CHAPTER -3 UNDERSTANDING QUADRILATERAL
Plane Surface:It is a flat surface in which a straight line joining any two
points lies completely on that surface only.
Example: Paper
Plane Curve:It is a curve that lies in a single plane or a plane surface.
Simple curves: Curve that does not intersect itself at any point.
Open curves: Curve whose end points do not meet.
Closed curves: Curve that completely encloses an area and has no
endpoints.
[GURUKUL GANDHINAGAR] Page 1
What are Polygons?
A Polygon is a closed figure made up of lines segments (not curves) in two-
dimensions.
A minimum of three line segments are required for making a closed figure,
thus a polygon with a minimum of three sides is known as Triangle.
Diagonals: It is a line segment which connects two non-consecutive
vertices of a polygon.
For the figure shown above, SQ and PR are the two diagonals for the given
quadrilateral.
Interior and Exterior of a closed curve:
(a) Interior of a closed curve:The area which is covered inside the closed
[GURUKUL GANDHINAGAR] Page 2
curve is called its interior.
In the fig; the shaded portion reflects the interior of the given closed
curve.
(b) Exterior of a closed curve:The area which is covered outside the
closed curve is called its exterior.
In the fig; the shaded portion reflects the exterior of the given closed
curve.
Depending on the sides and angles, the polygons are classified into
different types, namely
Regular Polygon
Irregular Polygon
Convex Polygon
Concave polygon
Regular Polygon: If all the sides and interior angles of the polygon are
equal, then it is known as a regular polygon.For example square,
triangle, etc
[GURUKUL GANDHINAGAR] Page 3
Irregular Polygon: If all the sides and the interior angles of the polygon
are of different measure, then it is known as an irregular polygon.For
example rectangle, kite, etc
Convex Polygon: If all the interior angles of a polygon are strictly less
than 180 degrees, then it is known as a convex polygon.
Concave Polygon: If one or more interior angles of a polygon are more
than 180 degrees, then it is known as a concave polygon.
Angle sum property of polygon: (n-2)1800
Angle sum property of triangle:The sum of the three angles of a triangle
is 180o.
Sum of the measure of the Exterior Angles of a Polygon:The sum on the
measures of the external angles of any polygon is 360o.
[GURUKUL GANDHINAGAR] Page 4
Quadrilaterals: A Quadrilateral is a polygon having the number of sides
equal to four.
Types of Quadrilaterals:
Trapezium:
It is a quadrilateral with a pair of parallel sides.
In all these figures, there is at least one pair of parallel side.
[GURUKUL GANDHINAGAR] Page 5
Kite:It is another special type of quadrilateral in which there will be
exactly two distinct consecutive pairs of sides of equal length.
Parallelogram:It is a quadrilateral whose opposite sides are parallel.
Properties of parallelogram:
(i) The opposite sides of a parallelogram are of equal length.
(ii) The opposite angles of a parallelogram are of equal measure.
(iii) The adjacent angles in a parallelogram are supplementary.
(iv) The diagonals of a parallelogram bisect each other.
Some special Parallelograms:
Rhombus: It is a quadrilateral which has all sides of equal length.
A rhombus has all the properties of a parallelogram as well as that of a
kite.
Property: The diagonals of rhombus are perpendicular bisectors of each
other.
Rectangle:It is a parallelogram with equal angles.
Opposite sides of rectangle are equal and each angle 900.
[GURUKUL GANDHINAGAR] Page 6
Property: The diagonals of a rectangle are of equal length as well as they
bisect each other.
Square:All the sides of squares are equal and each angle 900.
Property: The diagonals of a square are equal and perpendicular
bisectors of each other.
The table below gives the comparison of Opposite sides, angles, and
diagonals of different Quadrilateral.
Quadrilateral Sides All Opposite Diagonal
Opposite sides angles
Equal equal
Parallel Equal Equal Perpendicular
Rectangle ✔ ✔ ✖ ✔ ✔ ✔
Parallelogram ✔ ✔ ✖ ✔ ✖ ✖
Rhombus ✔ ✔ ✔ ✔ ✖ ✔
Trapezium ✖ ✖ ✖ ✖ ✖ ✖
(Only one
side)
Square ✔ ✔ ✔ ✔ ✔ ✔
[GURUKUL GANDHINAGAR] Page 7
