Sometimes similar but it depends
sometimes, the altitude of isosceles triangles resting on their base and equilateral triangles are angle bisectors
Yes, two equilateral triangles are always similar to each other. They have the same number of sides, the same angles and the lengths of the sides between the two triangles are always in the same ratio.
Since triangles have three sides and must have interior angles equal to pi or 180o, the three interior angles in all equilateral triangles will be pi/3 or 60o.However, equilateral triangles must also have equal side lengths. Since equilateral triangles with interior angles of 60o will not necessarily have the same side lengths, two equilateral triangles will sometimes be similar to each other.
never ever ever
If they are congruent they must be similar.