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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

- What is a trapezium?
- What is the difference between a trapezium and a parallelogram?
- In a trapezium $\text{ABCD}$, such that $\text{BC} || \text{AD}$, what is the value of
- $\angle \text{ABC} + \angle \text{DAB}$
- $\angle \text{BCD} + \angle \text{CDA}$
- 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

### What is a trapezium shape?

A two-dimensional quadrilateral that has a pair of non-adjacent parallel sides and a pair of non-parallel sides is referred to as a trapezium shape. It looks like a triangle that is cut from the top.

### Is trapezium always a parallelogram?

No, a trapezium is not a parallelogram. In a trapezium, only one pair of opposite sides are parallel, whereas, in the case of a parallelogram, both pairs of opposite sides are parallel and equal.

### Is a trapezium a rhombus?

No, a trapezium is not a rhombus. In a rhombus, all four sides are equal and both pairs of opposite sides are parallel, whereas, in the case of a trapezium, one pair of opposite sides is parallel and equal.

### Why is a kite not a trapezium?

In kites, none of the sides is parallel, whereas, in the case of a trapezium, one pair of opposite sides are parallel and equal.

### Is trapzoid the same as trapezium?

A trapezoid and a trapezium are actually the same shapes. In North America, the shape is called a trapezoid, but it is known as a trapezium in other English-speaking countries around the world. The definition of both is the same.

## 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)
- Properties of Triangle – Theorems & Examples
- 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
- Types of Triangles – Definition & Examples
- What is Triangle in Geometry – Definition, Shapes & Examples
- Pair of Angles – Definition, Diagrams, Types, and Examples
- Construction of Angles(Using Protractor & Compass)
- Types of Angles in Maths(Acute, Right, Obtuse, Straight & Reflex)
- What is an Angle in Geometry – Definition, Properties & Measurement
- How to Construct a Tangent to a Circle(With Steps & Pictures)
- Tangent of a Circle – Meaning, Properties, Examples
- Angles in a Circle – Meaning, Properties & Examples
- Chord of a Circle – Definition, Properties & Examples
- How to Draw a Circle(With Steps & Pictures)
- What is a Circle – Parts, Properties & Examples
- How to Construct a Perpendicular Line (With Steps & Examples)
- How to Construct Parallel Lines(With Steps & Examples)
- How To Construct a Line Segment(With Steps & Examples)
- What are Collinear Points in Geometry – Definition, Properties & Examples
- What is a Transversal Line in Geometry – Definition, Properties & Examples
- What are Parallel Lines in Geometry – Definition, Properties & Examples
- What is Concurrent lines in Geometry – Definition, Conditions & Examples
- What is Half Line in Geometry – Definition, Properties & Examples
- What is a Perpendicular Line in Geometry – Definition, Properties & Examples
- Lines in Geometry(Definition, Types & Examples)