Exercise 39 Page 614

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

According to the Converse of the Base Angles Theorem, if two angles of a triangle are congruent, then the sides opposite them are congruent. Using this theorem we can show that $\overline{A B} ≅ \overline{A C},$ $\overline{A C} ≅ \overline{A B},$ and finally $\overline{A C} ≅ \overline{B C} .$

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

Alternative Solution

Two-Column Proof

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

0. Statement	0. Reason
1. $△ A B C$ is equiangular	1. Given
2. $∠ B ≅ ∠ C ∠ A ≅ ∠ C ∠ A ≅ ∠ B$	2. Definition of Equilateral Triangle
3. $\overline{A B} ≅ \overline{A C} \overline{A B} ≅ \overline{B C} \overline{A C} ≅ \overline{B C}$	3. Converse of the Base Angles Theorem
4. $△ A B C$ is equilateral	4. Definition of Equilateral Triangle.