Quadrilaterals are four-sided polygons with unique properties. From parallelograms to kites, each type has distinct characteristics that define its shape and angles. Understanding these properties helps solve geometric problems and classify shapes.

Theorems about quadrilaterals provide powerful tools for proving and problem-solving. By applying these theorems, you can determine missing measurements, establish relationships between sides and angles, and identify specific types of quadrilaterals in various scenarios.

Quadrilateral Classification and Properties

Characteristics of quadrilaterals

• A polygon with four sides and four angles
• Sum of the angles always equals 360°
• Types include parallelograms, rhombuses, rectangles, squares, trapezoids, and kites
  • Parallelograms have opposite sides parallel and congruent (equal in length)
  • Rhombuses have all sides congruent and opposite sides parallel (equilateral parallelogram)
  • Rectangles have opposite sides parallel and congruent with all 90° angles (equiangular parallelogram)
  • Squares have all sides congruent, opposite sides parallel, and all 90° angles (regular quadrilateral)
  • Trapezoids have one pair of opposite sides parallel (bases)
  • Kites have two pairs of adjacent sides congruent and perpendicular diagonals

Classification of quadrilaterals

• Parallelograms
  • Opposite sides parallel and congruent
  • Opposite angles congruent
  • Consecutive angles supplementary (add up to 180°)
  • Diagonals bisect each other (divide into two equal parts)
• Rhombuses
  • All sides congruent
  • Opposite sides parallel
  • Opposite angles congruent
  • Diagonals bisect each other at right angles (90°)
• Rectangles
  • Opposite sides parallel and congruent
  • All angles measure 90°
  • Diagonals congruent and bisect each other
• Squares
  • All sides congruent
  • Opposite sides parallel
  • All angles measure 90°
  • Diagonals congruent, bisect each other, and perpendicular
• Trapezoids
  • One pair of opposite sides parallel (bases)
  • Isosceles trapezoids have congruent legs and base angles
• Kites
  • Two pairs of adjacent sides congruent
  • One pair of opposite angles congruent
  • Diagonals perpendicular with one bisecting the other

Theorems for quadrilateral properties

• Parallelogram theorems
  • If both pairs of opposite sides parallel, then it's a parallelogram
  • If both pairs of opposite sides congruent, then it's a parallelogram
  • If one pair of opposite sides both parallel and congruent, then it's a parallelogram
  • If both pairs of opposite angles congruent, then it's a parallelogram
  • If diagonals bisect each other, then it's a parallelogram
• Rhombus theorems
  • If a parallelogram has all sides congruent, then it's a rhombus
  • If the diagonals of a parallelogram are perpendicular, then it's a rhombus
• Rectangle theorems
  • If a parallelogram has one right angle (90°), then it's a rectangle
  • If a parallelogram has congruent diagonals, then it's a rectangle
• Square theorems
  • If a parallelogram has all sides congruent and one right angle, then it's a square
  • If a rhombus has one right angle, then it's a square
  • If a rectangle has all sides congruent, then it's a square

Applications of quadrilateral properties

• Use properties to:
  1. Find missing side lengths
  2. Find missing angle measures
  3. Determine relationships between sides and angles
  4. Prove a quadrilateral belongs to a specific category
• Problem-solving strategies
  • Use the sum of angles in a quadrilateral ($360°$) to find missing angle measures
  • Apply properties of parallel lines and transversals to find missing angle measures (alternate interior angles congruent, corresponding angles congruent, etc.)
  • Use properties of congruent sides and angles to find missing measurements
  • Employ properties of diagonals in specific quadrilaterals (bisecting, perpendicular, congruent) to solve problems