Polygon

 

 

We know a polygon is a simple closed figure made up of only line segments.  We can classify polygonsaccording to the number of sides or vertices.
The simple polygon we know is a triangle. A trianglehas three sides and, thus, is a three-sided polygon. 
A four-sided polygon is called a quadrilateral.  
A five sided polygon is called a pentagon.
In this manner, we can obtain a six-sided polygoncalled a hexagon, a seven-sided polygon, called aheptagon, and so on.
If we have an n-sided polygon, it is called an n-gon.

 

CONCAVE & CONVEX POLYGONS:
We know that each side of a polygon is connected by two consecutive vertices of the polygon.
A diagonal is a line segment that connects the non-consecutive vertices of a polygon.  
If a diagonal lies outside a polygon, then the polygon is called a concave polygon.
If all the diagonals lie inside the polygon, then the polygon is said to be a convex polygon.

 

REGULAR & IRREGULAR POLYGONS:
A regular polygon is equiangular and equilateral. The word equiangular means, the interior angles of the polygon are equal to one another. The word equilateral means, the lengths of the sides are equal to one another.
The polygon with unequal sides and unequal angles is called an irregular polygon.

ANGLE SUM PROPERTY:
The sum of all interior angles of a polygon is called the angle sum.
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At one vertex, we extend a side. This side makes an angle with its consecutive side. This angle is called the exterior angle.The interior angle and the exterior angles are adjacent angles.  These angles form a linear pair. Hence the sum of the exterior angles of any polygon is 

 

Kinds of Quadrilaterals

 

 

 

We know that a quadrilateral is a four sided polygon. We can differentiate quadrilaterals based upon the sides and the angles between sides.
A quadrilateral with a pair of opposite sides parallel is called a trapezium.
If the non parallel sides of the trapezium have the same length, then the trapezium is called an isosceles trapezium.
A quadrilateral having exactly two distinct consecutive pairs of sides of equal length is called kite.
A quadrilateral with opposite sides parallel and equal to each other is called a parallelogram.   
Elements of a Parallelogram: In a Parallelogram
The sides which have a common vertex are calledadjacent sides.
Angles having one arm in common are called adjacent angles.
Angles at the opposite vertices are called opposite angles.
The lines joining the opposite vertices of the parallelogram are called diagonals.
The diagonals of a parallelogram are not equal. 

Property: The opposite sides of a parallelogram are equal.

Property: The opposite angles of a parallelogram are of equal measure.

Property: The diagonals of a parallelogram bisect each other.

 

A quadrilateral is a closed figure which has four sides, four angles and four vertices. There are different types of quadrilaterals such asParallelogram, Rectangle, Square, Rhombus,Trapezium and Kite. 
A parallelogram is a quadrilateral in which

• opposite sides are parallel and equal,

• opposite angles are equal,

• the diagonals bisect each other,

• each diagonal divides it into two congruent triangles,

• adjacent angles are supplementary.

A square is a parallelogram in which

• all the sides are equal,

• each angle measures 900,

• diagonals are equal and bisect at right angles.

A rectangle is a parallelogram in which

• diagonals are equal and bisect each other,

• each angle measures 900.

A rhombus is a parallelogram in which

• all four sides are equal,

• diagonals bisect each other at right angles.

 

Theorem:  A diagonal of a parallelogram divides it into two congruent triangles.

 

Theorem: If each pair of opposite sides is equal in a quadrilateral, then it is a parallelogram.

 

 

Theorem: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.