# Proof that Equilateral Triangles are Equiangular

In the previous post, we have proved the Isosceles Triangle Theorem. The theorem states that the angles opposite to the two congruent sides of an isosceles triangle are congruent.  In this post, we use the said theorem to prove that equilateral triangles are equiangular.

Theorem

Equilateral triangles are equilangular.

Given

Equilateral triangle $PQR$

What To Show

$\angle P \cong \angle Q \cong \angle R$.

Proof

$\overline{PQ} \cong \overline{PR}$ since all sides of an equilateral triangle are congruent.

$\angle Q \cong \angle R$ the angles opposite to the two congruent sides of a triangle are congruent (Isosceles Triangle Theorem)

$\overline{PQ} \cong \overline{QR}$  since all sides of an equilateral triangle are congruent.

$\angle R \cong \angle P$, again, by the Isosceles Triangle Theorem

Now, since $\angle Q \cong \angle R$ and $\angle R \cong \angle P$, by the Transitivity Property of Equality, $\angle Q \cong \angle P$.

Therefore, $\angle P \cong \angle Q \cong \angle R$.

So, equilateral triangles are equiangular.