10. The following is a list of triangles: I. Right triangles, II. Isosceles triangles, III. Equilateral triangles. 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 having all three sides equal and all three angles equal to 60 degrees. Because of this consistency in angle measures and side lengths, any two equilateral triangles are always similar to each other.
A) I only
Right triangles can have different angle measures and side lengths, which means they are not necessarily similar. For example, a 3-4-5 right triangle is not similar to a 5-12-13 right triangle, as they do not have the same angles.
B) II only
Isosceles triangles have at least two equal sides, but the angles can vary. Therefore, two isosceles triangles can have different angle measures and are not guaranteed to be similar. For instance, an isosceles triangle with angles 70-70-40 is not similar to one with angles 80-80-20.
C) III only
Equilateral triangles are always similar because they have the same angle measures (60 degrees each) and proportional side lengths. Thus, any pair of equilateral triangles will be similar by definition.
D) I and III only
While equilateral triangles are similar, right triangles do not guarantee similarity due to varying angles and side lengths. Therefore, this option is incorrect as it includes right triangles which do not have to be similar.
Conclusion
Equilateral triangles are the only set that must be similar due to their defining characteristics of equal angles and side lengths. All other options fail to meet the criteria for similarity, as they allow for variations in angle measures and side proportions. Thus, option C is the definitive correct answer.