Exercise 37 Page 614

According to the Corollary to the Base Angles Theorem, if a triangle is equilateral, then it is equiangular. To show this using the Base Angles Theorem, we will begin by drawing an equilateral triangle.

The Base Angles Theorem states if two sides of a triangle are congruent, then the angles opposite them are congruent. Using this theorem we can show that $∠ A ≅ ∠ C,$ $∠ B ≅ ∠ C,$ and finally $∠ A ≅ ∠ B .$

Thus, by using the Base Angles Theorem, we were able to show that all angles are congruent. Therefore, we can now claim that $△ A B C$ is equiangular.

Alternative Solution

Two Column Proof

Let's show this as a two-column proof.

Statement	Reason
$△ A B C$ is equilateral	Given
$\overline{A B} ≅ \overline{A C} \overline{A B} ≅ \overline{B C} \overline{A C} ≅ \overline{B C}$	Definition of Equilateral Triangle
$∠ B ≅ ∠ C ∠ A ≅ ∠ C ∠ A ≅ ∠ B$	Base Angles Theorem
$△ A B C$ is equiangular	Definition of Equiangular Triangle.