# Isosceles Triangle

Triangle with two equal sides are known as Isosceles Triangle

In Latin, the word “Isosceles” mean “Equal Legged”
Hence ,in Isosceles triangle, two of the legs are equal in length

## Base and Legs of Isosceles Triangle

The sides which are equal in length is known as Legs.
And the third side is known as Base

The above triangle ABC is an Isosceles Triangle with two equal sides.
The equal sides AB & AC is called Legs
The other side BC is called Base of the triangle

## Angles of Isosceles Triangle

The angle between the two equal sides are known as Vertex angle.

Angle that rests on the base of Isosceles triangle is known as Base Angle

The above image shown Isosceles Triangle ABC, where side AB = AC
\angle A is Vertex angle since it is between the two equal sides.
\angle B and \angle C are base angle since it rest on base BC.

In Isosceles triangle, base angles are equal to each other (explanation is given below)
\angle B = \angle C

## Examples of Isosceles Triangle

In all the above examples, note the two sides (marked in red) which are equal to each other.
And the third side (marked in Blue) which is different in length

## Properties of Isosceles Triangle

(01) In Isosceles Triangle, the angle opposite to equal sides are equal to each other

In the above Isosceles Triangle, side AB = AC
Angle opposite to side AB = \angle C
Angle opposite to side AC = \angle B

According to the Isosceles property, angle opposite to equal sides are equal.
\angle B=\angle C

(02) Perpendicular drawn from vertex angle to the base bisects the base and vertex Angle

Above image is of Isosceles Triangle ABC, where:
Side AB = AC
\angle A = 40 degree

When a perpendicular line AO is drawn from Vertex angle A to side BC then the line will bisect angle A and side BC.
You can observe that side BC and \angle A is divided into two equal halves

(03) Right Angled Isosceles Triangle

In Right angles isosceles triangle one angle is 90 degree and the other two angle will be 45 degree each.

Proof
Prove that in right angle Isosceles triangle, the other two angles are 45 degree

Solution
Given triangle ABC is right angled isosceles triangle, where
\angle B = 90 degree
And side AB = BC

Since side AB = BC, their opposite angles are also equal
\angle A = \angle C = x degree

We know that sum of angle of triangle is 180 degree
\angle A + \angle B + \angle C = 180
⟹ x + 90 + x = 180
⟹ 2x = 90
⟹ x = 45 degree

Hence, \angle A = \angle C = 45 degree

(04) In Right Angles Isosceles Triangle, the altitude on hypotenuse is half the length of hypotenuse

ABC is right angled isosceles triangle where AB = AC and AC is the hypotenuse
BO is the perpendicular line on hypotenuse AC

According to the property:

BO = \frac{AC}{2}

i.e. the altitude is half the hypotenuse

## Area of Isosceles Triangle

Given below are two formulas for calculation of area of isosceles triangle.
(a) Formula when base and height of triangle is given
(b) Formula when length of all sides are given

Area formula when base and height is given

The above image is of isosceles triangle with the following details:
Length of Base = b
Height from base = h

In this case,
Area of Triangle = \frac{1}{2} \ \times \ b\ \times h

Area Formula when length of sides are given

Given above is an isosceles triangle with side length “a” and base length “b”
In this case, the formula is:

Area of Triangle = \frac{b}{2}\sqrt{a^{2} -\frac{b^{2}}{4}}

## Isosceles Triangle Theorem Proof

Theorem
Angle opposite to equal sides are equal

Given:
ABC is an isosceles triangle with sides AB = AC
AD is an angle bisector such that \angle 1 = \angle 2

To Prove:
\angle B = \angle C

Proof:

Taking Triangle ABD and ACD
Side AB = AC
\angle 1 = \angle 2
AD = AD {Common side of both triangle}

Hence by SAS congruency, triangle ABD and ACD are congruent.
This means, \angle B = \angle C

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