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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
Kinds 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
Summary
Here is the Understanding Quadrilaterals Class 8 notes summary
