# Isosceles Triangle Theorems

You may have already learnt about the properties and types of triangles. One of the special types of a triangle is the isosceles triangle. An isosceles triangle is a triangle which has two equal sides, no matter in what direction the apex (or peak) of the triangle points. Some pointers about isosceles triangles are:

• It has two equal sides.
• It has two equal angles, that is, the base angles.
• When the third angle is 90 degree, it is called a right isosceles triangle.

In this article, we have given two theorems regarding the properties of isosceles trianglesÂ along with their proofs.

## Isosceles Triangle Theorems and Proofs

Theorem 1: Angles opposite to the equal sides of an isosceles triangle are also equal.

Proof: Consider an isosceles triangle ABC where AC = BC.

We need to prove that the angles opposite to the sides AC and BC are equal, that is, âˆ CAB = âˆ CBA.

We first draw a bisector of âˆ ACB and name it as CD.

Now in âˆ†ACD and âˆ†BCD we have,

AC = BCÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (Given)

âˆ ACD = âˆ BCDÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (By construction)

CD = CDÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (Common to both)

Thus,Â  âˆ†ACD â‰…âˆ†BCD Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (By SAS congruence criterion)

So, âˆ CAB = âˆ CBAÂ  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (By CPCT)

Hence proved.

Theorem 2: Sides opposite to the equal angles of a triangle are equal.

Proof: In a triangle ABC, base angles are equal and we need to prove that AC = BC orÂ âˆ†ABC is an isosceles triangle.

Construct a bisector CD which meets the side AB at right angles.

Now in âˆ†ACD and âˆ†BCD we have,

âˆ ACD = âˆ BCDÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (By construction)

CD = CD Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (Common side)

âˆ ADC = âˆ BDC = 90Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (By construction)

Thus, âˆ†ACD â‰… âˆ†BCD Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (By ASA congruence criterion)

So, AC = BC Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (By CPCT)

Or âˆ†ABC is isosceles.