Difficulty: Hard

Average Score: 36%

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

Triangles from group III must be similar to each other.

Group III consists of triangles that all have the same angle measures, which is a fundamental criterion for triangle similarity. Since similar triangles maintain proportional sides and the same shape regardless of size, the triangles in this group share the necessary properties to be classified as similar.

A (I only) YOUR ANSWER

Triangles in group I do not necessarily exhibit the same angles or proportional sides. Without knowing the specific angle measures or side ratios, we cannot conclude that the triangles in this group are similar. Thus, this option is incorrect as it lacks sufficient similarity criteria.

B (II only) YOUR ANSWER

Group II may contain triangles that vary in their angle measures and side lengths. Similarity requires that all corresponding angles are equal and sides are proportional, which is not guaranteed in this group. Therefore, selecting this option would misrepresent the conditions for triangle similarity.

C (III only) CORRECT

The triangles in group III share identical angle measures, meeting the criteria for similarity. By the Angle-Angle (AA) similarity postulate, if two triangles have equal corresponding angles, they are similar. This makes group III the only correct choice for pairs of triangles that must be similar.

D (I and III only) YOUR ANSWER

Although group III contains similar triangles, group I does not provide sufficient information to suggest similarity. Since the triangles in group I may not share the same angles or proportional sides, this option cannot be correct. This misinterpretation of triangle properties invalidates the selection.

Conclusion

Triangles are classified as similar when they have equal angles and their sides are in proportion. Group III fulfills these requirements by consistently sharing angle measures, establishing similarity among its triangles. In contrast, groups I and II do not meet the necessary criteria, emphasizing the importance of angle congruence in determining triangle similarity.