Quick Revision Notes & Formula Sheet

Class: 8th | Subject: Mathematics
Chapter: Understanding Quadrilaterals (चतुर्भुजों को समझना)
Board: MP Board (NCERT)

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1. Basic Concepts & Definitions (मूलभूत अवधारणाएं)

• Plane Curve (समतल वक्र): A figure drawn on a flat surface without lifting the pencil.
• Polygon (बहुभुज): A simple closed curve made up of only line segments.
  • Minimum sides required to form a polygon = 3 (Triangle).

Classification of Polygons based on Sides:

Sides ($n$)	Name (English)	नाम (Hindi)
3	Triangle	त्रिभुज
4	Quadrilateral	चतुर्भुज
5	Pentagon	पंचभुज
6	Hexagon	षट्भुज
7	Heptagon	सप्तभुज
8	Octagon	अष्टभुज
$n$	$n$-gon	$n$-भुज

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2. Types of Polygons (बहुभुज के प्रकार)

1. Convex Polygon (उत्तल बहुभुज):

   • All diagonals lie completely inside the polygon.
   • Every interior angle is less than $180^\circ$.

2. Concave Polygon (अवतल बहुभुज):

   • At least one diagonal lies outside the polygon.
   • At least one interior angle is greater than $180^\circ$ (reflex angle).

3. Regular Polygon (सम बहुभुज):

   • A polygon which is both equiangular (all angles equal) and equilateral (all sides equal).
   • Examples: Equilateral triangle, Square.

4. Irregular Polygon (विषम बहुभुज):

   • A polygon whose sides or angles are not all equal.
   • Examples: Rectangle, Scalene triangle.

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3. Important Formulas for Polygons (महत्वपूर्ण सूत्र)

Let $n$ be the number of sides of a polygon:

• Sum of Interior Angles of an $n$-sided polygon:
  $$\text{Sum} = (n - 2) \times 180^\circ$$

• Sum of Exterior Angles of ANY polygon:
  $$\text{Sum of exterior angles} = 360^\circ$$

• Number of Diagonals (विकर्णों की संख्या) in a polygon:
  $$\text{Number of diagonals} = \frac{n(n - 3)}{2}$$

For a Regular Polygon with $n$ sides:

• Each Interior Angle:
  $$\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}$$

• Each Exterior Angle:
  $$\text{Exterior Angle} = \frac{360^\circ}{n}$$

• Number of sides ($n$) when exterior angle is given:
  $$n = \frac{360^\circ}{\text{Exterior Angle}}$$

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4. Quadrilateral & Angle Sum Property

• Quadrilateral (चतुर्भुज): A polygon having 4 sides, 4 angles, and 4 vertices.
• Angle Sum Property of a Quadrilateral:
  The sum of all four interior angles of a quadrilateral is always $360^\circ$. $$\angle A + \angle B + \angle C + \angle D = 360^\circ$$

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5. Types of Quadrilaterals & Their Properties (चतुर्भुज के प्रकार और गुण)

A. Trapezium (समलंब)

• Has one pair of parallel opposite sides.
• Isosceles Trapezium: A trapezium whose non-parallel sides are equal in length.

B. Kite (पतंग)

• Has two distinct pairs of equal length adjacent sides.
• Diagonals intersect each other at $90^\circ$ (right angles).
• One of the diagonals bisects the other.

C. Parallelogram (समांतर चतुर्भुज)

A quadrilateral with both pairs of opposite sides parallel.

• Properties:
  1. Opposite sides are equal ($AB = CD$ and $AD = BC$).
  2. Opposite angles are equal ($\angle A = \angle C$ and $\angle B = \angle D$).
  3. Adjacent angles are supplementary ($\angle A + \angle B = 180^\circ$).
  4. Diagonals bisect each other ( seasonal midpoint is same).

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6. Special Parallelograms (विशेष समांतर चतुर्भुज)

A. Rhombus (समचतुर्भुज)

A parallelogram with all sides of equal length.

• Properties:
  • All properties of a parallelogram.
  • All 4 sides are equal.
  • Diagonals are perpendicular bisectors of each other (intersect at $90^\circ$).

B. Rectangle (आयत)

A parallelogram with a right angle ($90^\circ$).

• Properties:
  • All properties of a parallelogram.
  • Each interior angle is equal to $90^\circ$.
  • Diagonals are equal in length.

C. Square (वर्ग)

A rectangle with all sides equal (or a rhombus with right angles).

• Properties:
  • All 4 sides are equal and all 4 angles are $90^\circ$.
  • Diagonals are equal in length AND bisect each other at right angles ($90^\circ$).

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7. Quick Summary Table: Properties of Special Quadrilaterals

Quadrilateral	All Sides Equal?	Opposite Angles Equal?	Diagonals Equal?	Diagonals Perpendicular ($90^\circ$)?	Diagonals Bisect Each Other?
Parallelogram	No	Yes	No	No	Yes
Rhombus	Yes	Yes	No	Yes	Yes
Rectangle	No	Yes	Yes	No	Yes
Square	Yes	Yes	Yes	Yes	Yes
Kite	No	Only 1 pair	No	Yes	No (only 1 bisected)
Trapezium	No	No	No	No	No

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8. Key Tips for Exam Solving

1. Exterior Angle + Adjacent Interior Angle = $180^\circ$ (Linear Pair).
2. To find the number of sides of a regular polygon given an exterior angle, use:
   n = 360° / Exterior Angle
3. Always check if a quadrilateral is a parallelogram first before applying properties like opposite angles being equal.