What is a Trapezium – Definition, Types, Properties & Examples

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The word ‘trapezium’ originated from the Greek word ‘trapeza’ which means table. A trapezium is a quadrilateral that has one pair of parallel opposite sides. 

Let’s understand the trapezium definition and the properties of trapezium with examples.

Trapezium Definition

A trapezium is a 2D shape and a quadrilateral in which of two pairs of opposite sides, only one pair is parallel. The opposite parallel sides are referred to as the base and the non-parallel sides are referred to as the legs of the trapezium. 

The above figure shows a trapezium.

In the above figure, $\text{XY}$ and $\text{WZ}$ are the bases while $\text{XW}$ and $\text{YZ}$ are the legs of the trapezium. 

Types of Trapezium

Broadly trapeziums can be divided into three types. These are

• Isosceles Trapezium
• Scalene Trapezium
• Right Trapezium

Isosceles Trapezium

The trapezium in which the two legs are of equal length. In the figure, $\text{ABCD}$ is an isosceles trapezium, where $\text{BC} = \text{DA}$.

Scalene Trapezium

The trapezium whose neither the sides nor the angles are equal is a scalene trapezium.

Right Trapezium

The trapezium which has right angles in a pair is known as a right trapezium.

In the above figure, $\text{PQRS}$ is a right trapezium, where $\text{PS} || \text{QR}$, and $\text{PQ} \perp \text{PS}$,  $\text{PQ} \perp \text{QR}$.

Properties of Trapezium

The following are the basic properties of trapeziums that help you identify them. 

• In trapezium, exactly one pair of opposite sides are parallel
• The bases of a trapezium are parallel to each other.
• The diagonals of a trapezium always intersect each other.
• Two pairs of adjacent interior angles sum up to $180^{\circ}$.
• The sum of all the interior angles in a trapezium is always $360^{\circ}$.
• In the isosceles trapezium, the length of both diagonals is equal.
• The non-parallel sides in the trapezium are unequal except in the isosceles trapezium
• The line that joins the mid-points of the non-parallel sides is always parallel to the bases or parallel sides which is equal to half of the sum of parallel sides
• The mid-points of the sides are collinear to the intersection point of diagonals

Diagonals of an Isosceles Trapezium Are Equal

The diagonals of an isosceles trapezium are equal, i.e., in an isosceles trapezium $\text{PQRS}$, where $\text{PQ} = \text{RS}$, $\text{PR} = \text{QS}$.

Let’s see how to prove the above statement.

To prove it, the first step is to do a little construction.

Construction is to draw two perpendiculars, $\text{PA}$ and $\text{QB}$ on the side $\text{SR}$.

Now, in $\triangle \text{PSA}$ and $\triangle \text{QRB}$, we have

$\text{PS} = \text{PS}$ (Given)

$\angle \text{PAS} = \angle \text{QBR} = 90^{\circ}$ (By construction)

$\text{PA} = \text{QB}$ (Perpendicular distance between two parallel lines)

Therefore, $\triangle \text{PSA} \cong \triangle \text{QRB}$ (By RHS congruence criterion)

Thus, $\angle \text{PSA} = \angle \text{QRB}$ (Corresponding Parts of Congruent Triangles)

$⇒\angle \text{PSR} = \angle \text{QRS}$

Now, in $\triangle \text{PSR}$ and $\triangle \text{QRS}$, we have

$\text{PS} = \text{QR}$ (Given)

$\angle \text{PSR} = \angle \text{QRS}$ (Proved above)

$\text{SR} = \text{RS}$ (Common)

Therefore, $\triangle \text{PSR} \cong \triangle \text{QRS}$ (By SAS congruence criterion)

Thus, $\text{PR} = \text{QS}$ (Corresponding Parts of Congruent Triangles)

Line Joining Mid Points of Non-Parallel Sides is Parallel to Parallel Sides

To prove that the line joining midpoints of non-parallel sides of a trapezium is parallel to the parallel sides, let’s consider a trapezium $\text{ABCD}$, where $\text{AC}$ and $\text{BD}$ are the diagonals of the trapezium.

Further let $\text{M}$ and $\text{N}$ are the midpoints of the diagonals $\text{AC}$ and $\text{BD}$.

Now, we need to prove that $\text{MN} || \text{AB} || \text{CD}$.

Let’s do one simple construction. Join $\text{CN}$ and extend it to meet $\text{AB}$ at $\text{E}$.

Now, consider $\triangle \text{CDN}$ and $\text{EBN}$,

Since $\text{N}$ is the midpoint of $\text{BD}$

$\text{DN} = \text{BN}$

Since the alternate interior angles are equal, therefore,

$\angle \text{DCN} = \angle \text{NEB}$

and $\angle \text{CDN} = \angle \text{NBE}$

Therefore, $\triangle \text{CDN} \cong \text{EBN}$ (ASA Congruence Criterion)

Thus,  $\text{DC} = \text{EB}$ and $\text{CN} = \text{NE}$

Now, consider $\triangle \text{CAE}$

$\text{M}$ and $\text{N}$ are the midpoints of $\text{AC}$ and $\text{CE}$

$\text{MN} || \text{AE}$

Therefore, by midpoint theorem, $\text{MN} || \text{AB} || \text{CD}$.

Practice Problems

1. What is a trapezium?
2. What is the difference between a trapezium and a parallelogram?
3. In a trapezium $\text{ABCD}$, such that $\text{BC} || \text{AD}$, what is the value of
4. $\angle \text{ABC} + \angle \text{DAB}$
5. $\angle \text{BCD} + \angle \text{CDA}$
6. State True or False
   • In an isosceles trapezium, parallel sides are equal
   • In an isosceles trapezium, non-parallel sides are equal
   • In an isosceles trapezium, a line joining midpoints of non-parallel sides of a trapezium is parallel to the parallel sides
   • In every trapezium, a line joining midpoints of non-parallel sides of a trapezium is parallel to the parallel sides

FAQs

Conclusion

A trapezium is a 2D shape and a quadrilateral in which of two pairs of opposite sides, only one pair is parallel. You can find many real-world examples of objects shaped trapezium such as lamp shades, popcorn tubs, handbags, bathtubs, sheds, etc.

Recommended Reading

• How to Construct a Triangle(With Steps, Diagrams & Examples)
• Median of a Triangle(Definition & Properties)
• Altitude of a Triangle(Definition & Properties)
• Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
• Similarity of Triangles Criteria – SSS, SAS, AA
• Angle Bisector of a Triangle – Definition, Properties & Examples
• What is Quadrilateral in Math(Definition, Shape & Examples)
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• Median of a Triangle(Definition & Properties)
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• Congruence of Triangles Criteria – SSS, SAS, ASA, RHS
• Similarity of Triangles Criteria – SSS, SAS, AA
• Types of Triangles – Definition & Examples
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