Geometry

Question Analysis:
This question evaluates understanding of triangle congruency by identifying which statement is false.
The focus is on distinguishing between properties that always hold for congruency and those that do not.

Key Concept Explanation:
Congruent Triangles: Identical in shape and size (all corresponding sides and angles equal).
Equilateral Triangles: All sides and angles are equal, but side lengths can vary (e.g., a 2-2-2 triangle vs. a 5-5-5 triangle).
SAS Theorem: If two sides and the included angle are equal, the triangles must be congruent.
Rigid Transformations: Preserve size and shape (translations, rotations, reflections).

Step-by-Step Solution:
Evaluate Option A:
Statement: "All equilateral triangles are congruent."
Analysis: False. Equilateral triangles can have different side lengths (e.g., 2-2-2 and 3-3-3). They are similar but not necessarily congruent.

Evaluate Option B:
Statement: "Two triangles that satisfy the SAS theorem must be congruent."
Analysis: True. SAS is a valid congruency postulate.

Evaluate Option C:
Statement: "If tw...