# Why are equilateral triangles equiangular?

Nov 24, 2015

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!

Nov 24, 2015

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

$a = b = c$

So...

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \angle C$

becomes

${a}^{2} = {a}^{2} + {a}^{2} - 2 a \cdot a \cdot \cos \angle A$

$- {a}^{2} = - 2 {a}^{2} \cos \angle A$

$1 = 2 \cos \angle A$

$\frac{1}{2} = \cos \angle A$

$\textcolor{b l u e}{\angle A = {60}^{\circ}}$

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 $\angle C$, and since sides $a = b = c$, we have $\angle A = \angle B = \angle C$.