Understanding Quadrilaterals

• Basic Concepts

  • Plane Curve: Curve formed without lifting the pencil
    • Simple Closed Curve: Closed curve that does not cross itself
  • Polygon: Simple closed curve made up of only line segments
    • Classification: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), n-gon (n)
  • Diagonals: Line segment connecting two non-consecutive vertices
    • Formula: Number of diagonals = $\frac{n(n-3)}{2}$

• Types of Polygons

  • Convex Polygon: No portions of diagonals lie in the exterior
  • Concave Polygon: At least one diagonal lies in the exterior
  • Regular Polygon: Both Equilateral (all sides equal) and Equiangular (all angles equal)
  • Irregular Polygon: Sides or angles are not all equal

• Angle Properties

  • Interior Angle Sum
    • Triangle = $180^\circ$
    • Quadrilateral = $360^\circ$
    • Formula: $(n - 2) \times 180^\circ$
  • Exterior Angles
    • Sum of exterior angles of any polygon = $360^\circ$
    • Each exterior angle (Regular Polygon) = $\frac{360^\circ}{n}$
    • Number of sides ($n$) = $\frac{360^\circ}{\text{Exterior Angle}}$

• Types of Quadrilaterals

  • Trapezium
    • Exactly one pair of parallel sides
    • Isosceles Trapezium: Non-parallel sides are equal
  • Kite
    • Two distinct consecutive pairs of equal sides
    • Diagonals are perpendicular to each other
    • One diagonal bisects the other
  • Parallelogram
    • Both pairs of opposite sides are parallel
    • Opposite sides are equal
    • Opposite angles are equal
    • Adjacent angles are supplementary ($180^\circ$)
    • Diagonals bisect each other

• Special Parallelograms

  • Rhombus
    • Parallelogram with all four sides equal
    • Diagonals are perpendicular bisectors of each other
  • Rectangle
    • Parallelogram with four right angles ($90^\circ$)
    • Opposite sides are equal
    • Diagonals are equal in length and bisect each other
  • Square
    • Parallelogram with all sides equal and all angles $90^\circ$
    • Has properties of a Parallelogram, Rhombus, and Rectangle
    • Diagonals are equal and perpendicular bisectors of each other