Kind of Quadrilaterals: Types, Definitions, and Properties

When we look around us, we see many closed figures with four sides, which are of different shapes, different lengths and breadth. Those four-sided figures are nothing but quadrilaterals. Thus, a quadrilateral is a simple closed plane figure bounded by four sides. The quadrilateral with different shapes have different properties, and because of some properties, they become unique. But, the sum of the interior angles of all quadrilaterals will be the same. Squares, rectangles, parallelograms, rhombuses, trapeziums, kites are all different kinds of quadrilaterals. In this article, we will learn about all the kinds of quadrilaterals, their shapes, and their properties.

What are Quadrilaterals?

A quadrilateral is a four-sided figure.

In the above figure, \(ABCD\) is a quadrilateral. It has:

– \(4\) sides \(AB,\,BC,\,CD\) and \(AD\)
– \(4\) Interior angles \( – \angle A,\,\angle B,\,\angle C\) and \(\angle D\)
– \(4\) Vertices \( – A,\,B,\,C\) and \(D\)
– \(2\) Diagonals \( – AC\) and \(BD\)

In quadrilateral \(ABCD\),

– sides \(AB,\,BC;\,BC,\,CD;\,CD,\,DA;\,DA,\,AB\) are adjacent sides.
– sides \(AB,\,CD\) and \(BC,\,DA\) are pairs of opposite sides.
– angles \(\angle A,\,\angle B;\,\angle B,\,\angle C;\,\angle C,\,\angle D;\,\angle D,\,\angle A\) are pairs of adjacent angles.
– angles, \(\angle A,\,\angle C\) and \(\angle B,\,\angle D\) are pairs of opposite angles.

The above diagrams show some quadrilaterals in our daily lives. Let us learn each kind of quadrilateral and its properties in detail.

Learn About Properties of Quadrilaterals

Kind of Quadrilaterals with Properties

Let us learn about the definitions, properties of different kinds of quadrilaterals in detail.

Trapezium

A quadrilateral in which only one pair of opposite sides is parallel is called a trapezium. The parallel sides are known as the bases of the trapezium. The line segment joining mid-points of non-parallel sides is called it’s median.

In the above figure, \(AB\parallel CD\) whereas \(AD\) and \(BC\) are non-parallel, so \(ABCD\) is a trapezium. \(AB\) and \(CD\) are its bases, and \(EF\) is its median, where \(E\) and \(F\) are mid-points of \(AD\) and \(BC\), respectively.

Properties of a Trapezium

1. It is a polygon.
2. It is a two-dimensional figure.
3. The diagonals of a trapezium intersect each other.
4. The sum of adjacent angles is equal to \({180^{\rm{o}}}\).
5. A trapezium has non-parallel sides unequal, except for an isosceles trapezium.
6. Only one pair of opposite sides are parallel.
7. The sum of internal angles is equal to \({360^{\rm{o}}}\).
8. The mid-points of sides are collinear to the intersection point of diagonals.

Isosceles Trapezium

If the sides are not parallel and are equal in a trapezium, it is called an isosceles trapezium. Here, \(AB\parallel DC,\,AD\) and \(BC\) are non-parallel and \(AD = BC\).

Properties of an Isoceles Trapezium

1. \(\angle A = \angle B\) and \(\angle C = \angle D\)
2. \(\angle A + \angle D = {180^{\rm{o}}}\) and \(\angle B + \angle C = {180^{\rm{o}}}\).
3. \(\angle A + \angle C = {180^{\rm{o}}}\) and \(\angle B + \angle D = {180^{\rm{o}}}\)
4. Diagonal \(AC = \) Diagonal \(BD\)

Parallelogram

A parallelogram is a type of unique quadrilateral in which both the pairs of opposite sides are parallel

In the above-given figure, \(AB\parallel DC\) and \(AD\parallel BC\), so \(ABCD\) is a parallelogram.

Properties of a Parallelogram

1. Opposite sides are equal.
2. Opposite angles are equal.
3. Adjacent angles are  equal to \({180^{\rm{o}}}\)
4. If one of the angles is a right angle, then all the other angles are also right angles.
5. The diagonals bisect each other.
6. Each diagonal separates it into two congruent triangles.

Rectangle

A rectangle is a quadrilateral in which each angle is equal to \({90^{\rm{o}}}\) and whose opposite sides are parallel and equal to each other. If one of the angles of a parallelogram is a right angle, it is called a rectangle.

Properties of a Rectangle

1. The opposite sides are congruent, i.e., \(AB = CD\) and \(AD = BC\)
2. The opposite sides are parallel, i.e., \(AB\parallel CD\) and \(AD\parallel BC\)   
3. All the angles are equal, i.e., \(\angle A = \angle B = \angle C = \angle D = {90^{\rm{o}}}\)
4. The diagonals of a rectangle are of equal length, i.e., \(AC = BD\)
5. Diagonals of a rectangle bisect each other.

Rhombus

A rhombus is a quadrilateral in which all sides are equal.

In the above-given figure, \(AB = BC = CD = DA\), so \(ABCD\) is a rhombus. Every rhombus is a parallelogram. A parallelogram becomes a rhombus when all of its sides are equal.

Properties of a Rhombus

1. All sides are equal, i.e., \(AB = BC = DC = DA\).
2. Diagonals of a rhombus bisect at \({90^{\rm{o}}}\) .i.e., diagonals \(BD\) and \(AC\) bisect each other at \({90^{\rm{o}}}\)
3. Opposite sides are parallel, i.e., \(AB\parallel CD\)  and \(AD\parallel BC\).
4. Opposite angles are equal, i.e., \(\angle A = \angle C\) and \(\angle B = \angle D\)
5. Adjacent angles add up to \({180^{\rm{o}}}\), i.e., \(\angle A + \angle B = {180^{\rm{o}}},\,\angle B + \angle C = {180^{\rm{o}}},\,\angle C + \angle D = {180^{\rm{o}}}\) and \(\angle D + \angle A = {180^{\rm{o}}}\).

Square

If two adjacent sides of a rectangle are equal, then it is called a square. Alternatively, if one angle of a rhombus is a right angle, it is called a square.

In the above-given figure, \(AB = AD\) so \(ABCD\) is a square. Of course, the remaining sides are also equal. A rectangle, rhombus and square are all special cases of a parallelogram. A parallelogram becomes a square if all its sides are equal and any angle is \({90^{\rm{o}}}\).

Properties of a Square

1. All sides are congruent, i.e., \(AB = BC = CD = DA\).
2. The opposite sides are parallel, i.e., \(AB\parallel CD\) and \(AD\parallel BC\).
3. All the angles are equal, i.e., \(\angle A = \angle B = \angle C = \angle D = {90^{\rm{o}}}\).
4. The diagonals of a square are of equal length, i.e., \(AC = BD\).
5. Diagonals bisect each other at \({90^{\rm{o}}}\) in a square.
6. The diagonal divide it into two similar isosceles triangles in a square.

A square is a type of rectangle and a rhombus, so it has all the rectangle properties and the properties of a rhombus.

Kite

A kite or diamond is a type of quadrilateral in which two pairs of adjacent sides are equal. In the below-given figure, \(AD = AB\) and \(DC = BC\), so \(ABCD\) is a kite.

Properties of a Kite

1. The \(2\) angles are equal where the unequal sides meet.
2. A kite has \(2\) diagonals that intersect each other at \({90^{\rm{o}}}\).
3. A kite is symmetrical about its main diagonal.
4. The diagonals are perpendicular to each other.
5. In the quadrilateral kite, the longer diagonal bisects the shorter diagonal.

Solved Examples – Kind of Quadrilaterals

Q.1. Two angles of a quadrilateral are \({90^{\rm{o}}}\) each. Is this quadrilateral a square? Give reason.

Ans: No, the given quadrilateral is not a square. From the given figure, except for the two given right angles, the other two are not \({90^{\rm{o}}}\) or right angles. Hence, the given quadrilateral is not a square.

Q.2. Opposite sides of a quadrilateral are parallel, and one of its angles is \({90^{\rm{o}}}\). Is it a rectangle? Give reason.
Ans: If the opposite sides of a quadrilateral are parallel and one angle is \({90^{\rm{o}}}\), then, yes, it is a rectangle. Opposite sides are parallel means the quadrilateral is a parallelogram. And if one of the angles of the parallelogram is \({90^{\rm{o}}}\), then the resulting figure is a rectangle.

Q.3. What are additional properties which a parallelogram must have to be a rectangle.
Ans: The additional properties that a parallelogram must have to be a rectangle are,
1. Diagonals must be equal.
2. Any angle measure should be equal to \({90^{\rm{o}}}\).
If any angle of a parallelogram is \({90^{\rm{o}}}\), then by the properties of angles of a parallelogram, each angle will become \({90^{\rm{o}}}\) it will be a rectangle.

Q.4. State two properties that make a given quadrilateral a rhombus.
Ans: The two properties are
1. Diagonal bisects each other at \({90^{\rm{o}}}.\)
2. All the sides should be equal.

Q.5. If all the angles of a quadrilateral are \({90^{\rm{o}}}\) each, is this quadrilateral a square? Give reason.
Ans: The given quadrilateral is not necessarily a square. Because when all the angles of a quadrilateral are \({90^{\rm{o}}}\) each, it is always a rectangle. And, if any pair of adjacent sides of this rectangle are equal, then only it will be a square.

Q.6. Is the rhombus a rectangle? Give reason.
Ans: No, a rhombus cannot be considered a rectangle. Because no angle of a rhombus is \({90^{\rm{o}}}\) and its diagonals are also not equal. Thus, the rhombus is not a rectangle.

Summary

In this article, we first had a quick view of a quadrilateral. And later, we learned about the kinds of quadrilateral, their definitions, their properties in detail. In addition to the leaning of a quadrilateral, we also solved some examples to strengthen our grip on the kinds of a quadrilateral.

Learn All the Concepts on Quadrilaterals

Frequently Asked Questions (FAQs)

Q.1. What are the \(6\) types of quadrilaterals?
Ans: The \(6\) types of quadrilaterals are trapezium, parallelogram, rhombus, kite, rectangle, and square.

Q.2. What are quadrilaterals examples?
Ans: The examples of different quadrilaterals are parallelogram, square, rectangle, rhombus, trapezium, isosceles trapezium and kite.

Q.3. What are special quadrilaterals?
Ans: Special quadrilaterals are those quadrilaterals that makes them unique in their segment. For example, the special type of a quadrilateral is a parallelogram.
The following properties make it unique:
1. Opposite sides are equal.
2. Opposite angles are equal.
3. Adjacent angles are equal to \({180^{\rm{o}}}\).
4. If one angle is a right angle, then all the other angles are also right angles.
5. In a parallelogram, the diagonals bisects each other.
6. Both the diagonals separates it into two congruent triangles.
7. A square, rectangle and rhombus are special cases of a parallelogram.

Q.4. Define quadrilateral.
Ans: A simple closed plane figure bounded by four sides is called a quadrilateral.

Q.5. What are the properties all quadrilaterals share in common?
Ans: The common properties every quadrilateral exhibit in itself is
1. They have \(4\) sides.
2. They all have \(4\) vertices.
3. They all have \(2\) diagonals.
4. Interior angles add upto \({360^{\rm{o}}}\).

We hope this detailed article on the kind of quadrilaterals helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!