# Isosceles Trapezium : Definition & Properties

## What is Isosceles Trapezium

Its a trapezium in which one pair of sides are parallel and other non parallel sides are equal.

Keyword for Isosceles Trapezium
(a) One pair of sides are parallel
(b) Non parallel sides are equal

## Example of Isosceles Trapezium

Given above is the example of Isosceles Trapezium.

⟹ AB & CD are parallel sides called Base

⟹ Non Parallel sides AD & BC are equal (AD = BC)

Non Parallel sides are also called legs

## Properties of Isosceles Trapezium

(01) In Isosceles Trapezium, the non parallel sides are equal

This property separates the Isosceles Trapezium from general Trapezium because in general trapezium all sides are unequal

In the above trapezium ABCD, the non parallel sides AD & BC are equal

(02) In Isosceles Trapezium, both adjacent and opposite angles are supplementary

Adjacent angles are supplementary

Similar to normal trapezium, the adjacent angles are supplementary in Isosceles Trapezium.

∠ 1 + ∠ 2 = 180 degree

∠ 3 + ∠ 4 = 180 degree

Opposite angles in Isosceles Trapezium are also supplementary

This property is special for Isosceles Trapezium

∠ 1 + ∠ 4 = 180 degree

∠ 3 + ∠ 2 = 180 degree

(03) In Isosceles Trapezium, the diagonals are equal to each other

ABCD is an Isosceles Trapezium in which AC & BD are diagonals.

According to diagonal property of Isosceles Trapezium
AC = BD

(04) The diagonals divides into equal proportion

In trapezium, diagonals intersect such that:

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

Also in case of Isosceles Trapezium
DO = OC and;
OB = OC

(05) Base angles are equal

Given above is the isosceles trapezium ABCD in which:
∠ ADC = ∠ BCD

Prove Base angles in Isosceles Trapezium are equal

Given
ABCD is an isosceles trapezium in which AB II CD and side AD = BC

To prove
∠ ADC = ∠ BCD

Solution
Draw diagonals AC & BD in the trapezium

Now take triangle ▵ DAC & ▵ CBD

(Separated the triangles for your understanding)

Side DA = CB (Legs of Isosceles Trapezium are equal)
AC = BD (Diagonals in Isosceles Trapezium are equal)
DC = DC (Common Side)

Hence by SSS congruency, ▵ DAC & ▵ CBD are congruent

so, ∠ ADC = ∠ BCD

Hence Proved

(06) Area of Isosceles Trapezium

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

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

(07) Converting Isosceles Trapezium into Rectangle

Given above is an Isosceles Trapezium ABCD.

Here we have plotted lines AM & BN as height of trapezium perpendicular to line DC

Comparing triangle ADM & BNC

AD = BC {equal sides
AM = BN {similar height}
∠ ADM = ∠ BCN

On comparison, we can say that:
DM = CN

Now cut the triangle AMD from dotted line and paste above triangle BCN as shown in below figure

Observe how we have transformed a trapezium into a rectangle.

This method is very helpful to solve problems related to area calculation

Example Question
If ABCD is an Isosceles Trapezium in which diagonal length AC = 10 cm and height 5 cm. Find the area of Trapezium

Now as we have seen in above concept, remove the triangle AOD & attach towards side BC

After all the deliberations, we got rectangle ADCO with breadth 5 cm and diagonal 10 cm

Using Pythagoras Theorem to find length of OC

\mathsf{AC^{2} =\ OA^{2} +\ OC^{2}}\\\ \\ \mathsf{10^{2} =\ 5^{2} +\ OC^{2}}\\\ \\ \mathsf{OC^{2} \ =\ 100\ -25}\\\ \\ \mathsf{OC^{2} \ =\ 75}\\\ \\ \mathsf{OC\ =\ 8.6\ cm} \\ \\

We know,
Area of Rectangle = OA x OC = 5 x 8.6 = 43 sq cm

Hence, 43 sq cm is the original area of trapezium

(08) Isosceles Trapezium’s Circumcircle

Isosceles Trapezium can be fully inscribed in circumcircle.

Circumcircle is a circle which inscribe the geometrical figure by passing through all the vertex.

Location of Circumcenter
Circumcenter location can be found by the intersection of perpendicular bisector of adjacent sides

Given above is Isosceles Trapezium ABCD

M & N are perpendicular bisectors of side AD & CD.

The bisectors meet at point O, which is circumcenter of the trapezium.

Take OC as radius and draw a circle.
Note that circle passes through all the vertex

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