Why are equilateral triangles equiangular?

Take an example ...

As another example, the sides opposite the base angles of an isosceles triangle have sides that are equal because the base angles are equal .

Finally, if all the sides of the triangle are equal, then the angles opposite those sides must also be equal. This is a equilateral or equiangular triangle!

We can prove this using the law of cosines with the SSS case.

So...

becomes

Notice how on every triangle you draw, a side is opposite to an angle. That means only one side corresponds to one particular angle.

Equilateral triangles are equiangular because all three interior angles of an equilateral triangle are congruent, meaning they have the same measure. This occurs because each angle of an equilateral triangle subtends an arc of the same length on the circle, and angles inscribed in the same arc are congruent. Since all sides of an equilateral triangle are congruent, the arcs subtended by each side are also congruent, resulting in angles of equal measure at each vertex. Hence, equilateral triangles are equiangular.