Quadrilaterals have four sides, four angles, and four vertices. They are utilized in our day-to-day lives in many ways. Books, electronics and road signs are examples of how quadrilaterals are used, and quadrilaterals can be found throughout various architectural applications.
Features common to all quadrilaterals:
– Four straight edges
– Four angles with a total of 360 degrees
– Four corner points
Types of Quadrilaterals and Their Properties
1. Square
– Definition: A quadrilateral that has equal sides, and all four angles are 90 degrees.
– Features:
– All four sides are equal
– All four angles are 90 degrees
– The diagonals cross at 90 degrees
– Common real-world application: the square tiles on chessboards and sticky notes.
2. Rectangle
– Definition: A quadrilateral with opposite sides equal and four right angles.
– Features:
– Opposite sides are equal and parallel.
– All four angles are 90 degrees
– The diagonals cross each other.
– Common real-world application: doors, books and smartphones.
3.Rhombus
Examples of Real-World Objects: Baseball diamond shape, and kite shape.
Definition: A type of quadrilateral where all four sides of the shape are of equal length; however, the angles do not measure 90 degrees.
Properties:
✅ All four sides have the same length.
✅ Opposite Angles are Equal.
✅ Diagonals of a rhombus cross at 90-degree angles.
4. Parallelogram:
* Definition: A paralleled shape (or quadrilateral) with one pair of opposite sides being both equal and parallel, while the remaining pair of sides do not measure 90-degrees.
* Properties:
✅ Opposite Side Lengths are Equal and Parallel.
✅ Opposite Angle Values are Equal.
✅ The diagonals of a parallelogram bisect each other equally (have equal length).
* Examples of Real-World Objects: Slanted roof tiles and erasers in a rectangular shape.
5. Trapezoid (also called Trapezium)
* Definition: A trapezoidal shape (or quadrilateral) with only one set of two opposite sides being parallel.
* Properties:
✅ One Pair of Opposite Sides (Bases) that are Parallel.
✅ The Remaining Two Sides (Non-Base) are called Legs.
✅ The Angles Between the Legs add up to 180 degrees.
* Examples of Real-World Objects: Two Sides of a Handbag and the supports of a bridge.
6. Rhomboid
* Definition: A type of parallelogram having two unequal lengths of adjacent sides that do not form right-angle corners with the base.
* Properties:
✅ The Opposite Sides are Equal and Parallel.
✅ The Opposite Angle Values are Equal.
✅ The Diagonals of a rhomboid cross but are not Perpendicular to each Other.
* Examples of Real-World Objects: Decorative tiles and pattern shapes used in some products.
Comparison Chart of Quadrilaterals
| Shape | Sides | Angles | Parallel Sides | Diagonals |
|---|---|---|---|---|
| Square | All equal | All 90° | 2 pairs | Equal, bisect at 90° |
| Rectangle | Opposite equal | All 90° | 2 pairs | Equal, bisect (not at 90°) |
| Rhombus | All equal | Not 90° | 2 pairs | Unequal, bisect at 90° |
| Parallelogram | Opposite equal | Not 90° | 2 pairs | Bisect each other |
| Trapezoid | Unequal | Varies | 1 pair only | Do not bisect |
| Rhomboid | Opposite equal | Not 90° | 2 pairs | Unequal, no 90° |
Examples:
- Square: A playground tile.
- Rectangle: A sheet of paper.
- Rhombus: A kite.
- Parallelogram: A slanted box top.
- Trapezoid: A road sign with slanted sides.
Exercises:
Exercise 1 — Classifying Quadrilateral Shapes
1) The shape of a square is defined by four equal-length sides that meet at four right angles (90 degrees).
Good Practice Tip: Think of the angles in a rectangle; they are all 90 degrees. This means that a square has exactly the same angle measure as a square.
Exercise 2 — Identify Whether Each Statement is True or False
1) Every square has 4 equal-length sides and 4 right angles. (True)
2) Every Rectangle has 4 equal-length sides. (False)
3) A Rhombus has 2 equal opposite sides and 2 equal opposite angles. (True)
4) A Trapezoid (UK) was previously called a “trapezium”. A Trapezoid has at least one side that is parallel to another pair of opposite sides. (False)
5) A Parallelogram has equal angle measures for both pairs of opposite angles. (True)
Exercise 3: Match the Quadrilaterals.
Read through the following description of each Quadrilateral. Match the numeric value of each Quadrilateral to the correct description:
1. A four sided figure which has four equal sides and four angles equal to 90º.
2. A four sided figure which the opposite sides are parallel and equal in length, but none of the angles are equal to 90º.
3. A four sided figure that has only one pair of parallel sides.
4. A four sided figure that has equal sides but no angles equal to 90º.
5. A four sided figure that has opposite sides parallel and equal in length.
Options: a. Square, b. Rectangle, c. Rhombus, d. Parallelogram, e. Trapezoid
Quadrilaterals are unique and useful. Here are some helpful hints to remember:
1. Squares and rectangles have 90° angles.
2. Rhombuses and parallelograms are made up of equal-length squares, but with angles that do NOT measure 90°.
3. Trapezoids only have one pair of parallel sides.
Identify Trapezoids
A trapezoid, also called a trapezium in some places, is a quadrilateral with exactly one pair of parallel sides. The other two sides are non-parallel, which makes it different from parallelograms, rectangles, and squares that have two pairs of parallel sides.
Important facts about a trapezoid
1. One pair of parallel sides (bases)
These are the two sides that are parallel to each other (the bases). The longer base is commonly referred to as the major base and the shorter base as the minor base.
2. Non-parallel sides (legs)
The other two sides are referred to as legs. If the legs are equal in length, the trapezoid is called an isosceles trapezoid.
3. Angles
The angles at each end of the legs are supplementary to each other and will add to 180 degrees. In an isosceles trapezoid, the angles at the bases (angles adjacent to the bases) are equal to each other.
Types of Trapezoids
1. General Trapezoid
* Has 1 pair of parallel sides
* Legs are not equal in length
2. Isosceles Trapezoid
* Has congruent non-parallel sides (legs)
* Base angles are both equal
* Has equal diagonals
3. Right Trapezoid
* One leg forms a right angle with the bases
How To Identify A Trapezoid:
To determine if a quadrilateral is a trapezoid, do the following:
* Check For Parallel Sides:
* Slope calculations (if using coordinates) will show only 1 pair of parallel sides
* Measure the length of the four sides. If 2 sides don’t intersect when continued, then those 2 are parallel to each other
* Check the lengths of the non-parallel sides (for isosceles trapezoid)
* Measure the non-parallel sides: If they are equal, then this indicates an isosceles trapezoid.
* Check the Angles (for right trapezoid)
* If one leg is perpendicular to both bases and forms right angles with each, then this indicates a right trapezoid.
Real-Life Examples of Trapezoids
- The sides of a bridge support (often shaped like an isosceles trapezoid).
- The structure of a handbag (some designs have trapezoidal shapes).
- Architectural elements, such as windows or door frames with a sloping top.
Identify Rhombuses
A rhombus (plural: rhombi or rhombuses) is a special type of quadrilateral with unique properties that distinguish it from other four-sided shapes. It is a type of parallelogram but with additional symmetry and equal side lengths.
Key Properties of a Rhombus
- All Sides Equal in Length:
- Unlike general parallelograms (where only opposite sides are equal), a rhombus has all four sides of the same length.
- Opposite Sides Parallel:
- Like all parallelograms, opposite sides are parallel (they never intersect).
- Opposite Angles Equal:
- The angles opposite each other are congruent (equal in measure).
- Adjacent Angles Supplementary:
- Consecutive (adjacent) angles add up to 180°.
- Diagonals Perpendicular and Bisect Each Other:
- The diagonals of a rhombus intersect at 90° (right angles).
- They also bisect each other (cut each other exactly in half).
- Additionally, the diagonals bisect the angles of the rhombus.
- Symmetry:
- A rhombus has two lines of symmetry (along its diagonals) and rotational symmetry of 180°.
The Real World Example of Rhombuses:
Playing card diamond shapes
Traditional kite shapes
Baseball infield designs, i.e., baseball diamond shapes (square, rhombus)
Mosaic tile/dynamic tile and design patterns, i.e., mosaics that depict rhombuses.
Classification of Quadrilaterals
Classification of Quadrilaterals – What Does it Mean?
Definition: Classification of quadrilaterals refers to the categorization of quadrilaterals into classes based upon unique characteristics, including side lengths, angles, and number of parallel sides.
Quadrilaterals are four-sided polygons, and thus each quadrilateral type possesses distinctive traits that make categorization easier.
Quadrilaterals can be classified by their lengths or the number of right angles in each quadrilateral.
Length Classification
1) The length of each side can be used to classify a quadrilateral:
A quadrilateral is a square if all sides are equal; it is a rhombus if all four sides are equal. A quadrilateral is a rectangle if it has two opposite sides equal and two adjacent sides equal. The quadrilateral is a parallelogram. A quadrilateral is an isosceles trapezoid if it has non-parallel sides of equal length. A quadrilateral is an irregular quadrilateral if no two sides are equal.
2) The angles of a quadrilateral can be used to classify a quadrilateral:
If all angles are right angles or 90°, then a quadrilateral is either a square or a rectangle. If two opposite angles are equal, a quadrilateral is a rhombus or parallelogram. Trapezoids have one pair of supplementary angles (the sum of the two angles equals 180°).
3. By Parallel Sides
Identify how many pairs of sides are parallel.
- Two pairs of parallel sides: Square, Rectangle, Rhombus, Parallelogram
- Only one pair of parallel sides: Trapezoid
- No parallel sides: Irregular Quadrilateral (sometimes called a “kite” if two distinct pairs of adjacent sides are equal)
Examples of Classifying Quadrilaterals
| Shape | Sides | Angles | Parallel Sides |
|---|---|---|---|
| Square | All equal | All 90° | 2 pairs |
| Rectangle | Opposite equal | All 90° | 2 pairs |
| Rhombus | All equal | Opposite equal | 2 pairs |
| Parallelogram | Opposite equal | Opposite equal | 2 pairs |
| Trapezoid | No restrictions | No restrictions | 1 pair |
| Kite | Two distinct pairs of equal adjacent sides | One pair of equal opposite angles | 0 pairs |
This guide will outline the steps required to draw common quadrilaterals.
1. To start with a square:
-Properties: All four sides are the same length, and all four angles measure 90 degrees. A square has two sets of parallel sides.
-To draw a square:
-Draw a straight line segment on your paper that is 5 centimeters long (or whatever measurement you choose).
-From each endpoint, draw a line straight up that is the same length (5 centimeters).
-Connect the two new lines together to complete your square.
2. Next up we will draw a rectangle:
-Properties: Opposite sides are equal in length, and all four angles measure 90 degrees. A rectangle has two sets of parallel sides.
-To draw a rectangle:
-Draw a straight line that is longer horizontally than vertically (ex: 6 centimeters) across your paper.
-Then from either endpoint of that line segment, draw a straight line that is shorter than the original line segment (3 centimeters) vertically up or down to form the two shorter sides.
-Connect the two new lines to complete the rectangular shape.
3. Now we will make a parallelogram:
-Properties: Opposite sides are equal in length and also parallel, and opposite angles are equal.
-To draw a parallelogram:
-Start off by making a horizontal line that is 5 centimeters long, similar to what we did with the previous shapes.
-Then on an angle draw another line that is the same length (5 centimeters) as the first one.
-Lastly, connect both ends of the first and second lines together to complete the parallelogram shape.
4. How to create a rhombus:
-Properties: All four sides are equal in length. Opposite angles are equal, and diagonals of the rhombus will intersect perpendicularly.
-To draw a rhombus:
-Begin by drawing two perpendicular lines (diagonals) forming a cross.
-Connect each endpoint of one diagonal to the other end of the opposite diagonal to complete the diamond shape that defines a rhombus.
5. Trapezoid
- Trapezoids are quadrilaterals that contain only one pair of parallel sides. When drawing a trapezoid, follow these steps:
- 1. Create a horizontal base that has a length of around 6 cm.
2. From one endpoint, draw a line extending in the upward direction (not parallel) to form a non-parallel side.
3. Draw another (parallel) shorter line on top of that non-parallel side.
4. Finally, draw a connecting line between the endpoints of both the top and bottom lines.