# Isosceles Triangle

## Definition

A triangle whose lengths of any two of three sides are equal, is called an isosceles triangle.

A triangle can have two equal length sides. The meaning of isosceles is two equal sides. According to this, the triangle which contains two equal length sides is known as an isosceles triangle. Not only lengths of two sides are equal and two interior angles are equal as well but they are the opposite angles of the equal length sides.

An isosceles triangle is symbolically represented as a triangle with two equal length sides. Double minor lines are displayed perpendicularly at middle point of two equal length sides and a single minor line is displayed perpendicular at middle point of the remaining side.

### Properties

The $\Delta HIJ$ is a fundamental example to understand the properties of the isosceles triangle.

#### Sides

$\overline{HI}$, $\overline{IJ}$ and $\overline{JH}$ are three sides of the triangle. The lengths of $\overline{IJ}$ and $\overline{JH}$ are equal but the length of $\overline{HI}$ is different.

$HI \neq IJ = JH$

The triangle $HIJ$ is known as an isosceles triangle due to two equal length sides.

#### Angles

$\angle HIJ$, $\angle IJH$ and $\angle JHI$ are three interior angles of $\Delta HIJ$. The sides $\overline{JH}$ and $\overline{IJ}$ are equal length sides and due to this, their opposite angles ($\angle JHI$ and $\angle JIH$) are equal angles but third interior angle ($\angle IJH$) is different.

$\angle HIJ = \angle JHI \neq \angle IJH$

In isosceles triangle, any two of three interior angles are equal and they are the opposite angles of equal length sides.

##### Construction

An isosceles triangle can be constructed geometrically using geometrical tools to study the properties.

1. Draw a horizontal line of any length using a ruler. In this example, $10$ centimetres line is drawn horizontally and its endpoints are called as point $K$ and point $L$.
2. Take compass and consider a ruler to set the distance between pencil lead’s point and need point to any length (In this example, $7$ centimeters is set).
3. Draw an arc above the horizontal line from $K$ and also draw another arc above the line from point $L$. The two arcs intersect each other at a point, called point $M$.
4. Take ruler and join points $K$ and $M$, and then $L$ and $M$ to get a triangle.

A $\Delta KLM$ is constructed geometrically as the result of this geometrical procedure. The points $K$, $L$ and $M$ are called vertices. $\overline{KL}$, $\overline{LM}$ and $\overline{MK}$ are sides of the triangle.

1
###### Lengths of sides

The lengths of all three sides are known.

1. The length of side $\overline{KL}$ is $KL = 10$ centimetres
2. The length of side $\overline{LM}$ is $LM = 7$ centimetres.
3. The length of side $\overline{MK}$ is $MK = 7$ centimetres.

The lengths of the sides $\overline{LM}$ and $\overline{MK}$ are equal but the length of $\overline{KL}$ is not equal to both of them. Hence, the triangle $\Delta KLM$ become an example to an isosceles triangle.

2
###### Angles

The interior angles of triangle $KLM$ are unknown. So, measure them by using a protractor.

1. The angle ($\angle MKL$ (or) $\angle LKM$ (or) $\angle K$) is $44.5^\circ$
2. The angle ($\angle KLM$ (or) $\angle MLK$ (or) $\angle L$) is $44.5^\circ$
3. The angle ($\angle LMK$ (or) $\angle KML$ (or) $\angle M$) is $91^\circ$

$\angle MKL$ and $\angle KLM$ are equal angles because they are opposite angles of equal length sides $\overline{KM}$ and $\overline{LM}$.