Introduction to Quadrilaterals

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.

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Introduction to Quadrilaterals-PDF

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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:

1. Parallelogram
2. Trapezium
3. Rhombus
4. Kite
5. Rectangle
6. Square

Different Types and Properties of Quadrilateral

1. Parallelogram

A Quadrilateral in which opposite sides are parallel and equal is called a Parallelogram.

Properties of Parallelogram

• In a Parallelogram,

1.  Opposite sides are equal.
2.  Opposite angles are equal.
3.  Each diagonal bisects the parallelogram.
4. 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.

2. 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,

1. The adjacent angles of the parallel sides are equal.
2. 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.

3. 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.

4. 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.

5. 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.

6. 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

1. In a parallelogram,
   1. Opposite sides are equal.
   2.  Opposite angles are equal.
   3.  Each diagonal bisects the parallelogram.
2. The diagonal of the parallelogram bisect each other.
3. The diagonals of the rectangle are equal and bisect each other.
4. The diagonals of the rhombus bisect each other at 90⁰.
5. 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

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