Plane Geometry, Isosceles and Equilateral Triangles

Scalene, Isosceles, Equilateral

In a scalene triangle, all three sides have different lengths.
In an isosceles triangle, at least two sides are the same length.
In an equilateral triangle, all three sides are the same length.

picture of triangles

Base Angles

Place the short side of an isosceles triangle at the base and reflect the picture through a mirror.  The two triangles, the original and its reflection, are congruent by SSS, hence the base angles of an isosceles triangle are equal.

Conversely, assume the base angles of a triangle are equal.  The triangle is congruent to its reflection by ASA, hence the two sides have the same length.  As a corollary, none of the angles in a scalene triangle are congruent.

An equalateral triangle is congruent to any of its rotations by SSS, hence all of the angles are congruent.  Since they sum to 180°, each angle is 60°.  Conversely, if all three angles are congruent, the triangle is isosceles in any orientation, hence all sides have the same length, and the triangle is equilateral.