QUADRILATERALS Mathematics - Class 8 Chapter 3 Unit 5

QUADRILATERALSMathematics - Class 8

Chapter 3 Unit 5

Module Objectives• Learn to identify quadrilaterals• Understand basic properties of quadrilaterals• Classify quadrilaterals into common types,

and recognize their specific properties

Basic Geometrical Figures• Line

• Bounded by 2 end points• All points are collinear i.e. lie on the same straight line

• Triangle• Plane figure bounded by three sides• Atleast 3 non-collinear points

• Quadrilateral• ‘Quad’ = four, ‘lateral’ = side• Formed by joining 4 points• Any 3 out of these 4 points are non-collinear

Quadrilaterals• Any closed figure having four sides formed by

joining four points and three of which are not collinear is called quadrilateral.

A

C

BD

P

Q

R

S

W

YZ

X

M

OP

N

Are these Quadrilaterals?

No, the sides cross each other

A

B

CD

A

B

C D

No, the sides are all not line segments

A

C

BD

Yes, it is a closed figure •formed by the union of four line segments •that join 4 points lying on the same plane•no three of which are collinear •and each segment meet exactly 2 other lines, each at their end point

Quadrilateral Notations• Let ABCD be a quadrilateral

• Vertices - Points A, B, C and D• Four sides - Segments AB, BC, CD and DA• Four angles - DAB, ABC, BCD and CDA• Two diagonals - Segments AC and BD

A

C

BD

• Naming Hint: If you join adjacent letters in the name, then there should not be any crossing of line segments

• Naming a quadrilateral e.g. ABCD• Refer to its vertices in a particular order• We cannot read it as ADBC or ADCB

Quadrilateral Notations• Adjacent or consecutive sides• Two sides of a quadrilateral have a

common end point• E.g. AB and AD, CD and CB

• Opposite sides• Two sides do NOT have a common end

point• E.g. AB and DC

A

C

BD

Quadrilateral Notations• Adjacent or consecutive angles• Two angles have a side common to

them• E.g. DAB and ABC, with AB being

the common side

• Opposite angles• Two angles do NOT have a common

side• E.g. DAB and ABC

A

C

BD

Properties of Quadrilaterals• Diagonal Property• Diagonal AC divides the quadrilateral

into 2 triangles ABC and ADC

• Angle Sum Property• The sum of the angles of a

quadrilateral is 360• ABC + BCD + CDA + DAB = 360

D

B

CA

A

C

BD

Types of Quadrilaterals• Convex Quadrilateral• Quadrilateral in which every

internal angle of the quadrilateral is lesser than 180.

• Concave Quadrilateral• A quadrilateral is concave if any

internal angle of the quadrilateral is greater than 180.

N

L

MK

P

Q

R

S

Special Kinds of Quadrilaterals• Classification based on nature of sides or angles

• Is Parallelogram a type of Trapezium? • Yes, it has parallel opposite sides

• Is Kite a type of Parallelogram?• No, it does not have 2 pairs of equal-length opposite sides

Type of Quadrilateral

Properties

Trapezium One pair of opposite sides are parallel

Parallelogram Both pairs of opposite sides are parallelOpposite sides are equal and opposite angles are equal

Kite Two pairs of equal-length adjacent sides

Trapezium

• Quadrilateral with a pair of opposite side that are parallel

• Isoceles Trapezium• Non-parallel sides are equal• Base angles are equal• Diagonal are equals• Adjacent angles corresponding to

parallel sides are supplementary

Parallelograms

• Both pairs of opposite sides are parallel

• Opposite sides are of equal length

• Special Kinds of Parallelograms• Rectangles• Rhombus• Square

W

YZ

X

Kinds of Parallelograms - Rectangle

• All angles are equal and right angles

• All sides are not equal• Diagonals are equal and

bisect each other

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ά

ά

άάά

Kinds of Parallelograms - Rhombus

• All sides are equal• All angles are not equal• Diagonals bisect each other at

right angles• Two diagonals divide the

rhombus into four congruent right angled triangles

• Angles are bisected by the diagonals

D

C

B

A

Kinds of Parallelograms - Square• Square• All its angles are equal and right

angles• All sides are equal• Both diagonals are equal• Diagonals bisect each other at

right angles

Kind of ParallelogramsParallel Sides All Sides All Angles Diagonals

Rectangle2 pairs of opposite sides

Equal Equal &Right angles

Equal &Bisect each other

Rhombus Equal Not equal Bisect each other at right angles

Square Equal Equal

Kite

• Type of quadrilateral but not a parallelogram

• Has 2 pairs of equal-length adjacent sides

• Two isoceles triangles are joined along the common base

• Rhombus is a special kind of Kite

A

ODB

C