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Problem : What are the measures of the angles in an equiangular triangle?

60 degrees

Problem : If two angles in a triangle have a measure of 60 degrees, what is true about the sides of that triangle?

The sides are all equal, because the triangle is equiangular, and therefore equilateral.

Problem : In triangle ABC, sides AB and BC are equal, and angle B is 60 degrees. What is true about triangle ABC?

It is an equilateral triangle, and therefore equiangular as well.

Problem : Can a trapezoid be divided into two equilateral triangles? A parallelogram? A rhombus? A square? A rectangle?

No. Yes. Yes. No. No. Only a rhombus can be divided into two equilateral triangles, and a rhombus is a parallelogram. A rhombus with two 60 degree angles and two 120 degree angles can be split by one of its diagonals (the one that bisects the 120 degree angles) into two congruent, equilateral triangles.

Problem : Is it ever possible for a triangle to be either equilateral or equiangular and not the other?

No, the statement goes both ways; if a triangle is equilateral, it is equiangular, and if a triangle is equiangular, it is equilateral.