In this chapter we will prove that angle opposite to equal side of isosceles triangle are equal.

### What is an Isosceles Triangle ?

The triangle in which **two sides are equal** are known as isosceles triangle.

In the above triangle ABC, side AB = AC.

Since two sides are equal, it is an isosceles triangle.

## Proving angle opposite to equal side in isosceles triangle are equal

Consider the below isosceles triangle ABC in which AB = AC.

We have to prove that angle opposite to equal side AB & AC are also equal.

i.e. ∠ABC = ∠ACB

**Given;**

ABC is an isosceles triangle such that AB = AC**Construction;**

Draw angle bisector line AM that bisect ∠A into two equal halves.

∠BAM = ∠CAM

**To prove;**

∠ABC = ∠ACB **Solution**

Take triangle **ABM and ACM**

AB = AC { given }

∠BAM = ∠CAM { AM is the angle bisector }

AM = MA { common sides }

By **SAS congruency condition**, the triangles ABM and ACM are congruent.

i.e. \mathtt{\triangle ABM\ \cong \triangle ACM}

Since both the triangles are congruent, we can say that, **∠ABC = ∠ACB **

Hence, we proved that in isosceles triangle, angle opposite to equal sides are equal.

## Isosceles Exterior angle property

In Isosceles triangle, the **exterior angle produced after extending the equal angles are also equal**.

**Given**:

ABC is an isosceles triangle

In the above triangle;

AB = AC

∠1 = ∠2

Here ∠3 and ∠4 are the exterior angles produced by extending equal angles ∠1 & ∠2.**To prove:**

∠3 = ∠4 **Prove**Since line AM is a straight line, we can say that;

∠1 + ∠3 = 180 degree

Similarly AN is a straight line, we can say that;

∠2 + ∠4 = 180 degree

Subtract the two equations;

∠1 + ∠3 – ∠2 – ∠4 = 180 – 180

Since ∠1 = ∠2, the expression will cancel out.

∠3 – ∠4 = 0

∠3 = ∠4

Hence in Isosceles triangle, the exterior angles produced by extending equal angles are equal.

**Next chapter :** **Isosceles triangle side property**