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?

Rationale

Equilateral triangles are similar to each other.

All equilateral triangles, by definition, have three equal sides and three equal angles, specifically 60 degrees each. This inherent property ensures that any equilateral triangle is similar to any other equilateral triangle, as their corresponding angles are congruent and the ratios of their sides are consistent.

A (I only) YOUR ANSWER

Right triangles do not necessarily share similarity with each other, as they can have different angles and side lengths. For example, a right triangle with angles of 30-60-90 degrees is not similar to a right triangle with angles of 45-45-90 degrees. Thus, not all right triangles are similar, making this choice incorrect.

B (II only) YOUR ANSWER

Isosceles triangles have at least two equal sides, but they can vary in the length of the base and the angles opposite the non-equal side. This variability means that not all isosceles triangles are similar to one another; for instance, one could have angles of 70-70-40 degrees while another has angles of 80-80-20 degrees. Therefore, this choice is not correct.

C (III only) CORRECT

Equilateral triangles are defined by having all three sides equal and all angles equal. This uniformity guarantees that any two equilateral triangles are similar because their corresponding sides and angles will always maintain the same ratio and degree, making this choice correct.

D (I and III only) YOUR ANSWER

While equilateral triangles are indeed similar to each other, not all right triangles share this property. Since right triangles can vary significantly in their dimensions and angles, this choice is incorrect as it inaccurately includes right triangles in the similarity requirement.

Conclusion

Similarity in triangles requires that corresponding angles be equal and sides be in proportion. Among the given groups, only equilateral triangles consistently exhibit these properties, ensuring that all equilateral triangles are similar to each other. Right triangles and isosceles triangles do not guarantee similarity due to their variability in dimensions and angles. Thus, equilateral triangles stand out as the only group where similarity is inherent.