Session 5: Quadrilaterals – Definition, Types, Properties, Interactives and Examples

What is a Quadrilateral?

The prefix “quad-” means “four”, and “lateral” is derived from the Latin word for “side”. So a quadrilateral is a four-sided polygon. Since it is a polygon, you know that it is a two-dimensional figure made up of straight sides. A quadrilateral also has four angles formed by its four sides.

AB, BC, CD and DA are the sides and A, B, C and D are the vertices of the quadrilaterals. 

Line segments AC and BD joining two non-consecutive vertices are called diagonals.

Two sides like AB and AD having a common end point are called adjacent sides.  

Two sides like AB and DC having no common endpoint are called non-adjacent sides. 

Angles like ∠A and ∠B having a common arm are called adjacent angles.

Angles like ∠A and ∠C having no common arm are called non-adjacent angles.   

Position of a Point with respect to a Quadrilateral

In the quadrilateral below, A, B and C are the three points. 

• The region consisting of all points lying outside a quadrilateral is called the exterior of the quadrilateral. Point A is an exterior point. 
• The region consisting of all points lying inside a quadrilateral is called the interior of the quadrilateral. Point B is an interior point. 
• The region consisting of all points which are on the quadrilateral is called the boundary of the quadrilateral. Point C is on the boundary of the quadrilateral. 

What is a Convex Quadrilateral?

A diagonal is a line segment that connects any two non-adjacent vertices of a quadrilateral. A quadrilateral is convex if all diagonals remain inside the quadrilateral. Most quadrilaterals that you study in this class will be convex.

What is a Concave Quadrilateral?

If a quadrilateral is not convex then it is concave (or non-convex). Some diagonals of a concave quadrilateral lie partly or wholly outside the quadrilateral.

Types of Quadrilaterals

There are many common special quadrilaterals that you should be familiar with. Below, these special quadrilaterals are described with their definitions and some properties.

Kite

A kite is a convex quadrilateral with two pairs of adjacent equal sides such that not all sides are equal. 

The angles between the equal sides are called vertex angles. The other angles are called non-vertex angles. 

Properties of a Kite:

1. Two pairs of adjacent sides are equal, i.e.,  and .

2. Non-vertex angles are equal, i.e., .

3. Diagonals intersect each other at right angles, i.e., . 

4. The longer diagonal of a kite bisects the shorter one, i.e., .

5. The diagonal through the vertex angles is the angle bisector for both angles, i.e.,  and .

6. One of the diagonals bisects the kite, i.e., divides it into two equal triangles, i.e., .

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Trapezium

A trapezium is a quadrilateral with exactly one pair of parallel sides.

Properties of a Trapezium:

1. One pair of opposite sides are parallel, i.e., .

2. The two pairs of adjacent angles along the sides are supplementary, i.e., and .

Note: Some texts leave out the word “exactly”, which means quadrilaterals with two pairs of parallel sides are sometimes considered trapeziums. Here, assume trapeziums have exactly one pair of parallel sides.

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Isosceles Trapezium

An isosceles trapezium is a trapezium with the non-parallel sides equal. An additional property of isosceles trapeziums is base angles are equal.

Properties of an Isosceles Trapezium:

1. One pair of opposite sides are parallel, i.e., . 

2. Two pairs of adjacent angles are supplementary, i.e., and .

3. Base angles are equal, i.e.,  and .

4. The diagonals are equal, i.e., .

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Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides.

Properties of a Parallelogram:

1. Opposite sides are parallel, i.e.,  and . 

2. Opposite sides are equal, i.e.,  and .

3. Opposite angles are equal, i.e., and .

4. Adjacent angles are supplementary, i.e.,, and . 

5. Diagonals bisect each other, i.e.,  and .

6. Each diagonal bisects the parallelogram, i.e., divides it into two equal triangles  and . 

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Rectangle

A rectangle is a quadrilateral with four right angles. All rectangles are parallelograms.

Properties of a Rectangle: 

1. Opposite sides are parallel, i.e., AB∥CD and BC∥DA. 

2. Opposite sides are equal, i.e., AB=CD and BC=DA.

3. All the four angles are equal and measure 90∘, i.e., ∠ABC=∠BCD=∠CDA=∠DAB=90∘.

4. Diagonals are equal, i.e., AC=BD.

5. Diagonals bisect each other, i.e., BO=OD and AO=OC. 

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Rhombus

A rhombus is a quadrilateral with four equal sides. All rhombuses are parallelograms.

Properties of a Rhombus:

1. Opposite sides are parallel, i.e., AB∥CD and BC∥DA. 

2. All four sides are equal, i.e., AB=BC=CD=DA. 

3. Opposite angles are equal, i.e., ∠ABC=∠CDA and ∠DAB=∠BCD. 

4. Diagonals are the interior angle bisectors, i.e., ∠BAC=∠DAC, ∠BDC=∠BDA, ∠DBC=∠DBA and ∠BCA=∠DCA.

5. Diagonals intersect each other at right angles, i.e., ∠AOB=∠BOC=∠COD=∠DOA=90∘.

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Square

A square is a quadrilateral with four right angles and four equal sides. All squares are rectangles and rhombuses.

Properties of a Square: 

1. All four sides are equal, i.e., AB=BC=CD=DA.

2. All the four angles are equal and measures 90∘, i.e., ∠ABC=∠BCD=∠CDA=∠DAB=90∘.

3. Diagonals are equal, i.e., AC=BD. 

4. Diagonals bisect each other at right angles, i.e., ∠AOB=∠BOC=∠COD=∠DOA=90∘. 

Hierarchy of Quadrilaterals 

Notice that the properties of quadrilaterals overlap. A square is not only a square, but also a rhombus, a rectangle, a parallelogram, and a quadrilateral. This means that a square will have all the same properties as rhombuses, rectangles, parallelograms, and quadrilaterals.

The following diagram shows the hierarchy of quadrilaterals. 

Quadrilaterals – Examples

Example 1

All squares are rectangles, but not all rectangles are squares. How is this possible?

Rectangles are defined as quadrilaterals with four right angles. Squares are defined as quadrilaterals with four right angles and four equal sides. Because all squares have four right angles and satisfy the definition of rectangles, they can all also be called rectangles. On the other hand, not all rectangles have four equal sides, so not all rectangles can also be called squares.

Example 2

A quadrilateral has four equal sides. What type of quadrilateral must it be? What type of quadrilateral could it be?

It must be a rhombus and therefore also a parallelogram. It could be a square.

Example 3

Is every square a rhombus?

A rhombus is defined as a quadrilateral with four equal sides. A square is defined as a quadrilateral with four right angles and four equal sides. Because all squares have four equal sides and satisfy the definition of rhombuses, they can also be called rhombuses. Also, the diagonals of a square intersect each other at a right angle as that in a rhombus and the measure of the opposite angle of the square is equal to that in a rhombus. On the other hand, not all rhombuses have four right angles, so not all rhombuses can be called squares.

Example 4

Is every rectangle a parallelogram?

A parallelogram is defined as a quadrilateral with two pairs of opposite, equal and parallel sides. A rectangle is defined as a quadrilateral with two pairs of opposite, equal and parallel sides. So, all rectangles satisfy the definition of parallelograms, they can also be called parallelograms. On the other hand, not all parallelograms have four right angles, so not all parallelograms can be called rectangles.

 Remember this!

• A quadrilateral is a four-sided polygon. 
• A quadrilateral in which the measure of each angle is less than 180∘ is called a convex quadrilateral. 
• A quadrilateral in which the measure of at least one of the angles is more than 180∘ is known as a concave quadrilateral. 
• If two sides have a common endpoint, they are called adjacent sides. 
• If two sides do not have a common endpoint, they are called non-adjacent sides. 
• The angles having a common arm are called adjacent angles. 
• A kite is a convex quadrilateral with two pairs of adjacent equal sides such that not all sides are equal. 
• A trapezium is a quadrilateral with exactly one pair of parallel sides. 
• An isosceles trapezium is a trapezium with the non-parallel sides equal. 
• A parallelogram is a quadrilateral with two pairs of parallel sides. 
• A rectangle is a quadrilateral with four right angles. All rectangles are parallelograms. 
• A rhombus is a quadrilateral with four equal sides. All rhombuses are parallelograms. 
• A square is a quadrilateral with four right angles and four equal sides. All squares are rectangles and rhombuses.