8. A pair of triangles from which of these groups must be similar to each other?
Answer: C
Equilateral triangles must be similar to each other.
Equilateral triangles are defined as triangles with all three sides of equal length and all three angles equal to 60 degrees. Since all equilateral triangles share these properties, any two equilateral triangles will be similar to each other.
A) I only
Right triangles do not have to be similar to each other, as they can have different side lengths and angles, provided that one angle is 90 degrees. Therefore, not all right triangles meet the criteria for similarity.
B) II only
Isosceles triangles have at least two equal sides and can vary in the length of their third side. This means that not all isosceles triangles are similar, as their angles can differ while still maintaining two equal sides.
C) III only
Equilateral triangles are always similar to each other, regardless of their size. They maintain the same angles and proportions between corresponding sides, confirming their similarity.
D) I and III only
While equilateral triangles are similar, right triangles can vary in shape and size. Therefore, this option is incorrect as it includes right triangles, which do not guarantee similarity.
Conclusion
Equilateral triangles are the only group among the options that ensures similarity due to their consistent side lengths and angles. The other options include triangle types that can vary significantly, making them not necessarily similar. Thus, option C is the definitive correct answer.