Quadrilaterals
🎯 In this topic you will
- Identify quadrilaterals.
- Describe quadrilaterals.
- Classify quadrilaterals.
- Sketch quadrilaterals.
🧠 Key Words
- bisect
- decompose
- parallel
- diagonal
- justify
- trapezia
Show Definitions
- bisect: To divide something into two equal parts.
- decompose: To break a shape or number into smaller parts that are easier to analyze.
- parallel: Lines that run in the same direction and never meet, even if extended indefinitely.
- diagonal: A line segment that connects two non-adjacent vertices of a polygon.
- justify: To give clear mathematical reasoning that proves why a statement or result is true.
- trapezia: The plural of trapezium, a quadrilateral with at least one pair of parallel sides.
Quadrilaterals in Everyday Tiles
Quadrilateral tiles are often used on kitchen and bathroom walls and floors. This is because they fit together exactly leaving no spaces (tessellate). You need to be able to identify and describe the different types of quadrilaterals. This is really important when you need to order tiles for your house.
❓ EXERCISES
$1$. Copy and complete these properties of a square. Show each one on a diagram. Diagrams a, b and e have been done for you.
a. A square is a ________.
b. It has ________ equal sides.
c. It has ________ pairs of parallel sides.
d. The sides meet at ________°.
e. The diagonals ________ each other at $90^\circ$.
f. It has ________ lines of symmetry.
👀 Show answer
a. A square is a quadrilateral.
b. It has $4$ equal sides.
c. It has $2$ pairs of parallel sides.
d. The sides meet at $90^\circ$.
e. The diagonals bisect each other at $90^\circ$.
f. It has $4$ lines of symmetry.
🧠 Reasoning Tip
The single curved lines show that these two angles are equal.
The double curved lines show that these two angles are equal.
$2$. Copy and complete these properties of a parallelogram. Show each one on a diagram. Diagrams a, d and f have been done for you.
a. A parallelogram is a ________.
b. It has ________ pairs of equal sides.
c. It has ________ pairs of parallel sides.
d. It has ________ pairs of equal angles.
e. The diagonals ________ each other.
f. It has ________ lines of symmetry.
👀 Show answer
a. A parallelogram is a quadrilateral.
b. It has $2$ pairs of equal sides.
c. It has $2$ pairs of parallel sides.
d. It has $2$ pairs of equal angles.
e. The diagonals bisect each other.
f. It has $0$ lines of symmetry.
🧠 Think like a Mathematician
Task: Answer these questions. The diagrams A to F show different trapezia.
Questions:
Group 1: trapezia that are isosceles
Group 2: trapezia that are not isosceles
i an isosceles trapezium.
ii a trapezium that is not isosceles.
Show each one on a diagram.
Tip:
Where have you met the word isosceles before?
What do you think is special about an isosceles trapezium?
👀 show answer
a) The isosceles trapezia are A, D, E, and F. The trapezia that are not isosceles are B and C.
b) i An isosceles trapezium has exactly one pair of parallel sides, and its non-parallel sides are equal in length. Because of this, the base angles are equal in pairs, and it has a line of symmetry.
b) ii A trapezium that is not isosceles still has one pair of parallel sides, but the non-parallel sides are not equal. Its base angles are not equal in matching pairs, and it does not have a line of symmetry.
c) A complete comparison should mention these key properties: every trapezium has one pair of parallel sides; isosceles trapezia also have equal non-parallel sides, equal base angles, and symmetry; non-isosceles trapezia do not have those extra equalities. The word isosceles also appears in isosceles triangles, where equal side lengths are the special feature.
❓ EXERCISES
3.
a. Make a sketch of each of the seven special quadrilaterals: square, rectangle, parallelogram, trapezium, isosceles trapezium, rhombus and kite. If the shape has any lines of symmetry, draw them onto your sketch.
b. Copy and complete this tick box table showing some of the properties of the seven special quadrilaterals. The parallelogram has been done for you.
| Quadrilateral | Square | Rectangle | Parallelogram | Trapezium | Isosceles trapezium | Rhombus | Kite |
|---|---|---|---|---|---|---|---|
| Four equal sides | ✓ | ||||||
| Two pairs of equal sides | ✓ | ✓ (opposite sides) | |||||
| One pair of equal sides | ✓ (non-parallel sides) | ✓ (adjacent sides) | |||||
| One pair of parallel sides | ✓ | ✓ | |||||
| Two pairs of parallel sides | ✓ | ✓ | ✓ | ✓ | |||
| All angles $90^\circ$ | ✓ | ✓ | |||||
| One pair of equal angles | ✓ (base angles) | ✓ (one pair of opposite angles) | |||||
| Two pairs of equal angles | ✓ | ✓ | ✓ (opposite angles) | ✓ (opposite angles) | |||
| Diagonals bisect each other | ✓ | ✓ | ✓ | ✓ | |||
| Diagonals meet at $90^\circ$ | ✓ | ✓ | ✓ |
👀 Show answer
3a. Sketches should show the seven shapes. Lines of symmetry:
• Square: $4$ lines
• Rectangle: $2$ lines
• Parallelogram: $0$ lines (unless it is a rectangle)
• Trapezium (general): $0$ lines
• Isosceles trapezium: $1$ line through midpoints of parallel sides
• Rhombus: $2$ lines (the diagonals)
• Kite: $1$ line through one diagonal.
3b. Completed table as shown above. Note: "Parallelogram" column is given, other entries are typical properties. Some quadrilaterals (like trapezium) may have variations; the table follows common definitions (trapezium = one pair parallel; isosceles trapezium = non‑parallel sides equal).
4. Zara and Sofia are looking at this question.
What shape am I? I am a quadrilateral. I have one pair of equal sides.
There is not enough information to work out what the shape is. I disagree. There is enough information to work out what the shape is.
Who is correct, Zara or Sofia? Explain your answer.
👀 Show answer
Zara is correct. The information “one pair of equal sides” is too vague. Many quadrilaterals have one pair of equal sides: an isosceles trapezium (non‑parallel sides equal), a kite (adjacent sides equal), or even a general quadrilateral that happens to have two equal sides without any other symmetry. Without further details (like parallel sides or angle properties) we cannot identify a unique shape.
5. Keon draws this parallelogram and isosceles trapezium. He labels the lines that make the shapes $a$, $b$, $c$, $d$, $e$, $f$, $g$ and $h$. He draws the shapes so that $a$ is parallel to $e$.
Write true or false for each of these statements. Justify your answer. The first one is done for you.
i. $a$ is parallel to $g$
True because we are told $a$ is parallel to $e$ and in the trapezium we know that $g$ is parallel to $e$, so $g$ must also be parallel to $a$.
ii. $b$ is parallel to $d$
iii. $c$ is parallel to $e$
iv. $f$ is parallel to $g$
👀 Show answer
ii.True. In a parallelogram, opposite sides are parallel: $b$ is opposite $d$, so $b \parallel d$.
iii.False. $c$ is a side of the parallelogram. $e$ is the top base of the isosceles trapezium. They are not necessarily parallel; $c$ is parallel to $a$ (in the parallelogram), and $a \parallel e$ is given, so $c \parallel e$ would be true if $c \parallel a$ and $a \parallel e$ (transitive property). Wait — correction: in a parallelogram $a \parallel c$ (opposite sides). So $a \parallel e$ (given) and $a \parallel c$, therefore $c \parallel e$. Actually that makes the statement true. Let's re-read: In the parallelogram, sides are usually labelled consecutively: a, b, c, d. So a and c are opposite, so a ∥ c. Given a ∥ e, then c ∥ e. So statement iii is actually True by transitivity.
iv.False. In an isosceles trapezium, the bases ($e$ and $g$) are parallel. $f$ and $h$ are the non‑parallel legs. $f$ is not parallel to $g$ (the bottom base); $f$ meets $g$ at an angle.
(Summary: i. True (given example), ii. True, iii. True, iv. False)
6. This is part of Shen's homework.
Question: Draw a diagram to show if parallelograms tessellate.
Solution: Yes they do. They fit together leaving no spaces.
a. Draw diagrams like Shen's to show if these shapes tessellate.
i rectangle
ii rhombus
iii trapezium
iv kite
b. Which of these statements is correct?
i Not all quadrilaterals will tessellate.
ii All quadrilaterals will tessellate.
🧠 Reasoning Tip Use squared paper to help you.
👀 Show answer
6a. Sketches should show that rectangles and rhombuses tessellate (like parallelograms). Trapeziums also tessellate by rotating 180° and fitting them together. Kites tessellate as well (e.g. by forming parallelograms). All four shapes can tessellate the plane.
6b. The correct statement is ii: All quadrilaterals will tessellate. Any quadrilateral can tessellate because you can always arrange copies rotated 180° around midpoints of sides to fill the plane (this is a known fact: every quadrilateral, convex or not, tessellates).
Work with a partner on this activity.
You are going to make a poster showing how you can decompose the special quadrilaterals into other shapes.
For example, three ways that you can decompose a square are like this:
There are lots of ways to decompose the special quadrilaterals, try to think of at least three for each one.
It is up to you how you present your poster showing the information.
When you have finished, compare your poster with others in your class.
Now that you have seen other posters, what do you think of your poster? Could you make it better or easier to understand? Who do you think made the best poster and why?