Isosceles Triangles

A triangle is said to be isosceles if 

1. Two sides in it have equal length, or 
2. Two angles in it are equal. 

The next two theorems show that the conditions (1) and (2) are equivalent. That is, each implies the other. 

Theorem 1. If  two sides in a triangle have equal lengths then the angles opposite them are equal.

Proof of theorem 1: 

 

  
Click on thumbnail to see proof.

Theorem 2. If two angles in a triangle are equal then the sides opposite them have equal length.

Proof of theorem 2: 

   

Click on thumbnail to see proof.

Equilateral and Equiangular Triangles

A triangle is said to be equilateral when and only when all three sides in it have the same length.  A triangle is said to be equiangular when and only when all three angles in it are equal.

Theorems 1 and 2 above imply the following.

Theorem 3. A triangle is equilateral when and only when it is equiangular.

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