Exercise 41 Page 614

We are given that $△ABC$ is an equilateral triangle and that three angles are congruent. Let's add this information to the diagram. According to the Corollary to the Base Angles Theorem, we can prove that $△DEF$ is equilateral by showing that it is equiangular. For this purpose, we will label the interior angles of $△DEF$ as $\angle 1,\angle 2$ and $\angle 3.$ Our mission, is to show that they are all $60^{\circ }.$

Again, by the Corollary to the Base Angles Theorem, we know that $△ABC$ is equiangular. Using the Angle Addition Postulate, we can claim that the three unmarked angles at each of the vertices on $△ABC$ are congruent. Let's show this.

If we examine the diagram, we see that the interior angles of $△DEF$ are exterior angles to $△ABD,$ $△BCE,$ and $△CAF.$ By the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Therefore we can write the following equations: Notice that the two marked angles at each vertex of $△ABC$ add up to $60^{\circ }.$ Therefore, since the nonadjacent interior angles to $\angle 1,$ $\angle 2$ and $\angle 3$ are made up of these marked angles, we can say that Since $△ABC$ is equiangular it is also equilateral.