Quick Revision Notes & Formula Sheet

Class: 9th Mathematics
Chapter: Quadrilaterals (चतुर्भुज)

### Introduction to Quadrilateral (चतुर्भुज का परिचय)

A quadrilateral is a closed figure formed by four intersecting line segments. It has:

• 4 sides (चार भुजाएँ)
• 4 angles (चार कोण)
• 4 vertices (चार शीर्ष)
• 2 diagonals (दो विकर्ण)

Angle Sum Property of a Quadrilateral:
The sum of all the four interior angles of a quadrilateral is always $360^\circ$.
$\angle A + \angle B + \angle C + \angle D = 360^\circ$

### Types of Quadrilaterals & Their Properties (चतुर्भुज के प्रकार और उनके गुण)

1. Parallelogram (समान्तर चतुर्भुज):

   • Both pairs of opposite sides are parallel and equal. ($AB \parallel CD, AD \parallel BC$ and $AB = CD, AD = BC$)
   • Opposite angles are equal. ($\angle A = \angle C, \angle B = \angle D$)
   • Consecutive angles (adjacent angles) are supplementary. ($\angle A + \angle B = 180^\circ$)
   • Diagonals bisect each other.

2. Rectangle (आयत):

   • A parallelogram with all interior angles equal to $90^\circ$.
   • Opposite sides are equal and parallel.
   • Diagonals are equal and bisect each other.

3. Square (वर्ग):

   • A rectangle with all four sides equal.
   • All interior angles are $90^\circ$.
   • Diagonals are equal, bisect each other at right angles ($90^\circ$).

4. Rhombus (समचतुर्भुज):

   • A parallelogram with all four sides equal.
   • Opposite angles are equal.
   • Diagonals bisect each other at right angles ($90^\circ$), but they are not necessarily equal.

5. Trapezium (समलंब चतुर्भुज):

   • A quadrilateral in which one pair of opposite sides is parallel. ($AB \parallel CD$)

6. Kite (पतंग):

   • A quadrilateral with two pairs of adjacent sides equal.
   • Diagonals intersect at $90^\circ$, and the longer diagonal bisects the shorter diagonal.

### Key Theorems (महत्वपूर्ण प्रमेय)

• Theorem 1: A diagonal of a parallelogram divides it into two congruent triangles.
• Theorem 2: In a parallelogram, opposite sides are equal. (Conversely, if each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram).
• Theorem 3: In a parallelogram, opposite angles are equal. (Conversely, if each pair of opposite angles is equal, then it is a parallelogram).
• Theorem 4: The diagonals of a parallelogram bisect each other.
• Theorem 5: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel.

### The Mid-Point Theorem (मध्य-बिंदु प्रमेय)

• Mid-Point Theorem:
  The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
  If $D$ and $E$ are mid-points of sides $AB$ and $AC$ respectively, then:
  $DE \parallel BC$ and $DE = \frac{1}{2} BC$

• Converse of Mid-Point Theorem:
  The line drawn through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

### Quick Tips for Exam (परीक्षा के लिए महत्वपूर्ण टिप्स)

• To prove a quadrilateral is a Parallelogram, show any one of the following:
  1. Both pairs of opposite sides are parallel.
  2. Both pairs of opposite sides are equal.
  3. One pair of opposite sides is equal and parallel.
  4. Diagonals bisect each other.
• Remember that the perimeter of any quadrilateral is the sum of all its four sides ($P = a + b + c + d$).
• Area formulas:
  • Rectangle: $\text{Length} \times \text{Breadth}$ ($l \times b$)
  • Square: $\text{Side}^2$ ($a^2$)
  • Parallelogram: $\text{Base} \times \text{Height}$ ($b \times h$)
  • Rhombus: $\frac{1}{2} \times d_1 \times d_2$ (where $d_1, d_2$ are diagonals)
  • Trapezium: $\frac{1}{2} \times (\text{Sum of parallel sides}) \times \text{Height}$