sides and two 50 cm sides has unequal sides, so it is not a rhombus.
The left and right sides each make a right angle with the bottom side, so both are perpendicular to the same side, and two sides perpendicular to the same side are parallel. The top and bottom sides are both perpendicular to the left side, so they are parallel too.