Following are the properties and theorems relating to isosceles and equilateral triangles.
Isosceles triangle
A triangle containing at least two equal-length sides and two equal interior angle measures.
Equilateral triangle
A triangle with three equal-length sides and three 60\degree interior angles. Equilateral triangles are a sub-class of isosceles triangles. Also known as an equiangular triangle.
Legs of an isosceles triangle
The sides of an isosceles triangle that are equal in length.
Base angles of an isosceles triangle
The angles that are opposite the legs of an isosceles triangle.
The base angles theorem and its converse describe the relationship between the base angles and congruent sides of an isosceles triangle.
Base angles theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
In this triangle, since \overline{AB} and \overline{AC} are legs of the triangle, the theorem tells us that the base angles \angle ABC and \angle ACB are congruent.
Converse of base angles theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
In this triangle, since \angle ABC and \angle ACB are congruent, the theorem tells us that AB=AC.
We can apply the base angles theorem to an equilateral triangle to prove that it is equiangular. Similarly, we can apply the converse of base angles theorem to an equiangular triangle to prove that it is equilateral.
Corollary to the base angles theorem
If a triangle is equilateral, then it is equiangular.
Corollary to the converse of base angles theorem
If a triangle is equiangular, then it is equilateral.
