Types and Properties of Quadrilaterals (Quadrangles)

Types and Properties of Quadrilaterals (Quadrangles)



Quadrilaterals Overview

A quadrilateral is a polygon with four sides (edges) or four vertices or four corners it can also be called as a TETRAGON

Properties of Quadrilaterals

  • Four Sides (Tetragon)
  • Four Angles (Quadrangle)
  • Four Vertices
  • Interior angles’ sum to 360°
  • Exterior angles’ sum to 360°

Important Terms Related to Quadrilaterals

Diagonals

Diagonals are line segments that join two opposite vertices of a quadrilateral (corners).

  • A quadrilateral can have only two diagonals

In the figure below line segment AD represents a diagonal of the Quadrilateral ABCD.

Adjacent Sides

Two sides of a quadrilateral are adjacent, if they share a common vertex.

In the quadrilateral shown above the 4 pair of adjacent sides are

  • AD and DC
  • DC and BC
  • BC and AB
  • AB and AD

Adjacent Angles

Two angles are of a quadrilateral adjacent, if they share a common side.

In the quadrilateral shown above the 4 pair of adjacent angles are

  • ∠1 and ∠3
  • ∠3 and ∠4
  • ∠2 and ∠4
  • ∠1 and ∠2

Types of Quadrilaterals

Convex Quadrilaterals

All interior angles are strictly less than 180 degrees

1. Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel and equal sides

Properties of Parallelogram
  • Theorem: Opposite sides are equal.
  • Theorem: Opposite angels are equal.
  • Theorem: Diagonals bisect each other.
  • Theorem: The diagonals of a parallelogram bisect each other to form two pairs of congruent triangles.
  • In a parallelogram if one angle is right angle, then all angles are right angle(i.e., It is a square).
  • Theorem: Parallelograms on the same base and between the same parallels are equal in area.

2. Rhombus (Rhomb)

A rhombus is a parallelogram whose all four sides are congruent.

Properties of Rhombus
  • All four sides are equal in length
  • Opposite sides are parallel(i.e., It’s a parallelogram).
  • Opposite angles are equal
  • Consecutive angles are supplementary (i.e., Sum of consecutive angle = 180°).
  • Diagonals are perpendicular bisectors of one another.
  • In case, if one angle is right angle, then all angles are right angle(i.e., It will be a square).
  • Four congruent triangles are formed by diagonals.

3. Rectangle [Equiangular Quadrangle]

A rectangle is a plane figure with four straight sides and four right angles.

Properties of Rectangle
  • Opposite sides that are equal and parallel (i.e., It’s a parallelogram).
  • Adjacent sides are of unequal length
  • Consecutive angles are supplementary (180 degree)
  • Diagonals are equal and bisect each other.

4. Square {Regular Quadrilateral}

A square is a rhombus whose all angles are right angles

Properties of Square
  • The diagonals are equal and bisect each other and at right angle
  • Opposite sides are parallel
  • All four angles are equal
  • All four sides are equal.

5. Kite

A kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

Properties of Kites
  • Two disjoint pairs of adjacent sides are equal (by definition)
  • One diagonal is the perpendicular bisector of the other diagonal.
  • One diagonal is a line of symmetry (it divides the quadrilateral into two congruent triangles that are mirror images of each other).
  • One diagonal bisects a pair of opposite angles

6. Trapezoid (Trapezium)

A quadrilateral with at least one pair of parallel sides is known as a trapezoid

Properties of Kites
  • The bases are parallel by definition
  • Each lower base angle is supplementary to the upper base angle on the same side.

Concave Quadrilaterals

At least one angle with a measure in between 180 degrees and 360 degrees

Cyclic Quadrilaterals

A cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle

Properties of Cyclic Quadrilaterals

  • The opposite angles of a cyclic quadrilateral sum to 180°
    • i.e. a + c = 180°
    • and b + d = 180°
  • The exterior angle of a cyclic quadrilateral is equal to the interior
    opposite angle
    • i.e. e = c

Pictorial Representation of Classification of Quadrilaterals

Quadrilateral Classification


Angle Sum Property of a Quadrilateral

The sum of the angles of a quadrilateral is 360°


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