Geometry

Question Analysis:
This question assesses the ability to determine when given information is sufficient to prove triangle congruency.
It requires evaluating each option to identify which pair of triangles lacks enough information for a valid congruency proof.

Key Concept Explanation:
To prove triangles congruent, we must satisfy one of the standard congruency postulates/theorems (SSS, SAS, ASA, AAS, or HL for right triangles).
The absence of sufficient corresponding parts means congruency cannot be established.

Option-by-Option Evaluation:
Option A: Two equilateral triangles with 6 cm sides
Analysis:
All sides in both triangles are equal (6 cm).
Congruency Method: SSS (Side-Side-Side).
Conclusion: Can be proven congruent.

Option B: Two isosceles triangles with 70° vertex angles
Analysis:
Both triangles have a 70° vertex angle and two equal base angles (each 55° since angles sum to 180°).
However, no side lengths are given. The triangles could be different sizes (e.g., legs of 5 cm vs. 10 cm).
Congruency Method: None. AAA (Angle-Angle-Angle) only proves similarity, not congruency.
Conclusion: Cannot be proven congruent with the given information.

Option C: Two right triangles with 5 cm and 12 cm legs
Analysis:
Both triangles have legs of 5 cm and 12 cm.
The included angle (90°) is implicitly equal.
Congruency Method: SAS (Side-Angle-Side).
Conclusion: Can be prove...