**CBSE Class 9 Maths Quadrilaterals Notes:-**Download PDF Here

Get the complete notes on quadrilaterals class 9 here. A quadrilateral is a shape which has four sides. In this article, we are going to discuss the different types of quadrilaterals such as square, rectangle, parallelogram properties with proofs.

To know more about Parallelogram, visit here.

### Parallelogram: Opposite sides of a parallelogram are equal

In Î”ABCÂ andÂ Î”CDA

AC=AC [Common / transversal]

âˆ BCA=âˆ DAC [alternate angles]

âˆ BAC=âˆ DCA [alternate angles]

Î”ABCâ‰…Î”CDAÂ [ASA rule]

Hence,

AB=DC and AD=BC [ C.P.C.T.C]

### Opposite angles in a parallelogram are equal

In parallelogram ABCD

ABâ€–CD; and AC is the transversal

Hence, âˆ 1=âˆ 3….(1) (alternate interior angles)

BCâ€–DA; and AC is the transversal

Hence, âˆ 2=âˆ 4….(2) (alternate interior angles)

Adding (1) and (2)

âˆ 1+âˆ 2=âˆ 3+âˆ 4

âˆ BAD=âˆ BCD

Similarly,

âˆ ADC=âˆ ABC

### Properties of diagonal of a parallelogram

– **Diagonals of a parallelogram bisect each other.**

InÂ Î”AOBÂ andÂ Î”COD,

âˆ 3=âˆ 5 [alternate interior angles]

âˆ 1=âˆ 2Â [vertically opposite angles]

AB=CD [opp. Sides of parallelogram]

Î”AOBâ‰…Î”COD [AAS rule]

OB=OD and OA=OC [C.P.C.T]

Hence, proved

Conversely,

–Â **If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.**

– **Diagonal of a parallelogram divides it into two congruent triangles.**

In Î”ABC and Î”CDA,

AB=CD [Opposite sides of parallelogram]

BC=AD [Opposite sides of parallelogram]

AC=AC [Common side]

Î”ABCâ‰…Î”CDA [by SSS rule]

Hence, proved.

#### For More Information On Properties Of Parallelogram, Watch The Below Video.

### Diagonals of a rhombus bisect each other at right angles

Diagonals of a rhombus bisect each – other at right angles

In Î”AODÂ andÂ Î”COD,

OA=OC [Diagonals of parallelogram bisect each other]

OD=OD [Common side]

AD=CD [Adjacent sides of a rhombus]

Î”AODâ‰…Î”COD [SSS rule]

âˆ AOD=âˆ DOC [C.P.C.T]

âˆ AOD+âˆ DOC=180 [âˆµ AOC is a straight line]

Hence, âˆ AOD=âˆ DOC=90

Hence proved.

#### For More Information On Properties Of Rhombus, Watch The Below Video.

### Diagonals of a rectangle bisect each other and are equal

In Î”ABCÂ andÂ Î”BAD,

AB=BA [Common side]

BC=AD [Opposite sides of a rectangle]

âˆ ABC=âˆ BAD [Each = 900Â âˆµ ABCD is a Rectangle]

Î”ABCâ‰…Î”BAD [SAS rule]

âˆ´AC=BDÂ [C.P.C.T]

ConsiderÂ Î”OADÂ andÂ Î”OCB,

AD=CBÂ [Opposite sides of a rectangle]

âˆ OAD=âˆ OCBÂ [âˆµ AD||BC and transversal AC intersects them]

âˆ ODA=âˆ OBCÂ [âˆµ AD||BC and transversal BD intersects them]

Î”OADâ‰…Î”OCB [ASA rule]

âˆ´OA=OCÂ [C.P.C.T]

Similarly we can prove OB=OD

#### For More Information On Properties Of Rectangle, Watch The Below Video.

### Diagonals of a square bisect each other at right angles and are equal

In Î”ABCÂ andÂ Î”BAD,

AB=BA [Common side]

BC=AD [Opposite sides of a Square]

âˆ ABC=âˆ BAD [Each = 900Â âˆµ ABCD is a Square]

Î”ABCâ‰…Î”BAD [SAS rule]

âˆ´AC=BDÂ [C.P.C.T]

ConsiderÂ Î”OADÂ andÂ Î”OCB,

AD=CBÂ [Opposite sides of a Square]

âˆ OAD=âˆ OCBÂ [âˆµ AD||BC and transversal AC intersects them]

âˆ ODA=âˆ OBCÂ [âˆµ AD||BC and transversal BD intersects them]

Î”OADâ‰…Î”OCB [ASA rule]

âˆ´OA=OCÂ [C.P.C.T]

Similarly we can prove OB=OD

In Î”OBAÂ andÂ Î”ODA,

OB=ODÂ Â [ proved above]

BA=DAÂ [Sides of a Square]

OA=OAÂ [ Common side]

Î”OBAâ‰…Î”ODA,Â [ SSS rule]

âˆ´âˆ AOB=âˆ AODÂ [ C.P.C.T]

But,Â âˆ AOB+âˆ AOD=1800Â [ Linear pair]

âˆ´âˆ AOB=âˆ AOD=900

### Important results related to parallelograms

Opposite **sides** of a parallelogram are **parallel** and **equal**.

AB||CD,AD||BC,AB=CD,AD=BC

Opposite **angles** of a parallelogram are **equal** adjacent angels are **supplementary**.

âˆ A=âˆ C,âˆ B=âˆ D,

âˆ A+âˆ B=1800,âˆ B+âˆ C=1800,âˆ C+âˆ D=1800,âˆ D+âˆ A=1800

A **diagonal** of parallelogram divides it into** two congruent triangles**.

Î”ABCâ‰…Î”CDA [With respect to AC as diagonal]

Î”ADBâ‰…Î”CBDÂ Â [With respect to BD as diagonal]

The diagonals of a parallelogram **bisect** each other.

AE=CE,BE=DE

âˆ 1=âˆ 5 (alternate interior angles)

âˆ 2=âˆ 6 (alternate interior angles)

âˆ 3=âˆ 7 (alternate interior angles)

âˆ 4=âˆ 8 (alternate interior angles)

âˆ 9=âˆ 11 (vertically opp. angles)

âˆ 10=âˆ 12 (vertically opp. angles)

## The Mid-Point Theorem

**The line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of the third side**

In Î”ABC,Â E – the midpoint of AB;Â F – the midpoint of AC

**Construction**: Produce EF to D such that EF=DF.

In Î”AEF and Î”CDF,

AF=CFÂ [ F is the midpoint of AC]

âˆ AFE=âˆ CFDÂ [ V.O.A]

EF=DF [ Construction]

âˆ´Î”AEFâ‰…Î”CDF [SAS rule]

Hence,

âˆ EAF=âˆ DCF….(1)

DC=EA=EBÂ [ E is the midpoint of AB]

DCâ€–EAâ€–AB [Since, (1), alternate interior angles]

DCâ€–EB

So EBCD is a parallelogram

Therefore, BC=ED and BCâ€–ED

Since, ED=EF+FD=2EF=BCÂ [ âˆµ EF=FD]

We have,EF=12BC and EF||BC

Hence proved.

To know more about Mid-Point Theorem, visit here.

## Introduction to Quadrilaterals

#### For More Information On Quadrilaterals, Watch The Below Videos.

### Quadrilaterals

Any four points in a plane, of which three are non-collinear are joined in order results into a four-sided closed figure called **‘quadrilateral’**

#### For More Information On Elementary Shapes – Quadrilaterals, Watch The Below Video.

To know more about Quadrilaterals, visit here.

### Angle sum property of a quadrilateral

Angle sum property – Sum of angles in a quadrilateral is 360

In â–³ADC,

âˆ 1+âˆ 2+âˆ 4=180 (Angle sum property of triangle)…………….(1)

In â–³ABC,

âˆ 3+âˆ 5+âˆ 6=180 (Angle sum property of triangle)………………(2)

(1) + (2):

âˆ 1+âˆ 2+âˆ 3+âˆ 4+âˆ 5+âˆ 6=360

I.e, âˆ A+âˆ B+âˆ C+âˆ D=360

Hence proved

## Types of Quadrilaterals

### Trapezium

A **trapezium** is a quadrilateral with any **one pair of opposite sides parallel**.

PQRS is a trapezium in which PQ||RS

### Parallelogram

A **parallelogram** is a quadrilateral, with both pair of **opposite sides parallel and equal**. In a parallelogram, diagonals bisect each other.

Parallelogram ABCD in which AB||CD,BC||ADÂ and AB=CD,BC=AD

### Rhombus

A **rhombus** is a parallelogram with **all sides equal.Â **In a rhombus, diagonals bisect each other perpendicularly

A rhombus ABCD in which AB=BC=CD=AD and Â ACâŠ¥BD

### Rectangle

A **rectangle** is a parallelogram with **all angles as right angles**.

A rectangle ABCD in which, âˆ A=âˆ B=âˆ C=âˆ D=900

### Square

A **square **is a special case of a parallelogram with **all angles as right angles and all sides equal.**

A**Â **square ABCD in which âˆ A=âˆ B=âˆ C=âˆ D=900 and AB=BC=CD=AD

### Kite

A **kite** is a quadrilateral with **adjacent sides equal**.

A kite ABCD in which AB=BC and AD=CD

#### For More Information On Types Of Quadrilaterals, Watch The Below Videos.

To know more about Different Types of Quadrilateral, visit here.