Quadrilaterals ||Maths|| Chapter 8 NCERT Notes

 Quadrilaterals ||Maths|| Chapter 8 NCERT Notes

 

1. Basic Definitions and Properties of Quadrilaterals

  • Quadrilateral: A closed figure with four sides, four vertices, and four angles.

    Some basic properties include:

    • The sum of the interior angles of a quadrilateral is always 360°.
    • A quadrilateral can be convex or concave, depending on whether all the interior angles are less than 180° (convex) or one of the angles is greater than 180° (concave).

Common Types of Quadrilaterals:

  • Parallelogram: A quadrilateral where opposite sides are parallel and equal.
  • Rectangle: A parallelogram where all angles are 90°.
  • Square: A rectangle with all sides equal.
  • Rhombus: A parallelogram with all sides equal.
  • Trapezium (Trapezoid): A quadrilateral with only one pair of parallel sides.
  • Kite: A quadrilateral with two pairs of adjacent sides equal.

2. Angle Sum Property of a Quadrilateral

The sum of the interior angles of any quadrilateral is 360°. This can be proved by dividing a quadrilateral into two triangles. Since the sum of the angles in a triangle is 180°, the sum of the angles in two triangles is 360°.

∠A + ∠B + ∠C + ∠D = 360°

3. Parallelogram

A parallelogram is a quadrilateral where both pairs of opposite sides are parallel and equal.

Properties of a Parallelogram:

  1. Opposite sides are equal and parallel.
  2. Opposite angles are equal.
  3. The diagonals bisect each other.
  4. Each diagonal divides the parallelogram into two congruent triangles.
  5. Adjacent angles are supplementary (sum equals 180°).

Important Theorems for Parallelograms:

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

4. Special Types of Parallelograms

Rectangle:

A rectangle is a parallelogram where each angle is 90°.

Properties:

  1. Opposite sides are equal and parallel.
  2. All angles are 90°.
  3. The diagonals are equal and bisect each other.

Rhombus:

A rhombus is a parallelogram where all four sides are equal.

Properties:

  1. All sides are equal.
  2. Opposite angles are equal.
  3. The diagonals bisect each other at right angles (90°).
  4. The diagonals bisect the angles of the rhombus.

Square:

A square is a rectangle and a rhombus at the same time. It has all the properties of both rectangles and rhombuses.

Properties:

  1. All sides are equal.
  2. All angles are 90°.
  3. The diagonals are equal, bisect each other at right angles, and bisect the angles.

5. Trapezium (Trapezoid)

A trapezium is a quadrilateral with one pair of opposite sides parallel. If the non-parallel sides are equal, it is called an isosceles trapezium.

Properties:

  1. The sum of the angles between a pair of parallel sides is 180°.
  2. In an isosceles trapezium, the non-parallel sides are equal, and the diagonals are equal.

6. Midpoint Theorem

The Midpoint Theorem states that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of its length.

This theorem can be extended to quadrilaterals to form important results about their diagonals and sides.

7. Important Theorems

Theorem 1: Diagonals of a Parallelogram Bisect Each Other

If a quadrilateral is a parallelogram, then the diagonals bisect each other.

Theorem 2: Converse of Theorem 1

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Theorem 3: Diagonals of a Rhombus are Perpendicular

In a rhombus, the diagonals are perpendicular to each other.

Theorem 4: Diagonals of a Rectangle are Equal

In a rectangle, the diagonals are equal.

Summary:

  • A quadrilateral is a four-sided polygon with various types like parallelogram, rectangle, square, rhombus, trapezium, and kite.
  • The sum of the interior angles of a quadrilateral is always 360°.
  • Parallelograms have specific properties like equal opposite sides, equal opposite angles, and diagonals that bisect each other.
  • Special types of parallelograms include rectangles, rhombuses, and squares, each with its unique properties.
  • Theorems related to parallelograms, like the diagonal bisecting property, are essential for solving problems.