Quadrilaterals and polygons
🎯 In this topic you will
- Identify the symmetry of regular polygons
- Identify and describe the hierarchy of quadrilaterals
🧠 Key Words
- hierarchy
- lines of symmetry
- quadrilateral
- regular polygon
- rotational symmetry
Show Definitions
- hierarchy: A system of classifying shapes into categories based on their properties, where more specific shapes are subtypes of more general ones.
- lines of symmetry: Imaginary lines that divide a shape into two identical halves that are mirror images of each other.
- quadrilateral: A polygon with four sides and four angles, including rectangles, squares, parallelograms, and trapezoids.
- regular polygon: A polygon with all sides equal in length and all interior angles equal in measure.
- rotational symmetry: A property where a shape can be rotated by some angle less than 360° and still look identical to its original position.
A regular polygon has all sides equal and all angles equal. For example, a regular pentagon has $5$ equal sides, $5$ lines of symmetry and rotational symmetry of order $5$.
A quadrilateral is a polygon with four sides. There are seven types of quadrilaterals that we will look at: square, rectangle, parallelogram, rhombus, kite, trapezium, isosceles trapezium.
A square has all sides equal, two pairs of parallel sides and angles of $90^\circ$.
All quadrilaterals have $4$ sides and $4$ angles. The sum of the interior angles of any quadrilateral is $360^\circ$.
❓ EXERCISES
| Name of regular polygon | Number of sides | Number of lines of symmetry | Order of rotational symmetry |
|---|---|---|---|
| Triangle | $3$ | $3$ | $3$ |
| Square | $4$ | $4$ | $4$ |
| Pentagon | $5$ | $5$ | $5$ |
| Hexagon | $6$ | $6$ | $6$ |
| Heptagon | $7$ | ||
| Octagon | $8$ | ||
| Nonagon | $9$ | ||
| Decagon | $10$ |
1. Complete the table.
a. What is the relationship between the number of sides and the number of lines of symmetry?
b. What is the relationship between the number of sides and the order of rotational symmetry?
c. A regular polygon has $15$ lines of symmetry. How many sides does it have?
d. A regular polygon has rotational symmetry of order $24$. How many sides does it have?
e. A regular polygon has $18$ sides. How many lines of symmetry does it have? What is the order of rotational symmetry?
👀 Show answer
1. Completed table:
| Name | Sides | Lines of symmetry | Rotational symmetry |
|---|---|---|---|
| Triangle | $3$ | $3$ | $3$ |
| Square | $4$ | $4$ | $4$ |
| Pentagon | $5$ | $5$ | $5$ |
| Hexagon | $6$ | $6$ | $6$ |
| Heptagon | $7$ | $7$ | $7$ |
| Octagon | $8$ | $8$ | $8$ |
| Nonagon | $9$ | $9$ | $9$ |
| Decagon | $10$ | $10$ | $10$ |
a. The number of lines of symmetry equals the number of sides.
b. The order of rotational symmetry equals the number of sides.
c. $15$ sides.
d. $24$ sides.
e. $18$ lines of symmetry and rotational symmetry of order $18$.
❓ EXERCISES
2. Look at rectangle $ABCD$. Write true or false for each statement. If the statement is false, write the correct statement.
a. $AC = BD$
b. $AB$ is parallel to $AC$
c. $BD$ is parallel to $AB$
d. All angles are $90^\circ$
👀 Show answer
a. True. In a rectangle, the diagonals are equal in length.
b. False. $AB$ is parallel to $DC$, not to $AC$. The correct statement is: $AB$ is parallel to $DC$.
c. False. $BD$ is a diagonal and $AB$ is a side; they are not parallel. The correct statement is: $BD$ is a diagonal of the rectangle.
d. True. By definition, all angles in a rectangle are $90^\circ$.
❓ EXERCISES
Word Bank:
opposite two all
3. Copy and complete the table for each quadrilateral. Use the words in the box to help you.
| Quadrilateral | sides the same length | pairs of parallel sides | opposite angles equal | all angles equal |
|---|---|---|---|---|
| Square | yes | yes | ||
| Rectangle | yes | yes | ||
| Parallelogram | yes | no | ||
| Rhombus | yes | no |
👀 Show answer
Completed table:
| Quadrilateral | sides the same length | pairs of parallel sides | opposite angles equal | all angles equal |
|---|---|---|---|---|
| Square | all | two | yes | yes |
| Rectangle | opposite | two | yes | yes |
| Parallelogram | opposite | two | yes | no |
| Rhombus | all | two | yes | no |
Explanation:
- Square: All sides equal (all), two pairs of parallel sides, opposite angles equal, all angles equal ($90^\circ$)
- Rectangle: Opposite sides equal (opposite), two pairs of parallel sides, opposite angles equal, all angles equal ($90^\circ$)
- Parallelogram: Opposite sides equal (opposite), two pairs of parallel sides, opposite angles equal, angles not necessarily equal
- Rhombus: All sides equal (all), two pairs of parallel sides, opposite angles equal, angles not necessarily equal
🧠 Think like a Mathematician
Question: What are the hierarchical relationships between different types of quadrilaterals?
Method:
- Analyze the definitions of each quadrilateral type: square, rectangle, parallelogram, and rhombus.
- Consider the properties that define each shape.
- Determine which shapes are special cases of others.
- Create a hierarchy diagram showing the relationships.
Follow-up Questions:
Show Answers
- a: Yes. A square is a special type of rectangle where all sides are equal.
- b: No. A rectangle is not necessarily a square because it only requires opposite sides to be equal, not all sides.
- c: Yes. A square is a special type of rhombus where all angles are $90^\circ$.
- d: No. A rhombus is not necessarily a square because it only requires all sides to be equal, not all angles to be $90^\circ$.
❓ EXERCISES
5. Zara is describing a square to Marcus. Has Zara given Marcus enough information for him to work out that the quadrilateral is a square? Explain your answer.
My quadrilateral has two pairs of parallel sides and all the angles are $90^\circ$. What is the name of my quadrilateral?
👀 Show answer
❓ EXERCISES
6. In isosceles trapezoid ABCD, AB is parallel to DC. (True/False)
👀 Show answer
7. In isosceles trapezoid ABCD, ∠A = ∠B. (True/False)
👀 Show answer
8. In isosceles trapezoid ABCD, the diagonals AC and BD are equal in length. (True/False)
👀 Show answer
9. In isosceles trapezoid ABCD, AD = BC. (True/False)
👀 Show answer
10. For the yellow trapezoid:
a. Number of pairs of parallel sides: _____
b. Number of pairs of equal sides: _____
c. Number of pairs of equal angles: _____
👀 Show answer
10. For the blue isosceles trapezoid:
a. Number of pairs of parallel sides: _____
b. Number of pairs of equal sides: _____
c. Number of pairs of equal angles: _____
👀 Show answer
11. For the light blue kite:
a. Number of pairs of parallel sides: _____
b. Number of pairs of equal sides: _____
c. Number of pairs of equal angles: _____
👀 Show answer
a. $0$ pairs
b. $2$ pairs (adjacent sides)
c. $1$ pair (between equal sides)
🧠 Think like a Mathematician
Question: What are the defining properties of different quadrilaterals, and how do they relate to each other?
Method:
- Consider the definition of each quadrilateral type: trapezoid, kite, and parallelogram.
- Analyze the specific properties that distinguish each shape.
- Determine whether one shape is always a special case of another.
- Justify your conclusions with geometric reasoning.
Follow-up Questions:
Show Answers
- a: No. A trapezoid is defined as a quadrilateral with exactly one pair of parallel sides. An isosceles trapezoid is a special case where the non-parallel sides are equal in length and base angles are equal. Not all trapezoids meet these additional conditions.
- b: No. A kite is defined as a quadrilateral with two distinct pairs of adjacent sides that are equal in length. A rhombus is a quadrilateral with all sides equal. While a rhombus can be considered a special kite, not all kites are rhombuses because kites don't require all sides to be equal.
- c: No. A parallelogram is defined as a quadrilateral with two pairs of parallel sides. A rectangle is a special parallelogram where all angles are $90^\circ$. Not all parallelograms have right angles, so not all parallelograms are rectangles.
❓ EXERCISES
13. Marcus describes his quadrilateral to Zara as "having two pairs of sides of equal length and two pairs of equal angles". What is the name of this quadrilateral?
👀 Show answe
14. Use the classification flowchart below to identify Marcus's quadrilateral. Select the correct option from a to g.
a. Square
b. Rectangle
c. Rhombus
d. Parallelogram
e. Isosceles Trapezoid
f. Trapezoid
g. General Quadrilateral
👀 Show answer
❓ EXERCISES
15a. A square is a special type of rectangle. (True/False)
👀 Show answer
b. A rectangle is a special type of square. (True/False)
👀 Show answer
c. A parallelogram is a special type of trapezium. (True/False)