Chapter 13 · Two-dimensional figures
Subcategories: Always, Sometimes or Never True?
Learn why a subcategory keeps every attribute of its category. Then judge statements about figures as always, sometimes or never true, and fix common mistakes.
You already sort figures (flat shapes) by their attributes. An attribute is a feature of a figure that you can check, such as how many sides it has, how long its sides are or how big its angles are. A category is a group of figures that all share certain attributes.
Some categories sit inside bigger ones. A subcategory is a category inside a bigger category, so every figure in the subcategory also belongs to the bigger category. Think of the Venn diagrams you have used to sort figures. There, a subcategory is a ring drawn inside a bigger ring.
In this lesson you will use subcategories to explain why figures have the attributes they have, to judge statements about figures and to catch mistakes.
| Category | Definition | It is a subcategory of |
|---|---|---|
Quadrilateral | A closed figure (no gaps) with four straight sides | No category in this table |
Trapezoid | A quadrilateral with at least one pair of parallel sides | Quadrilateral |
Parallelogram | A quadrilateral with two pairs of parallel sides | Trapezoid |
Rectangle | A parallelogram with four right angles | Parallelogram |
Rhombus | A parallelogram with four equal sides | Parallelogram |
Square | A rectangle with four equal sides | Rectangle and rhombus |
Triangle | A closed figure with three straight sides | No category in this table |
Isosceles triangle | A triangle with at least two equal sides | Triangle |
Equilateral triangle | A triangle with three equal sides | Isosceles triangle |
Look at how each definition builds on another row. A rectangle is a parallelogram. It just has one more attribute: four right angles. A square is a rectangle. It just has one more attribute: four equal sides. So the table holds these chains:
- Every square is a rectangle, and every rectangle is a parallelogram.
- Every square is a rhombus, and every rhombus is a parallelogram.
Why is a square a rhombus? A square is a rectangle, so it is a parallelogram. It also has four equal sides. A parallelogram with four equal sides is a rhombus.
Now look at the words at least in the table. At least one means one or more. A parallelogram is a quadrilateral with two pairs of parallel sides, and , so it has at least one pair. That makes every parallelogram a trapezoid. Every rectangle, rhombus and square is a parallelogram, so each of them is a trapezoid as well.
Triangles work the same way. At least two means two or more. An equilateral triangle is a triangle with three equal sides, and , so it has at least two equal sides. That makes every equilateral triangle an isosceles triangle.
Key concept
A subcategory keeps every attribute of its category
If every figure in a category has an attribute, then every figure in each of its subcategories has that attribute too. The reason is that each figure in a subcategory is also a figure in the bigger category.
For example, every rectangle has four right angles. Every square is a rectangle. So every square has four right angles.
This idea runs one way: from the bigger category down to its subcategories. An attribute that every figure in a subcategory has may or may not belong to every figure in the bigger category. Every square has four equal sides, but not every rectangle does.
Worked example
Passing an attribute down a chain
You know only one thing about a figure: it is a square. Use the table to explain why it must have two pairs of parallel sides.
Start at the square row. It says a square is a rectangle with four equal sides. So your figure is a rectangle.
Move to the rectangle row. It says a rectangle is a parallelogram with four right angles. So your figure is a parallelogram too.
Use the parallelogram row. Every parallelogram has two pairs of parallel sides. Your figure is a parallelogram, so it has this attribute. The attribute passed down two steps: from parallelograms to rectangles, then from rectangles to squares.
Answer
The square must have two pairs of parallel sides, because every square is a rectangle, every rectangle is a parallelogram and every parallelogram has two pairs of parallel sides.
Now you can judge statements about figures. A statement is a sentence that makes a claim, such as A rhombus is a trapezoid. Read it as a claim about the whole category it names first. Here that is every rhombus. A figure fits the statement when the statement is true for that figure. Then choose one word.
- Always true: every figure in that category fits.
- Sometimes true: at least one figure in that category fits, and at least one does not.
- Never true: no figure in that category fits.
To justify always or never, use an attribute that every figure in the category has. One example cannot prove always, because a different figure in the category might not fit. To justify sometimes, give two examples: one figure that fits and one that does not.
| Statement | Always, sometimes or never? | Why |
|---|---|---|
A rhombus is a trapezoid. | Always | Every rhombus is a parallelogram, so it has two pairs of parallel sides. Two pairs is at least one pair. |
A trapezoid is a rhombus. | Sometimes | A square fits. A trapezoid with parallel sides of 4 cm and 8 cm does not, because its sides are not all equal. |
An equilateral triangle is isosceles. | Always | It has three equal sides, and three equal sides is at least two equal sides. |
An isosceles triangle is equilateral. | Sometimes | A triangle with sides of 5 cm, 5 cm and 5 cm fits. One with sides of 5 cm, 5 cm and 8 cm does not. |
A triangle is a quadrilateral. | Never | Every triangle has three sides. A quadrilateral needs four sides. |
A rectangle has an angle of . | Never | Every rectangle has four right angles, so each of its angles measures . |
Worked example
Always, sometimes or never?
Decide whether this statement is always, sometimes or never true, and justify your choice. Statement: A rectangle is a rhombus.
Find what the statement asks for. The table says a rhombus is a parallelogram with four equal sides. So ask: is every rectangle a parallelogram with four equal sides?
Check the parallelogram part. A rectangle is a parallelogram with four right angles, so every rectangle is a parallelogram. Nothing in the table says that a rectangle's sides must all be equal, so test that part with examples.
Look for a rectangle that fits. A rectangle with sides of 4 cm, 4 cm, 4 cm and 4 cm is a parallelogram with four equal sides, so it is a rhombus. This rules out the word never.
Look for a rectangle that does not fit. A rectangle with sides of 3 cm, 5 cm, 3 cm and 5 cm has sides of two different lengths. Its four sides are not equal, so it is not a rhombus. This rules out the word always.
Choose the word that is left. At least one rectangle fits and at least one does not, and that is what sometimes means.
Answer
The statement is sometimes true. A rectangle with four sides of 4 cm is a rhombus, but a rectangle with sides of 3 cm, 5 cm, 3 cm and 5 cm is not.
Worked example
Find and fix the mistakes
A quadrilateral has two pairs of parallel sides, four right angles and four sides that are each 6 cm long. Someone has turned it so that one corner points straight down. A student says it looks like a diamond, so it is a rhombus and cannot be a square. Find the student's mistakes and correct them.
List the attributes you are given, because attributes decide the category. The figure has two pairs of parallel sides, four right angles and four sides of 6 cm. Turning a figure does not change any of these.
Match the attributes to the table. A quadrilateral with two pairs of parallel sides is a parallelogram. A parallelogram with four right angles is a rectangle. A rectangle with four equal sides is a square.
Name the first mistake. The student judged the figure by how it looks. The way a figure is turned is not an attribute in any row of the table, so it cannot decide the category.
Name the second mistake. The figure really is a rhombus, but the student is wrong that a rhombus cannot be a square. A square is a parallelogram with four equal sides, so every square is a rhombus. This figure is both.
Answer
The figure is a square, because it is a rectangle with four equal sides of 6 cm. It is also a rhombus. The student should check the attributes, not the way the figure is turned.
Here is the idea to keep. A subcategory keeps every attribute of the category it sits inside. Every square is a rectangle, so every square has whatever every rectangle has.
When you judge a statement, think about every figure in the category. Use attributes to show always or never. Use two examples, one that fits and one that does not, to show sometimes. And when someone flips a statement or judges a figure by its looks, go back to the attributes.