Equilateral Triangle is Equiangular

Theorem

Let $\triangle ABC$ be an equilateral triangle.

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

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 $AB = AC$.

$\angle ABC = \angle ACB$

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 $60 \degrees$.

$\blacksquare$

1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.12$: Corollary $2$
1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.5$: Theorems and Proofs: Problem Set $\text A.5$: $32$
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equilateral polygon
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): polygon
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): triangle
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equilateral polygon
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): polygon
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): triangle
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): equilateral