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.

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

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

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

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

Introduction to Quadrilaterals

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

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

Venn diagram for different types of quadrilaterals

Quadrilaterals Class 9-18

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