Why are equilateral triangles equiangular?

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Why are equilateral triangles equiangular?

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Answer 1 · 2015-11-24T02:48:44

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 · 2015-11-24T02:56:44

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 $/_A = /_B = /_C$ .

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Why are equilateral triangles equiangular?

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