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Understanding Quadrilaterals



What is Closed curve and Open curve

Closed curve is a figure in the plane with no end points. It completely encloses an area
 

 
 
Open curve is a figure in the place with end points
 

 

What are polygons

A simple closed curve made up of only line segments is called a polygon.

 

Classification of polygons

 
We classify polygons according to the number of sides( or vertices)
Number of sides
Classification
3
Triangle
4
Quadrilateral
5
Pentagon
6
Hexagon
7
Heptagon
8
Octagon
9
Nonagon
 

Diagonal in Polygons

A diagonal is a line segment connecting two non-consecutive vertices of a polygon
 
 
 

Convex and Concave Polygons

Convex Polygon
We have all the diagonals inside the Polygon

 
Concave Polygon
We don’t have all the diagonals inside the Polygon

We will be concentrating on Convex Polygon in this chapter

Regular and Irregular Polygons

A regular polygon is both ‘equiangular’ and ‘equilateral’.
So all the sides and angles should be same
a) So square is a regular polygon but rectangle is not
b) Equilateral triangle is a regular polygon

Angle Sum in the Polygons

The Sum of the angles in the polygon is given by
=(n-2) × 1800
For Triangle, n=3
So Total =1800
For quadrilateral, n=4
So total =3600
 
Example
Find the value of angle x

Solution
We know in the quadrilateral, sum of interior angle is 3600
So
50+130+120+x=360
x=600
Watch this tutorial on how to solve angle problems in quadrilateral

Sum of the Measures of the Exterior Angles of a Polygon

The sum of the measures of the external angles of any polygon is 360°.
This property is very useful is finding number of sides of the polygons

Example:
Find the number of sides of a regular polygon whose each exterior angle
has a measure of 60°.
Solution: Total measure of all exterior angles = 360°
Measure of each exterior angle = 60°
Therefore, the number of exterior angles =360/60=6
The polygon has 6 sides.
Watch this tutorial on how to solve exterior angles problems

What is Quadrilaterals

A quadrilateral is a four sides Polygon. It has four angles
 

 
Angle Sum Property
Sum of angles of the Quadrilaterals =3600
Exterior Angle Property
Sum of exterior angles of the Quadrilaterals =3600

Types of Quadrilaterals

 

Trapezium


Trapezium is a quadrilateral with a pair of parallel sides.

 

Isosceles trapezium


Trapezium when non-parallel sides of it are of equal length
 

Kite


It is a quadrilaterals having exactly two distinct consecutive pairs of sides of equal length
Here ABCD is a Kite

AB=BC
AD=CD
 

Parallelogram


It is a quadrilateral whose opposite sides are parallel.

Here ABCD is a Parallelogram
AD || BC, AB ||CD
Property 1
The opposite sides of a parallelogram are of equal length.
AD=BC , AB=CD
Property 2
The opposite angles of a parallelogram are of equal measure
∠A= ∠C , ∠B= ∠D
Property 3
The adjacent angles in a parallelogram are supplementary.
∠A+ ∠D=180 , ∠B+ ∠C=180
Property 4
The diagonals of a parallelogram bisect each other
Example

Find the value of the angles x,y,z in the parallelograms given above
Solution
Angle B and Angle D are opposite angle and are equal
So y=1000
Angle C and Angle B are adjacent angle and are supplementary
So x+100=180
x=800
Angle A and Angle C are opposite angle and are equal
So z=800

Rhombus


A rhombus is a quadrilateral with sides of equal length
A rhombus has all the properties of a parallelogram and also that of a kite.
 
 
Special Property
The diagonals of a rhombus are perpendicular bisectors of one another

Rectangle


A rectangle is a parallelogram with equal angles
 
Property 1
Each angle is of 900
Property 2
The diagonals of a rectangle are of equal length

Square


A square is a rectangle with equal sides.
So it has the same property of Rectangle in addition of 4 equal sides

Property
The diagonals of a square are perpendicular bisectors of each other.
 
Watch this tutorial for understanding quadrilaterals


 

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