Equilateral Triangle is Equiangular

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\triangle ABC$ be an equilateral triangle.

Then $\triangle ABC$ is also equiangular, with each internal angle equal to $60 \degrees$.


Proof

Let $\triangle ABC$ be an equilateral triangle.

By definition of equilateral triangle, any two of the sides of $\triangle ABC$ are equal.

Without loss of generality, let $AB = AC$.

Then by Isosceles Triangle has Two Equal Angles:

$\angle ABC = \angle ACB$

As the choice of equal sides was arbitrary, it follows that every two of the internal angles of $\triangle ABC$ are equal.

Hence all $3$ internal angles of $\triangle ABC$ are equal.

$\Box$


From Sum of Angles of Triangle equals Two Right Angles, the internal angles of $\triangle ABC$ all add up to $180 \degrees$.

Hence it follows that one such internal angle is $\dfrac {180 \degrees} 3$, or $60 \degrees$.

$\blacksquare$


Also see


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Equilateral_Triangle_is_Equiangular&oldid=724870"