Chapter 13 · Two-dimensional figures

Describe and Justify Classifications of Figures

Use properties to say how types of quadrilaterals are related, give reasons for a figure's names, and find and fix classification mistakes.

14 min read

A square has more than one name. It is also a rectangle, a rhombus, a parallelogram and a quadrilateral, which is a closed figure with 4 straight sides. How can one figure have so many names?

Each name is a category: a group of figures that share certain properties. A property, also called an attribute, is a fact about a figure's sides or angles that you can check. You have already sorted these categories into a hierarchy, where each smaller group sits inside a bigger one.

Now you will use that hierarchy to do three jobs:

describe how two categories are related
give reasons for the names you choose
find and fix mistakes

Properties table: what each category needs

Category	Properties it must have	Sits directly under
Quadrilateral	4 sides	Nothing here. It is the top category.
Parallelogram	4 sides and 2 pairs of parallel sides	Quadrilateral
Rectangle	4 sides, 2 pairs of parallel sides and 4 right angles	Parallelogram
Rhombus	4 sides, 2 pairs of parallel sides and 4 equal sides	Parallelogram
Square	4 sides, 2 pairs of parallel sides, 4 right angles and 4 equal sides	Rectangle and Rhombus

Look down the table. Each category keeps all the properties of the category it sits under, and then adds one more. That is what lets you say how two categories are related.

Take the sentence "every rhombus is a parallelogram." It makes a big promise. It says: pick any rhombus at all, and it will have everything a parallelogram needs.

To test a promise like that, ask one question: Does every figure in the first category have all the properties the second category needs? Then swap the two categories and ask again. The two answers can be different, so always ask both ways.

Key concept

A lower category keeps every property above it

A figure belongs to a category when it has every property that category needs. In a hierarchy, one category sits under another only when all of its figures have all the properties of the one above.

So the link runs one way. From the lower category to the higher one, you can say every: every square is a rectangle. From the higher one to the lower one, you can only say some: some rectangles are squares.

Worked example

Rhombus and parallelogram

How are rhombuses and parallelograms related? Use the properties table.

Write down what each category needs, because the properties decide everything. A parallelogram needs 4 sides and 2 pairs of parallel sides. A rhombus needs those and 4 equal sides as well.
Ask the question one way: does every rhombus have what a parallelogram needs? Yes. Every rhombus must have 4 sides and 2 pairs of parallel sides, so every rhombus passes.
Swap the categories: does every parallelogram have what a rhombus needs? Try a parallelogram with sides 7 cm, 4 cm, 7 cm and 4 cm. Its sides are not all equal, so it is not a rhombus. One figure that fails is enough to say "not every."
$7\text{ cm} \neq 4\text{ cm}$

Answer

Every rhombus is a parallelogram, but not every parallelogram is a rhombus. A parallelogram is a rhombus only when it has 4 equal sides.

The same two-way test on four more pairs

Two categories	How they are related	The reason
Square and rectangle	Every square is a rectangle. Not every rectangle is a square.	A rectangle does not need 4 equal sides.
Square and rhombus	Every square is a rhombus. Not every rhombus is a square.	A rhombus does not need 4 right angles.
Rectangle and parallelogram	Every rectangle is a parallelogram. Not every parallelogram is a rectangle.	A parallelogram does not need 4 right angles.
Rectangle and rhombus	Neither is always the other. A figure is both only when it is a square.	A rectangle does not need 4 equal sides. A rhombus does not need 4 right angles.

To classify a figure is to say which categories it belongs to. To justify a classification is to give the reason for it. A good reason names properties, not looks.

"It looks like a diamond" is not a reason, because turning a figure changes how it looks but not what it is. "It has 2 pairs of parallel sides and 4 equal sides" is a reason. Those are exactly the properties that make a quadrilateral a rhombus.

So a justification has two parts: the name, then the properties that earn it. To show that a name does not fit, point to one property the figure is missing.

Worked example

Justify the names for figure Q

Figure Q is a quadrilateral with 2 pairs of parallel sides. Every side is 5 cm long. Its angles measure $70^\circ$ , $110^\circ$ , $70^\circ$ and $110^\circ$ . Which of the five names in the properties table fit figure Q? Justify each answer with properties.

Start at the top of the properties table. Figure Q has 4 sides, so it is a quadrilateral. It has 2 pairs of parallel sides as well, so it is a parallelogram.
Check the sides. A rhombus needs 4 equal sides on top of what a parallelogram needs. All four sides of figure Q are 5 cm long, so figure Q is a rhombus.
$5\text{ cm} = 5\text{ cm} = 5\text{ cm} = 5\text{ cm}$
Check the angles. A rectangle needs 4 right angles, and a right angle measures 90°. The angles of figure Q are 70° and 110°, so it is not a rectangle.
$70^\circ \neq 90^\circ \text{ and } 110^\circ \neq 90^\circ$
A square needs 4 right angles too. Figure Q has none, so it is not a square, even though it has 4 equal sides.

Answer

Figure Q is a quadrilateral, a parallelogram and a rhombus, because it has 4 sides, 2 pairs of parallel sides and 4 equal sides. It is not a rectangle or a square, because it has no right angles.

Now turn the same checks on someone else's work. Classification mistakes usually come in two kinds.

A missing name. The names in the work are true, but the work leaves out a name that fits. To catch it, test every category in the properties table, not only the first one that fits.
A wrong place in the hierarchy. The work puts a category under one whose properties it does not always have. To catch it, read the link as an "every" sentence and look for one figure that breaks it.

Worked example

Find and fix Mia's mistakes

Mia's quadrilateral has 2 pairs of parallel sides and 4 right angles. Its sides are 8 cm, 3 cm, 8 cm and 3 cm. She writes: "From the properties table, only two names fit: rectangle and quadrilateral." In her hierarchy she also puts Rectangle directly under Rhombus. Find and fix her two mistakes.

Check the names Mia wrote. Her figure has 4 sides, so quadrilateral fits. It also has 2 pairs of parallel sides and 4 right angles, so rectangle fits. Both of her names are true.
Look for a missing name. A parallelogram needs only 4 sides and 2 pairs of parallel sides. Mia's figure has them, so parallelogram fits as well. Mia left it out.
Test the last two names. A rhombus and a square both need 4 equal sides. This figure has sides of 8 cm and 3 cm, so neither name fits.
$8\text{ cm} \neq 3\text{ cm}$
Check her link. Rectangle under Rhombus says "every rectangle is a rhombus." Mia's own figure breaks that sentence: it is a rectangle, but its sides are not all equal. The link is wrong.
Fix the hierarchy. Every rectangle has what a parallelogram needs, so Rectangle goes directly under Parallelogram, beside Rhombus. Square sits under both of them.

Answer

Mia left out the name parallelogram, so the full list from the properties table is quadrilateral, parallelogram and rectangle. Rectangle does not belong under Rhombus, because a rectangle does not need 4 equal sides. It belongs directly under Parallelogram.

Here is the idea to keep. The names of quadrilaterals are not separate labels. They are categories linked by properties. A lower category has every property of the one above it, so you can say "every square is a rectangle" but only "some rectangles are squares."

When you describe a relationship, justify a name, or check someone's work, do the same thing each time: list the properties the category needs, then check them one by one.