Why are equilateral triangles equiangular?
Equal sides implies equal angles
Take an example ...
A 30-60-90 triangle will have its smallest side opposite the
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.
#a = b = c#
#c^2 = a^2 + b^2 - 2abcos/_C#
#a^2 = a^2 + a^2 - 2a*a*cos/_A#
#-a^2 = -2a^2cos/_A#
#1 = 2cos/_A#
#1/2 = cos/_A#
#color(blue)(/_A = 60^@)#
Notice how on every triangle you draw, a side is opposite to an angle. That means only one side corresponds to one particular angle.
Since only one side