Table of Contents
1. | Introduction |
2. | What is a quadrilateral? |
3. | Types and Properties of Quadrilaterals |
5. | Theorems relating to Quadrilaterals |
6. | Example Problems |
7. | Summary |
8. | FAQs |
29 January 2021
Read time: 5 minutes

“Geometry is an art of correct reasoning from incorrectly drawn figures.”
-Henri Poincare
The word “Quadrilateral” is derived from the Latin language, in which “Quadri” means a variant of four and “Latus” means “side”.
Introduction to Quadrilaterals-PDF
A quadrilateral is a polygon with four edges (sides) and four vertices (corners). Let us understand the properties and applications of quadrilaterals. Here is a downloadable PDF to explore more.
📥 | Introduction to Quadrilaterals-PDF |
What is a Quadrilateral?
A Quadrilateral is a flat-surfaced polygon with four sides and four corners. The interior angles of the Quadrilateral always add up to 360°.
Quadrilaterals are categorized into 6 types depending on their sides. They are:
- Parallelogram
- Trapezium
- Rhombus
- Kite
- Rectangle
- Square
Different Types and Properties of Quadrilateral
-
Parallelogram
A Quadrilateral in which opposite sides are parallel and equal is called a Parallelogram.
Properties of Parallelogram
- In a Parallelogram,
- Opposite sides are equal.
- Opposite angles are equal.
- Each diagonal bisects the parallelogram.
- The opposite sides and angles are congruent.
- The consecutive angles of the parallelogram are supplementary i.e., they add up to 1800.
- The diagonals bisect each other.
- Area of the Parallelogram = Base\(\times\)Height
Where Base is any side of the Parallelogram.
Height is the perpendicular length from the opposite vertex to the base.
-
Trapezium
A Quadrilateral in which one pair of opposite sides are parallel but the other two sides of it are non-parallel is called Trapezium. A trapezium is also called Trapezoid.
Properties of Trapezium
- The two angles of the adjacent sides of the Trapezium are supplementary.
- The diagonals of the Trapezium do not bisect each other at right angles.
- If the non-parallel sides of the Trapezium are equal, it is called an Isosceles Trapezium.
- In Isosceles Trapezium,
- The adjacent angles of the parallel sides are equal.
- The diagonals are equal.
- Area of the Trapezium \(=\frac{1}{2}\times\) Sum of the parallel sides \(\times\) Height
Where Height is the perpendicular distance between two parallel sides.
-
Rhombus
The quadrilateral in which all the sides are equal is called a Rhombus.
Properties of Rhombus
- The diagonals of the rhombus bisect each other at 900.
- Any pair of adjacent sides are equal.
- The opposite sides of the Rhombus are parallel to each other.
- The opposite angles of the Rhombus are equal to each other.
- The sum of the two adjacent angles is equal to 1800.
- Area of the Rhombus \(= \frac{(d_1 \times d_2)}{2}.\)
Where, \(d_1\) and \(d_2\) are the diagonals of the Rhombus.
-
Rectangle
A quadrilateral in which the opposite sides are equal is called a Rectangle.
Properties of Rectangle
- Each interior angle of the Rectangle is 900.
- The diagonals of the rectangle are equal.
- The diagonals bisect each other.
- The opposite sides of the Rectangle are parallel to each other.
- Area of the Rectangle \(=\) Length \(\times\) Breadth.
-
Square
A quadrilateral in which all the sides and angles are equal is called a Square.
Properties of Square:
- All the sides of the Square are congruent to each other.
- The opposite sides of the Square are parallel to each other.
- The interior angle of the Square is 900.
- The diagonals of the square are equal.
- The diagonals bisect each other at 900.
- The diagonal cut the Square into two equal isosceles triangles.
- The length of the diagonal is greater than the length of its side.
- Area \(= a \times a\)
Where a is represented as the side of the Square.
-
Kite
A quadrilateral in which the two pairs of disjoint adjacent sides are equal is called a Kite.
Properties of Kite
- The kite is symmetrical about its long diagonal.
- Angles between the unequal sides of the Kite is equal.
- Diagonals of the Kite intersect each other at 900.
- The longer diagonal of the kite is the perpendicular bisector of the shorter diagonal of the kite.
- The shorter diagonal in the kite divides the kite into two isosceles triangles.
- Area of the Kite \(= \frac{1}{2}\times (D_1 \times D_2)\)
Where, \(D_1\) is the longer diagonal of the Kite.
\(D_2\) is the shorter diagonal of the Kite.
Theorems related to Quadrilaterals
- In a parallelogram,
- Opposite sides are equal.
- Opposite angles are equal.
- Each diagonal bisects the parallelogram.
- The diagonal of the parallelogram bisect each other.
- The diagonals of the rectangle are equal and bisect each other.
- The diagonals of the rhombus bisect each other at 900.
- The diagonals of the square bisect each other at 900.
Example Problems
Example 1 |
If the angles of the quadrilateral are X, X+20, X-40 and 2X. What is the difference between the greatest angle and smallest angle?
(a) \(70^\circ\)
(b) \(90^\circ\)
(c) \(80^\circ\)
(d) None of these
Answer: All the interior angles of the quadrilateral \(= 360^o\)
\(\begin{align}X+X+20+X-40+2 x&=360^{\circ} \\[6pt]5 X-20&=360^{\circ} \\[6pt]5 &X=360^{0}+20^{\circ} \\[6pt]5 X&=380^{\circ} \\[6pt]X&=76^{0}\end{align}\)
Therefore, the answer is Option (d)
Example 2 |
The diagonals of a rectangle PQRS intersect at O. If \(∠ROQ=60^\circ\), then find ∠OSP.
(a) \(70^\circ\)
(b) \(50^\circ\)
(c) \(60^\circ\)
(d) \(80^\circ\)
Answer:
\(∠ROQ=60^\circ\)
\(∠POS = 60^\circ\) (Vertically Opposite Angles)
In Triangle POS,
OP=OS (Diagonals of the Rectangle bisect each other)
Therefore, \(∠OSP = ∠SPO\)
So,
\[\begin{align}\angle OSP+\angle SPO+\angle POS&=180^{\circ} \qquad\left(\text { Sum of Interior Angles of triangle is } 180^{\circ}\right)\\[6pt]\angle OSP+\angle OSP+\angle POS&=180^{\circ}\\[6pt]
2 \angle OSP&=180^{\circ}- \angle POS\\[6pt]2 \angle OSP=180^{0}-60^{0}&=120^{\circ}\\[6pt]\angle OSP&=60^{\circ}\end{align}\]
Therefore, the answer is option (c).
Example 3 |
The bisectors of the angle of the parallelogram form ___________________
(a) Trapezium
(b) Rectangle
(c) Rhombus
(d) Kite
Answer: Rectangle. Therefore, the answer is option(b).
Example 4 |
The measure of all the angles of the parallelogram, if an angle adjacent to the smallest angle is 240 less than twice the smallest angle is.
(a) \(37^{\circ}, 143^{\circ}, 37^{\circ},143^{\circ}\)
(b) \(108^{\circ}, 72^{\circ}, 108^{\circ},72^{\circ}\)
(c) \(68^{\circ}, 112^{\circ}, 68^{\circ},112^{\circ}\)
(d) None of these
Answer: Let the smallest angle of the parallelogram be \(x^\circ\).
The adjacent angle is \(2x^{\circ}-24^{\circ}\)
The sum of the adjacent angles of the parallelogram is \(180^{\circ}.\)
\(\begin{align}\mathrm{x}^{\circ} +2 \mathrm{x}^{\circ}-24^{0}&=180^{\circ} \\[6pt]3 \mathrm{x}^{\circ}&=180^{0}+24^{\circ}\\[6pt]3 \mathrm{x}^{\circ}&=204^{0} \\[6pt]\mathrm{x}^{\circ}&=68^{\circ}\end{align}\)
The smallest angle is 680
The largest angle is \(2x^{\circ}-24^{\circ}=2 \times 68^{\circ}-24^{\circ}=112^{\circ}\)
Therefore, the answer is option (c)
Example 5 |
In a quadrilateral ABCD, the line segments bisecting ∠C and ∠D meet at E. Then ∠A+∠B is equal to____________.
(a) ∠CED
(b) \(\frac{1}{2}∠CE\)
(c) 2∠CED
(d) None of these
Answer: In Quadrilateral ABCD,
\(∠A+∠B+∠C+∠D=360^{\circ} \qquad eq (1)\)
Sum of all the interior angles of the quadrilateral is \(360^{\circ}\).
In Triangle CDE,
\(∠CED+∠DCE+∠CDE=180^{\circ}\)
Sum of all the interior angles of triangle is \(180^{\circ}\).
\(∠CDE+\frac{1}{2} \times ∠C+\frac{1}{2} \times ∠D = 180^{\circ}\)
Line segment bisects the angle
\(2∠CDE+∠C+∠D=360^{\circ} \qquad \dots (2) \quad \text{(Multiply by 2)}\)
Comparing the equation (1) and (2)
\(\begin{align}∠A+∠B+∠C+∠D&=2∠CDE+∠C+∠D\\[6pt]∠A+∠B&=2∠CDE\end{align}\)
Therefore, the answer is option (c).
Example 6 |
In the given figure, ABCD is a parallelogram in which \(∠DAB= 70^{\circ}\) and \(∠DBC= 65^\circ\), then find the angle \(∠CDB\)?
(a) \(45^\circ\)
(b) \(65^\circ\)
(c) \(80^\circ\)
(d) \(55^\circ\)
Answer:
\(∠A+∠B= 180^\circ\)
Sum of the Adjacent angles of the parallelogram is \(180^\circ\)
\(\begin{align}70^{\circ}+\angle {B}& =180^{\circ} \\[6pt]\angle {B}&=110^{\circ} \\[6pt]\angle {ABD}+\angle {DBC}&=110^{\circ} \\[6pt] \angle {ABD}+65^{\circ} &=110^{\circ} \\[6pt]\angle {ABD} &=45^{\circ} \\[6pt]\angle {ABD} &=\angle {CDB}=45^{\circ} \qquad (\text { Alternate Angles })\end{align}\)
Therefore, the answer is option (a).
Example 7 |
ABCD is a parallelogram in which \(∠ADC=75^\circ\) and side AB is produced to point E as shown in the figure. Find \( x+y?\)
(a) \(105^\circ\)
(b) \(210^\circ\)
(c) \(100^\circ\)
(d) \(220^\circ\)
Answer: In the above figure,
∠C and ∠D are the adjacent angles of the parallelogram ABCD
\(∠C+∠D=180^\circ\)
Sum of adjacent angles of the parallelogram is \(180^\circ\)
\(\begin{align}x+75^{\circ} &=180^{\circ} \\[6pt] x&=\angle C=105^{\circ}\\[6pt]x&=y=105^{\circ} \qquad \text { (Alternate Angles) }\end{align}\)
Therefore, \(x+y=105^\circ+105^\circ=210^\circ.\)
The answer is option (b).
Example 8 |
ABCD is a rhombus in which \(∠ACB=50^\circ\), then what is the measure of ∠ADB?
(a) \(50^\circ\)
(b) \(40^\circ\)
(c) \(45^\circ\)
(d) \(55^\circ\)
Answer: \(∠CAD = ACB = 500 \qquad (AD||BC, \text{Alternate Angles})\)
The diagonals of the rhombus bisect each other at \(90^{\circ}.\)
Therefore, \(∠AOD=90^{\circ}\)
In the triangle AOD,
\(∠OAD+∠AOD+∠ADO=180^{\circ}\)
(Sum of the interior angles of the triangle is 1800)
\(\begin{align}50^{0}+90^{\circ}+\angle A D O&=180^{\circ} \\\angle A D O&=180^{\circ}-\left(50^{\circ}+90^{\circ}\right) \\\angle A D O&=40^{\circ}=\angle A D B\end{align}\)
Therefore, the answer is option (b).
Example 9 |
The quadrilateral formed by joining the mid-points of the sides of a quadrilateral ABCD taken in order is a square only if
(a) ABCD is a rhombus
(b) Diagonals of ABCD are equal
(c) Diagonals of ABCD are equal and perpendicular
(d) Diagonals of ABCD are perpendicular
Answer: The answer is option (c).
Example 10 |
If angles P, Q, R and S of a quadrilateral PQRS, taken in order, are in the ratio 3: 7: 6: 4, then PQRS is a
(a) rhombus
(b) parallelogram
(c) trapezium
(d) kite
Answer: The quadrilateral PQRS is a Trapezium.
Learning Outcomes
- Understanding the types and basic properties of the quadrilateral.
- The above article helped us to determine the similarities and differences between the various types of quadrilaterals by analysing its sides, angles and the diagonal measures.
- The blog helped us to learn different theorems related to quadrilaterals.
-Written by Anusha Makam, Cuemath Teacher
About Cuemath
Cuemath, a student-friendly mathematics and coding platform, conducts regular Online Live Classes for academics and skill-development, and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills.
Frequently Asked Questions (FAQs) on Quadrilaterals
What are quadrilaterals?
Quadrilateral is a flat-surfaced polygon with four sides and four corners. The interior angles of the Quadrilateral always add up to \(360^\circ\).
How many types of quadrilaterals can be formed joining the vertices of a polygon of 12 sides?
\[\begin{align}12C4& = \frac{12!}{[4!*(12-4)!]}\\[6pt]12C4&=12\times11\times10\times9\times\frac{8!}{4!}\times8!\\[6pt]12C4&=495\end{align}\]
What is the sum of all the interior angles of the quadrilateral?
Sum of all the interior angles of the quadrilateral \(= 360^\circ\)