Why are equilateral triangles equiangular?

Question

Claire Akin · Accepted Answer

Equal sides implies equal angles

Take an example ... A 30-60-90 triangle will have its smallest side opposite the #30^o#. The largest side will be opposite the #90^o#. Finally, the side opposite the #60^o# will fall somewhere in between. 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!

Sadie Allen · Answer

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

#a = b = c#

So...

#c^2 = a^2 + b^2 - 2abcos/_C#

becomes

#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 #c# corresponds to only one #/_C#, and since sides #a = b = c#, we have #/_A = /_B = /_C#.

Savannah Aguilar · Answer

undefined 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.