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