What are Quadrilaterals?
Quadrilaterals are one type of polygon which has four sides and four vertices and four angles along with 2 diagonals. There are various types of quadrilaterals.
Types of Quadrilaterals
The classification of quadrilaterals are dependent on the nature of sides or angles of a quadrilateral and they are as follows:
A trapezium is a quadrilateral with a pair of parallel sides.
A parallelogram is a quadrilateral whose opposite sides are parallel and equal.
- A rhombus is a quadrilateral with sides of equal length.
- Since the opposite sides of a rhombus have the same length, it is also a parallelogram.
- The diagonals of a rhombus are perpendicular bisectors of one another.
Introduction to Curves
A curve is a geometrical figure obtained when a number of points are joined without lifting the pencil from the paper and without retracing any portion. It is basically a line which need not be straight.
The various types of curves are:
- Open curve: An open curve is a curve in which there is no path from any of its point to the same point.
- Closed curve: A closed curve is a curve that forms a path from any of its point to the same point.
A curve can be :
- A closed curve
- an open curve
- A closed curve which is not simple
A simple closed curve made up of only line segments is called a polygon.
Various examples of polygons are Squares, Rectangles, Pentagons etc.
The sides of a polygon do not cross each other.
For example, the figure given below is not a polygon because its sides cross each other.
Classification of Polygons on the Basis of Number of Sides / Vertices
Polygons are classified according to the number of sides they have. The following lists the different types of polygons based on the number of sides they have:
- When there are three sides, it is triangle
- When there are four sides, it is quadrilateral
- When there are fives sides, it is pentagon
- When there are six sides, it is hexagon
- When there are seven sides, it is heptagon
- When there are eight sides, it is octagon
- When there are nine sides, it is nonagon
- When there are ten sides, it is decagon
A diagonal is a line segment connecting two non-consecutive vertices of a polygon.
Polygons on the Basis of Shape
Polygons can be classified as concave or convex based on their shape.
- A concave polygon is a polygon in which at least one of its interior angles is greater than 90∘. Polygons that are concave have at least some portions of their diagonals in their exterior.
- A convex polygon is a polygon with all its interior angle less than 180∘. Polygons that are convex have no portions of their diagonals in their exterior.
Polygons on the Basis of Regularity
Polygons can also be classified as regular polygons and irregular polygons on the basis of regularity.
- When a polygon is both equilateral and equiangular it is called as a regular polygon. In a regular polygon, all the sides and all the angles are equal. Example: Square
- A polygon which is not regular i.e. it is not equilateral and equiangular, is an irregular polygon. Example: Rectangle
Introduction to Quadrilaterals
Angle Sum Property of a Polygon
According to the angle sum property of a polygon, the sum of all the interior angles of a polygon is equal to (n−2)×180∘, where n is the number of sides of the polygon.
As we can see for the above quadrilateral, if we join one of the diagonals of the quadrilateral, we get two triangles.
The sum of all the interior angles of the two triangles is equal to the sum of all the interior angles of the quadrilateral, which is equal to 360∘ = (4−2)×180∘.
So, if there is a polygon which has n sides, we can make (n – 2) non-overlapping triangles which will perfectly cover that polygon.
The sum of the interior angles of the polygon will be equal to the sum of the interior angles of the triangles = (n−2)×180∘
Sum of Measures of Exterior Angles of a Polygon
The sum of the measures of the external angles of any polygon is 360∘.
Properties of Parallelograms
Elements of a Parallelogram
- There are four sides and four angles in a parallelogram.
- The opposite sides and opposite angles of a parallelogram are equal.
- In the parallelogram ABCD, the sides ¯¯¯¯¯¯¯¯AB and ¯¯¯¯¯¯¯¯¯CD are opposite sides and the sides ¯¯¯¯¯¯¯¯AB and ¯¯¯¯¯¯¯¯BC are adjacent sides.
- Similarly, ∠ABC and ∠ADC are opposite angles and ∠ABC and ∠BCD are adjacent angles.
Angles of a Parallelogram
The opposite angles of a parallelogram are equal.
In the parallelogram ABCD, ∠ABC=∠ADC and ∠DAB=∠BCD.
The adjacent angles in a parallelogram are supplementary.
∴ In the parallelogram ABCD, ∠ABC+∠BCD=∠ADC+∠DAB=180∘
Diagonals of a Parallelogram
The diagonals of a parallelogram bisect each other at the point of intersection.
In the parallelogram ABCD given below, OA = OC and OB = OD.
Properties of Special Parallelograms
A rectangle is a parallelogram with equal angles and each angle is equal to 90∘.
- Opposite sides of a rectangle are parallel and equal.
- The length of diagonals of a rectangle is equal.
- All the interior angles of a rectangle are equal to 90∘.
- The diagonals of a rectangle bisect each other at the point of intersection.
A square is a rectangle with equal sides. All the properties of a rectangle are also true for a square.
In a square the diagonals:
- bisect one another
- are of equal length
- are perpendicular to one another