Revision Notes on Basic Geometrical Ideas

Learning Goal

Numbers and it’s types

 

Type of quadrilaterals (6)
Rectangle आयत	Square वर्ग	Parallelogram समान्तर चतुर्भुज	Rhombus सम चतुर्भुज	Trapezium समलम्ब चतुर्भुज	Kite पतंग

Quadrilaterals

A closed figure bounded by 4 sides is called quadrilaterals.

Four Sides: AB, BC, CD, DA

Four Vertices: A, B, C, D

Four Angle: ∠A, ∠B, ∠C, ∠D

Two Diagonals: AC and BD

Adjacent Sides: Two sides of a quadrilateral which has a common ends point are called adjacent sides.

Ex- AB and BC they have common ends B.

AB and AD, DA and DC, CD and CB

There are four pairs of adjacent sides.

Opposite Sides: Two sides of a quadrilateral which have no common ends point are called opposite sides.

Ex- AB and DC, BC and AD

There are two pairs of opposite sides.

Adjacent Angles: Two angles of a quadrilateral which have a common side are called Adjacent angles.

∠A and ∠B, ∠B and ∠C, ∠C and ∠D, ∠D and ∠A.

There are four pairs of adjacent angles.

Opposite Angles: Two angles of a quadrilateral which have no common side are called opposite angles.

∠B and ∠D, ∠A and ∠C

There are two pairs of opposite angles.

Rectangles: – A quadrilateral whose opposite sides are equal and each angle is 90°  is called rectangle.

Properties of a Rectangle

Opposite sides are parallel and congruent. (MN = OP and NO = PM)

A rectangle is a special type of parallelogram whose angles 90°

A rectangle has two lines of symmetry.

The order of rotational symmetry is 2.

The diagonals are congruent or equal and bisect each other.

Diagonals bisect the angles.                        

Each diagonal divides the rectangle into two congruent  right-angled triangles

The adjacent angles are supplementary. That is

∠P+∠O or∠O+∠N or ∠N+∠M or ∠M+∠P=180°

Formula

1. Length of the diagonal of a rectangle = √(L² + B²)
2. Area = L × B
3. Perimeter = 2(L+B)

Squares: –   A quadrilateral whose all sides are equal and each angle is 90° is called Square.

Properties of a square

All sides and angles are congruent.

Opposite sides are parallel to each other.

A square is parallelogram whose all angles and sides are equal.

A square has four lines of symmetry.

The order of rotational symmetry is 4.

The diagonals bisect each other at 90⁰.

The diagonals are equal and congruent.

Diagonals bisect the angles.

Each diagonal divides the square into two congruent isosceles right-angled triangles

The adjacent angles are supplementary. That is

∠D+∠E or∠E+∠F or ∠F+∠G or ∠G+∠D=180°

A parallelogram becomes a square when the diagonals are equal and right bisectors of each other.

Formula

Length of the diagonal

D = √2×side = √2× L

Area = (side)² = a²

Perimeter = 4×side = 4L

Parallelogram: –   A quadrilateral whose opposite sides are parallel and equal is called parallelogram.

 

 

 

 

Properties of a parallelogram

Opposite sides are parallel and congruent. (MN=OP and NO=PM)

Opposite angles are congruent. (∠M=∠O and ∠N=∠O )

Adjacent angles are supplementary.

(∠M+∠N=∠N+∠O=∠O+∠P=∠P+∠M = 180⁰)

Diagonals bisect each other.

Diagonal divides the parallelogram into two congruent triangles.

If one of the angles of a parallelogram is a right angle then all other angles are right and it becomes a rectangle.

Formula

Important formulas of parallelograms

Area = Base ×Height = L × H

Perimeter = 2(L+ B)

Rhombus: – A rhombus is a parallelogram whose diagonals are perpendicular to each other.

Properties of a Rhombus

All sides are congruent.

Opposite angles are congruent.

The diagonals bisect each other perpendicular.

Adjacent angles are supplementary (For ex- ∠A + ∠B = 180°).

Formula

Important formulas for a Rhombus

Perimeter = 4×side = 4L

Trapezium: – A quadrilateral whose one pair of opposite sides is parallel is called trapezium.

Properties of a Trapezium

The bases of the trapezium are parallel to each other (MN ⫽ OP).

No sides, angles and diagonals are congruent.

Important Formulas for a Trapezium

Area =  (Sum of parallel sides) × Height

Perimeter = Sum of all sides = L + L₁ + L₂ + L₃

Summary of properties

S.No.	Property	Parallelogram	Rectangle	Rhombus	Square
1	All sides are congruent	✕	✕	✓	✓
2	Opposite sides are parallel and congruent	✓	✓	✓	✓
3	All angles are congruent	✕	✓	✕	✓
4	Opposite angles are congruent	✓	✓	✓	✓
5	Diagonals are congruent	✕	✓	✕	✓
6	Diagonals are perpendicular	✕	✕	✓	✓
7	Diagonals bisect each other	✓	✓	✓	✓
8	Adjacent angles are supplementary	✓	✓	✓	✓

 

Convex and Concave Polygons

The polygons which have all the diagonals inside the figure are known as a Convex Polygon.

The polygons which have some of its diagonals outside the figure also are known as a Concave Polygon.

Regular and Irregular Polygons

Polygons which are equiangular and equilateral are called Regular Polygons i.e. a polygon is regular if-

It’s all sides are equal.

It’s all angles are equal.

Hence square is a regular polygon but a rectangle is not as its angles are equal but sides are not equal.

Angle Sum Property

The sum of all the interior angles of a polygon remains the same according to the number of sides regardless of the shape of the polygon.

The sum of interior angles of a polygon is-

(n – 2) × 180°

Where n = number of sides of the polygon

Example

Polygon	Number of Sides	Sum of Interior Angles
Triangle	3	(3 – 2) × 180° = 180°
Quadrilateral	4	(4 – 2) × 180° = 360°
n-gon	n	(n – 2) × 180°

Remark: This property is applicable to both convex and concave polygon.