Final answer:
The architect needs to know specific measurements of the existing building's triangular faces based on congruence theorems like SSS, SAS, and ASA. Accurate dimensions of sides and angles are crucial to ensure congruency. Identifying proportions and ratios is also important as they reflect the architectural principle that form follows function.
Step-by-step explanation:
To ensure that all triangular faces of the new building are congruent to the existing building, the architect must gather specific information about the existing building's triangular faces based on triangle congruence theorems and postulates. Since the triangles are not right triangles, the Pythagorean Theorem is not applicable. Instead, the architect must rely on other theorems such as Side-Side-Side (SSS), Side-Angle-Side (SAS), or Angle-Side-Angle (ASA).
- For SSS, the architect needs to know the lengths of all three sides of the triangular face.
- For SAS, it is necessary to know two sides and the included angle between them.
- For ASA and the less common Angle-Angle-Side (AAS), the architect requires the measure of two angles and the length of the intervening side or another side, respectively.
It is critical to have accurate measurements for these elements to ensure that the new building's faces will be congruent, reflecting the mathematical and architectural principle that form follows function. Additionally, the architect may consider the proportions and ratios used in the existing building, as they might also apply to the new structure's design.