Properties of trapeziums

key notes :

Definition of a Trapezium

  • A quadrilateral with at least one pair of parallel sides.
  • Also called a trapezoid in some regions.

Types of Trapeziums

  • Isosceles Trapezium: Non-parallel sides (legs) are equal in length, and base angles are equal.
  • Right-Angled Trapezium: Has one pair of right angles (90°).

Basic Properties of a Trapezium

  • Has four sides and four angles.
  • One pair of opposite sides is parallel (called bases).
  • The non-parallel sides are called legs.
  • The sum of the interior angles is 360°.

Angle Properties

  • Adjacent angles on the same leg are supplementary (sum to 180°).
  • In an isosceles trapezium, the base angles are equal.

Diagonal Properties

  • A trapezium has two diagonals that may or may not be equal.
  • In an isosceles trapezium, diagonals are of equal length.

Midsegment (Median) Theorem

  • The midsegment (line joining the midpoints of the non-parallel sides) is parallel to the bases.
  • Its length is the average of the two bases:

Midsegment = Base1 + Base2 / 2

Area of a Trapezium

  • The area is given by the formula:

Area = 1/2 × ( Base1 + Base2 ) × Height

  • Height is the perpendicular distance between the bases.

Symmetry and Reflection

  • An isosceles trapezium has one line of symmetry (through the midpoints of the bases).
  • A general trapezium may have no lines of symmetry.

Special Cases

  • A parallelogram can be seen as a trapezium where both pairs of opposite sides are parallel.
  • A rectangle and a square are also special types of trapeziums.

Learn with an example

If UX is not parallel to VW , what is VW?

VW= ____

UVWX is a quadrilateral with exactly one pair of parallel sides, UV and WX , and one pair of congruent base angles, ∠U and ∠V.

So, UVWX is an isosceles trapezium.

This means the legs are congruent. So, VW=UX=97.

If CD is not parallel to AB , what is CD?

CD =_____

ABCD is a quadrilateral with exactly one pair of parallel sides, AD and BC , and one pair of congruent base angles, ∠D and ∠A.

So, ABCD is an isosceles trapezium.

This means the legs are congruent. So, CD=AB=19.

If BE is not parallel to CD , what is BE?

BE = ____

BCDE is a quadrilateral with exactly one pair of parallel sides, BC and DE , and one pair of congruent base angles, ∠E and ∠D.

So, BCDE is an isosceles trapezium.

This means the legs are congruent. So, BE=CD=100.

Let’s practice!