**Trapezium (Geometry)**

Trapezium is a quadrilateral in which** two sides are parallel** and **other two sides are non parallel**.

It is one of the irregular quadrilateral which do have parallel opposite sides.

In fact the word **“Trapezium” means “Little Table”** in Greek.

**Keyword for Trapezium**

(a) one pair of parallel sides

(b) one pair of non-parallel sides

**Structure of Trapezium**

Trapezium is made up of following components

**(a) Base**

The parallel sides are known as Base.

**(b) Leg**

Sides which are non parallel are called Legs

**(c) Height**

Perpendicular distance between parallel lines is called height

**(d) There are four angles in trapezium**

**Examples of Trapezium**

**Types of Trapezium**

Trapezoids are classified on the basis of sides and angles.

**(01) Scalene Trapezoid**

Its a trapezium in which all angles and sides are of different measure

**(02) Isosceles Trapezium**

Its a trapezium in which the non parallel sides are equal

Given above is the trapezium ABCD in which AB & CD are parallel sides and AD & BC are non-parallel sides.

In Isosceles Trapezium, the non- parallel sides are equal

i.e. AD = BC

**(03) Right Trapezium**

A trapezium with at-least two right angles is called Right Trapezium

Given above is Right Trapezium ABCD in which, ∠A = ∠D = 90 degree

**Properties of Trapezium**

**(01) One pair of sides are parallel to each other**

In the above trapezium ABCD, side AB and CD are parallel

i.e. AB II CD

**(02) The adjacent angles between parallel lines are supplementary**

Given above is trapezium ABCD with line AB II CD

According to angle property of trapezium**∠1 + ∠ 2 = 180** and **∠3 + ∠ 4 = 180**

**Proof**

Line AB and CD are parallel lines and AD is a transversal

So,**∠1 + ∠ 2 = 180** { Same side interior angle property }

Similarly, consider AB & CD as parallel lines and BC as transversal

**∠3 + ∠ 4 = 180** {Same side interior angle property}

**(03) Diagonals of Trapezium**

Except Isosceles Trapezium, the diagonals do not bisect each other

**(04) In Trapezium, diagonals divide each other proportionally**

ABCD is a trapezium in which line AB is parallel to CD (AB II CD)

AC & BD are the two diagonals intersecting at O

According to trapezium property, diagonals divide each other proportionally.

** \mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}} **

**Proof**

Taking triangle AOB & DOC

Considering AB & CD as parallel lines and BD as transversal

∠ 1 = ∠ 3 {Alternate Interior angle}

Considering AB II CD and AC as transversal

∠ 4 = ∠ 2 {Alternate Interior angle}

∠ 5 = ∠ 6 { Vertically opposite angles}

Hence by AAA similarity, ▵AOB & ▵DOC are similar.

In similar triangles, the ratio of corresponding sides are same

\mathsf{\frac{DO}{OC} \ =\ \frac{OB}{OA}}

On changing sides we get:

\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}

**Hence Proved**

**(05) Trapezium Diagonal Property**

**Sum of square of diagonals = Sum of square of non parallel side + twice of the product of parallel sides**

Given is trapezium ABCD with parallel sides AB & CD.

AC & BD are the diagonals

According to Trapezium Diagonal property

** \mathsf{AC^{2} +BD^{2} =\ AD^{2} +BC^{2} \ +\ 2\ AB\ .\ CD} **

**(06) Midsegment Theorem of Trapezium**

The length of line joining the midpoint of non parallel side is equal to half of the sum of parallel sides

ABCD is a trapezium with AB & CD as parallel sides

E & F are the midpoints of non parallel sides AD & BC respectively

According to Midsegment Theorem**EF = 1/2 (AB + CD)**

**(07) Area formula for Trapezium**

Area of Trapezium is given by formula:

\mathsf{Area\ =\ \frac{1}{2} \ ( sum\ of\ parallel\ sides) \ \times \ height}\\\ \\ \mathsf{Area\ =\ \frac{1}{2} \ ( AB+CD) \ \times \ h}

**(08) Perimeter of Trapezium**

Perimeter of any figure is the total length of the boundary.

In case of Trapezium, the perimeter is the sum of all sides

**Perimeter = (a + b + c + d) cm **

**Questions on Trapezium**

**(01) ABCD is a trapezium in which AB || CD, AB = 16 cm and CD = 12 cm and distance between parallel sides is 10 cm. Find the area of trapezium **

Given

AB = 16 cm

CD = 12 cm

Height (h) = 10 cm

We know that;

Area of Trapezium = (1/2) x (Sum of parallel sides) x height

Area = (1/2) ( AB + CD ) x h

Area = (1/2) (16 + 12) x 10 = 140 sq. cm

**Hence 140 sq cm is the area of trapezium**

**(02) ABCD is a trapezium in which AB II CD. Diagonals intersect at point O.Its given that AO = 3, OC = x – 3, OD = 3x – 19, OB = x – 5Find the values of x**

Using diagonal property of Trapezium

\mathsf{\frac{DO}{OB} \ =\ \frac{CO}{OA}}

\mathsf{\frac{3x-19}{x-5} \ =\ \frac{x-3}{3}}\\\ \\ \mathsf{3( 3x-19) \ =( x-5) \ ( x-3)}\\\ \\ \mathsf{9x\ -57\ =\ x^{2} -8x+15}\\\ \\ \mathsf{x^{2} -17x+72\ =\ 0}\

On solving the quadratic equation we get;

x = 9 and x = 8

Hence both 8 & 9 are the solution for x

**Frequently Asked Questions – Trapezium**

**(01) How is Trapezium different from Parallelogram**

In Parallelogram, opposite sides are equal and parallel, while in Trapezium, only one pair of sides are parallel.

**(02) Are diagonals equal in Trapezium?**

NO!!

Neither they are equal, neither they bisect each other

**(03) In Trapezium, are opposite pair of angles equal?**

There is no such property of trapezium.

Angle related property related to trapezium are:

(a) Alternate angles are supplementary

(b) Sum of all angles add to 360 degree

**(04) Any other name of Trapezium?**

In some countries, they are also called Trapezoid.