Question

Why are equilateral triangles equiangular?

Answer 1

Equal sides implies equal angles

Explanation:

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!

Answer 2

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`.