CLASS-7
POLYGON - QUADRILATERALS

QUADRILATERALS -

A quadrilateral is a polygon with four sides, four vertices, and four angles. It is one of the most basic types of polygons and has a wide variety of forms, depending on the lengths of its sides and the measures of its angles.

Key Characteristics of Quadrilaterals:-

  1. Sides (Edges):- Four straight line segments that form the boundary.
  2. Vertices:- The four points where two sides meet.
  3. Angles:- The sum of the interior angles of any quadrilateral is always 360∘.

Types of Quadrilaterals:-

Quadrilaterals are classified based on the relationships between their sides and angles.

1. Parallelogram:-

  • Opposite sides are parallel and equal in length.
  • Opposite angles are equal.
  • The diagonals bisect each other.

2. Rectangle:-

  • A type of parallelogram where all angles are right angles (90∘).
  • Opposite sides are equal in length.
  • The diagonals are equal in length and bisect each other.

3. Square:-

  • A special type of rectangle where all sides are equal in length and all angles are right angles (90∘).
  • The diagonals are equal in length, bisect each other at right angles, and divide the square into four right-angled triangles.

4. Rhombus:-

  • A parallelogram with all four sides equal in length.
  • Opposite angles are equal.
  • The diagonals bisect each other at right angles and are not equal in length.

5. Trapezoid (US) / Trapezium (UK):-

  • A quadrilateral with at least one pair of parallel sides.
  • The non-parallel sides are called the legs.
  • Isosceles trapezoid:- A trapezoid where the non-parallel sides (legs) are equal in length, and the base angles are equal.

6. Kite:-

  • A quadrilateral with two pairs of adjacent sides that are equal in length.
  • The diagonals intersect at right angles, but only one diagonal is bisected by the other.

7. Irregular Quadrilateral:-

  • A quadrilateral that doesn’t fall into any of the specific categories above. Its sides and angles can have arbitrary lengths and measurements.

Properties of Quadrilaterals:-

  • Sum of Interior Angles:- The sum of the interior angles of any quadrilateral is always 360∘, regardless of its type. Sum of interior angles = 360∘
  • Diagonals:- Quadrilaterals have two diagonals, which are line segments connecting opposite vertices.

Examples:-

  • Square:- All sides and angles are equal. A square is both a rectangle and a rhombus.
  • Rectangle:- Opposite sides are equal, and all angles are 90∘.
  • Rhombus:- All sides are equal, and opposite angles are equal.
  • Trapezoid:- One pair of opposite sides is parallel.
  • Kite:- Two pairs of adjacent sides are equal.

Quadrilaterals play a significant role in both theoretical and practical applications of geometry, appearing frequently in architecture, engineering, and design due to their diverse properties.