Chapter 13 · Two-dimensional figures

One Shape, Many Names: The Quadrilateral Hierarchy

Find every name that fits a quadrilateral, place the six quadrilateral names in a hierarchy, and tell which names a shape must have and which it might have.

14 min read

Think about a square floor tile. You call it a square. But in math, that same tile has five more names, and every one of them is true!

How can one shape have six names? Each shape name stands for a category. A category is a group of shapes that all share certain attributes. An attribute is a feature you can check, such as how many sides a shape has or how big its angles are.

So a name works like a test. If a shape has every attribute the test asks for, it earns that name. One shape can pass several tests, so one shape can earn several names. In this lesson, you will find every name that fits a shape, and you will see how the names fit together.

Six names and the test for each one

Name	A shape earns this name when it is
Quadrilateral	a closed shape with 4 straight sides
Trapezoid	a quadrilateral with at least 1 pair of parallel sides
Parallelogram	a quadrilateral with 2 pairs of parallel sides
Rectangle	a parallelogram with 4 right angles
Rhombus	a parallelogram with 4 equal sides
Square	a parallelogram with 4 right angles and 4 equal sides

Read each test closely. Three ideas will help you.

At least 1 pair means 1 pair or more. A quadrilateral with 1 pair of parallel sides is a trapezoid. A quadrilateral with 2 pairs is a trapezoid too, because 2 pairs is more than 1 pair. So every parallelogram is also a trapezoid. Rectangles, rhombuses and squares are parallelograms, so they are trapezoids too.

A test asks only for what it says. The rectangle test asks for a parallelogram with 4 right angles. It says nothing about side lengths, so all 4 sides may be equal. The rhombus test asks for a parallelogram with 4 equal sides. It says nothing about angles, so all 4 angles may be right angles.

Some tests have another name inside them. To be a rectangle, a shape must first be a parallelogram. So when a shape passes the rectangle test, it has passed the parallelogram test as well.

To find every name for a shape, try all six tests. Do not stop at the first name that fits.

Worked example

Every name for one shape

A quadrilateral has 2 pairs of parallel sides and 4 right angles. Its sides are 8 cm, 3 cm, 8 cm and 3 cm long. Give every name that fits this shape.

Start at the top of the table. The shape has 4 straight sides, so it is a quadrilateral.
Check the parallel sides. The shape has 2 pairs. The trapezoid test asks for at least 1 pair, and 2 pairs is more than 1 pair, so the shape is a trapezoid. The parallelogram test asks for 2 pairs, so the shape is a parallelogram too.
$2 \text{ pairs} > 1 \text{ pair}$
Check the angles. The rectangle test asks for a parallelogram with 4 right angles. The shape is a parallelogram, and each of its 4 angles measures 90 degrees. So it is a rectangle.
$90^\circ,\ 90^\circ,\ 90^\circ,\ 90^\circ$
Check the side lengths. The rhombus test asks for a parallelogram with 4 equal sides. This shape has sides of 8 cm and 3 cm, and 8 cm is not equal to 3 cm. So it is not a rhombus.
$8 \text{ cm} \neq 3 \text{ cm}$
The square test asks for a parallelogram with 4 right angles and 4 equal sides. The shape has the 4 right angles but not the 4 equal sides. So it is not a square.

Answer

Four names fit: this shape is a quadrilateral, a trapezoid, a parallelogram and a rectangle. It is not a rhombus or a square, because its 4 sides are not all the same length.

Did you notice that some tests have another name inside them? That lets you put the six categories in order.

A hierarchy is a way to arrange categories in levels. The most general category goes at the top. General means it has the fewest rules, so the most shapes fit in it. The most specific category goes at the bottom. Specific means it has the most rules.

Here is the rule for placing a category: put it under another category when every shape in it has all the attributes of that other category.

Try it with parallelogram and trapezoid. Every parallelogram has 2 pairs of parallel sides. That is at least 1 pair, which is what a trapezoid needs. So parallelogram goes under trapezoid.

In the table below, the column Directly under names the next category up. Keep stepping up until you reach the top. Every category you pass is above the one where you started. Square has two categories directly above it, so step up from both. Categories on the same level sit side by side, so neither is above the other.

The quadrilateral hierarchy, from most general to most specific

Level	Category	Directly under	Why it goes there
1 (top)	Quadrilateral	Nothing. It is the top.	It has the fewest rules: only 4 straight sides.
2	Trapezoid	Quadrilateral	Every trapezoid has 4 straight sides.
3	Parallelogram	Trapezoid	Every parallelogram has 2 pairs of parallel sides, and that is at least 1 pair.
4	Rectangle	Parallelogram	Every rectangle has 2 pairs of parallel sides.
4	Rhombus	Parallelogram	Every rhombus has 2 pairs of parallel sides.
5 (bottom)	Square	Rectangle and rhombus	Every square has 4 right angles and 4 equal sides.

Worked example

Side by side, or one under the other?

Look at levels 4 and 5 of the hierarchy. Why do rectangle and rhombus sit side by side, and not one under the other? Why does square go under both of them?

Try putting rectangle under rhombus. That works only if every rectangle has 4 equal sides. Test the rectangle from the first example, with sides of 8 cm, 3 cm, 8 cm and 3 cm. Its sides are not all equal, so it is not a rhombus. Rectangle cannot go under rhombus.
$8 \text{ cm} \neq 3 \text{ cm}$
Try putting rhombus under rectangle. That works only if every rhombus has 4 right angles. Test a rhombus with four 7 cm sides and angles of 60, 120, 60 and 120 degrees. None of its angles is 90 degrees, so it is not a rectangle. Rhombus cannot go under rectangle.
$60^\circ \neq 90^\circ \qquad 120^\circ \neq 90^\circ$
Neither one goes under the other. But every rectangle is a parallelogram, and so is every rhombus. So both go directly under parallelogram, side by side on level 4.
Now place square. Every square is a parallelogram with 4 right angles, so it has all the attributes of a rectangle. Every square also has 4 equal sides, so it has all the attributes of a rhombus. Square goes under both.
Check with a square whose sides are all 6 cm. It is a parallelogram with 4 right angles, so it passes the rectangle test. Its 4 sides are equal, so it passes the rhombus test too.
$6 \text{ cm} = 6 \text{ cm} = 6 \text{ cm} = 6 \text{ cm}$

Answer

Rectangle and rhombus sit side by side, because some rectangles are not rhombuses and some rhombuses are not rectangles. Square goes under both, because every square has 4 right angles and 4 equal sides. So a square is always a rectangle and a rhombus too.

Key concept

Names above are a must, names below are a might

A quadrilateral has every name whose test it passes. In the hierarchy, every shape in a category has all the attributes of each category above it.

So when you know one name for a quadrilateral:

Every name above it is a name it must have. It fits every time.
Every name below it is a name it might have. It fits only when the shape also has the extra attributes that name asks for.

Rectangle and rhombus sit side by side. A shape has both names only when it is a square.

A square is at the bottom. So a square must also be a rectangle, a rhombus, a parallelogram, a trapezoid and a quadrilateral.

Worked example

Must have or might have?

You are told one thing about a shape: it is a rhombus. Which other names must it have? Which other names might it have?

Find rhombus in the hierarchy and step up. Rhombus is directly under parallelogram. Parallelogram is under trapezoid, and trapezoid is under quadrilateral. Every rhombus has 2 pairs of parallel sides, which is at least 1 pair, and it has 4 sides. So it must have all three names.
Look below rhombus. Square is under it. A square needs 4 right angles as well as 4 equal sides. A rhombus might have 4 right angles, or it might not. So square is a name it might have.
Look beside rhombus. Rectangle is on the same level. A rhombus is already a parallelogram, so it is a rectangle only when it has 4 right angles. That is the same extra attribute that makes it a square. So rectangle is a name it might have too.
Test two rhombuses with 7 cm sides. The first has angles of 60, 120, 60 and 120 degrees. It has no right angles, so it is not a rectangle or a square. The second has four angles of 90 degrees, so it is a rectangle and a square. Both are parallelograms, trapezoids and quadrilaterals.

Answer

A rhombus must also be a parallelogram, a trapezoid and a quadrilateral. It might also be a rectangle and a square. It is both of these when it has 4 right angles, and neither when it does not.

The big idea. A shape name is a test, and a quadrilateral earns every name whose test it passes. The hierarchy shows how the six names fit together: quadrilateral at the top, then trapezoid, then parallelogram, then rectangle and rhombus side by side, with square under both. Once you know one name for a quadrilateral, it must have every name above that one, and it might have the names below it.

Next, you will place the types of triangle in a hierarchy. You will also build and read hierarchy diagrams for two-dimensional figures.

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Describe and Justify Classifications of Figures

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