Final answer:
Triangles are classified by sides as equilateral, isosceles, and scalene, and by angles as acute, right, and obtuse. Equilateral triangles are always acute, whereas scalene can be right, but never equilateral and right at the same time. Thus, the correct classifications are equilateral and acute, and scalene and right.
Step-by-step explanation:
Classifying triangles by their sides and angles is fundamental in geometry, which allows us to understand their properties and apply theorems such as the Pythagorean theorem. A triangle can be classified by its sides into three categories: equilateral, isosceles, and scalene. An equilateral triangle has all three sides of the same length and therefore, all angles are also equal, which means they are each 60 degrees, making it an acute triangle. An isosceles triangle has two sides of the same length and can be acute, right, or obtuse, depending on the measure of the angles. A scalene triangle has all sides of different lengths, and the angles can also be acute, right, or obtuse.
Angles are classified as acute if all angles are less than 90 degrees, right if one angle is exactly 90 degrees, and obtuse if one angle is more than 90 degrees. Therefore, it is impossible for a triangle to be both equilateral and right, as a right triangle must have one 90 degree angle by definition, which does not fit the angle measures of an equilateral triangle.
Based on the classifications provided, the correct answer should include the possibility of an equilateral and acute triangle, and a scalene and right triangle, which corresponds to option 'd' from the given choices. Isosceles and obtuse is also a valid combination, but since it is not paired with either equilateral and acute or scalene and right, it is not a part of the correct answer.