Final answer:
The provided details are insufficient to determine congruence of triangles as none of the recognized congruency criteria (SSS, SAS, ASA, AAS, HL) are clearly met. The answer is b) No, the triangles' congruency cannot be proven with the given information.
Step-by-step explanation:
The information provided does not sufficiently prove congruency between any given sets of triangles. There is mention of triangles BAO and B₁A₁O being similar, alongside other fragmented notes about various algebraic expressions and relationships which appear unrelated to triangle congruency. To determine if triangles are congruent, specific criteria must be met, such as Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), or Hypotenuse-Leg (HL) for right triangles. None of these criteria can be conclusively identified in the information provided.
To prove congruency, we need clear statements about the equality of sides and angles within specific triangles. Without these, we cannot conclude that any pair of triangles are congruent. Thus, based on the provided information, the answer is b) No, the given details do not prove that the triangles are congruent.