Quadrilaterals Class 9 Notes - Chapter 8

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

Quadrilaterals Class 9-1

In ΔABC and ΔCDA

AC=AC [Common / transversal]

BCA=DAC [alternate angles]

BAC=DCA [alternate angles]

ΔABCΔCDA  [ASA rule]

Hence,

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

Opposite angles in a parallelogram are equal

Quadrilaterals Class 9-2

In parallelogram ABCD

ABCD; and AC is the transversal

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

BCDA; and AC is the transversal

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

Adding (1) and (2)

1+2=3+4

BAD=BCD

Similarly,
ADC=ABC

Properties of diagonal of a parallelogram

Diagonals of a parallelogram bisect each other.

Quadrilaterals Class 9-3

In  ΔAOB and ΔCOD,

3=5 [alternate interior angles]

1=2  [vertically opposite angles]

AB=CD [opp. Sides of parallelogram]

ΔAOBΔCOD [AAS rule]

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

Hence, proved

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

Diagonal of a parallelogram divides it into two congruent triangles.

Quadrilaterals Class 9-4

In ΔABC and ΔCDA,

AB=CD [Opposite sides of parallelogram]

BC=AD [Opposite sides of parallelogram]

AC=AC [Common side]

ΔABCΔCDA [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

Quadrilaterals Class 9-5

In ΔAOD and ΔCOD,

OA=OC [Diagonals of parallelogram bisect each other]

OD=OD [Common side]

AD=CD [Adjacent sides of a rhombus]

ΔAODΔCOD [SSS rule]

AOD=DOC [C.P.C.T]

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

Hence, AOD=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

Quadrilaterals Class 9-6

Rectangle ABCD

In ΔABC and ΔBAD,

AB=BA [Common side]

BC=AD [Opposite sides of a rectangle]

ABC=BAD [Each = 900  ABCD is a Rectangle]

ΔABCΔBAD [SAS rule]

AC=BD  [C.P.C.T]

Consider ΔOAD and ΔOCB,

AD=CB  [Opposite sides of a rectangle]

OAD=OCB  [ AD||BC and transversal AC intersects them]

ODA=OBC  [ AD||BC and transversal BD intersects them]

ΔOADΔOCB [ASA rule]

OA=OC  [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

Quadrilaterals Class 9-7

Square ABCD

In ΔABC and ΔBAD,

AB=BA [Common side]

BC=AD [Opposite sides of a Square]

ABC=BAD [Each = 900  ABCD is a Square]

ΔABCΔBAD [SAS rule]

AC=BD  [C.P.C.T]

Consider ΔOAD and ΔOCB,

AD=CB  [Opposite sides of a Square]

OAD=OCB  [ AD||BC and transversal AC intersects them]

ODA=OBC  [ AD||BC and transversal BD intersects them]

ΔOADΔOCB [ASA rule]

OA=OC  [C.P.C.T]

Similarly we can prove OB=OD

In ΔOBA and ΔODA,

OB=OD   [ proved above]

BA=DA  [Sides of a Square]

OA=OA  [ Common side]

ΔOBAΔODA,  [ SSS rule]

AOB=AOD  [ C.P.C.T]

But, AOB+AOD=1800  [ Linear pair]

AOB=AOD=900

Important results related to parallelograms

Quadrilaterals Class 9-8

Parallelogram ABCD

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.

A=C,B=D,

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

A diagonal of parallelogram divides it into two congruent triangles.

ΔABCΔCDA [With respect to AC as diagonal]

ΔADBΔCBD   [With respect to BD as diagonal]

The diagonals of a parallelogram bisect each other.

AE=CE,BE=DE

1=5 (alternate interior angles)

2=6 (alternate interior angles)

3=7 (alternate interior angles)

4=8 (alternate interior angles)

9=11 (vertically opp. angles)

10=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

Quadrilaterals Class 9-9

In ΔABCE – the midpoint of ABF – the midpoint of AC

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

In ΔAEF and ΔCDF,

AF=CF  [ F is the midpoint of AC]

AFE=CFD  [ V.O.A]

EF=DF [ Construction]

ΔAEFΔCDF [SAS rule]

Hence,

EAF=DCF….(1)

DC=EA=EB  [ E is the midpoint of AB]

DCEAAB [Since, (1), alternate interior angles]

DCEB

So EBCD is a parallelogram

Therefore, BC=ED and BCED

Since, ED=EF+FD=2EF=BC  [ 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’

Quadrilaterals Class 9-10

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

Quadrilaterals Class 9-11

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

In ADC,

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

In ABC,

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

(1) + (2):

1+2+3+4+5+6=360

I.e, A+B+C+D=360

Hence proved

Types of Quadrilaterals

Trapezium

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

Quadrilaterals Class 9-12

Trapezium

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.

Quadrilaterals Class 9-13

Parallelogram ABCD

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

Rhombus

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

Quadrilaterals Class 9-14

Rhombus ABCD

A rhombus ABCD in which AB=BC=CD=AD and  ACBD

Rectangle

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

Quadrilaterals Class 9-15

Rectangle ABCD

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

Square

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

Quadrilaterals Class 9-16

Square ABCD

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

Kite

A kite is a quadrilateral with adjacent sides equal.

Quadrilaterals Class 9-17

Kite ABCD

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.

Venn diagram for different types of quadrilaterals

Quadrilaterals Class 9-18

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