**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**AD = BC**

**(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