# Isosceles Triangle Angle property

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

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