Properties of Quadrilaterals

Quadrilaterals are four-sided polygons with unique properties and classifications. Understanding their angles, types, and relationships is essential in geometry, as it lays the groundwork for solving complex problems and exploring advanced concepts in the subject.

1. Definition of a quadrilateral

   • A quadrilateral is a polygon with four sides, four vertices, and four angles.
   • The sides can be of different lengths and the angles can vary in measure.
   • Quadrilaterals can be classified based on their properties and relationships between sides and angles.

2. Sum of interior angles

   • The sum of the interior angles of a quadrilateral is always 360 degrees.
   • This can be derived from the formula (n-2) × 180, where n is the number of sides.
   • Each angle can be calculated if the other three angles are known.

3. Types of quadrilaterals (parallelogram, rectangle, square, rhombus, trapezoid, kite)

   • Parallelogram: Opposite sides are parallel and equal in length.
   • Rectangle: A parallelogram with four right angles.
   • Square: A rectangle with all sides equal in length.
   • Rhombus: A parallelogram with all sides equal in length.
   • Trapezoid: At least one pair of parallel sides.
   • Kite: Two pairs of adjacent sides that are equal in length.

4. Properties of parallelograms

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

5. Properties of rectangles

   • All angles are right angles (90 degrees).
   • Opposite sides are equal and parallel.
   • The diagonals are equal in length.

6. Properties of squares

   • All sides are equal in length and all angles are right angles.
   • The diagonals are equal, bisect each other, and are perpendicular.
   • A square is both a rectangle and a rhombus.

7. Properties of rhombuses

   • All sides are equal in length.
   • Opposite angles are equal, and adjacent angles are supplementary.
   • The diagonals bisect each other at right angles.

8. Properties of trapezoids

   • At least one pair of opposite sides is parallel.
   • The angles on the same side of a leg are supplementary.
   • The median (midsegment) connects the midpoints of the non-parallel sides and is parallel to the bases.

9. Properties of kites

   • Two pairs of adjacent sides are equal in length.
   • One pair of opposite angles are equal (the angles between the unequal sides).
   • The diagonals intersect at right angles, with one diagonal bisecting the other.

10. Diagonals in quadrilaterals

    • Diagonals are line segments connecting non-adjacent vertices.
    • In a parallelogram, diagonals bisect each other.
    • In rectangles and squares, diagonals are equal in length.

11. Area formulas for different quadrilaterals

    • Parallelogram: Area = base × height.
    • Rectangle: Area = length × width.
    • Square: Area = side².
    • Rhombus: Area = (diagonal₁ × diagonal₂) / 2.
    • Trapezoid: Area = (base₁ + base₂) / 2 × height.
    • Kite: Area = (diagonal₁ × diagonal₂) / 2.

12. Cyclic quadrilaterals

    • A cyclic quadrilateral can be inscribed in a circle.
    • The opposite angles of a cyclic quadrilateral are supplementary.
    • The area can be calculated using Brahmagupta's formula.

13. Inscribed and circumscribed quadrilaterals

    • Inscribed quadrilaterals have all vertices on a circle.
    • Circumscribed quadrilaterals have all sides tangent to a circle.
    • The properties of angles and sides differ based on the type of quadrilateral.

14. Quadrilateral angle bisector theorem

    • The angle bisector of a vertex divides the opposite side into segments proportional to the adjacent sides.
    • This theorem helps in finding unknown lengths in a quadrilateral.

15. Quadrilateral midsegment theorem

    • The midsegment connects the midpoints of two sides of a quadrilateral.
    • The midsegment is parallel to the third side and its length is half the length of that side.
    • This theorem is useful for solving problems involving quadrilaterals.