Final answer:
The question contains a possible typo or error and lacks clear context or specific details needed to determine which pairs of triangles are congruent. Congruent triangles are identical in shape and size with equal corresponding angles and sides.
Step-by-step explanation:
The question appears to revolve around the concept of congruent triangles and their properties in geometry. When two triangles are congruent, it means that they are identical in shape and size, having corresponding sides and angles that are equal. The question presents a scenario with triangles BAO and B₁ A₁ O, which are stated to be similar triangles. It's important to remember that similar triangles have the same shape but not necessarily the same size, which means their corresponding angles are equal, and the lengths of their corresponding sides are proportional.
However, the question seems to include an error or typo, making it unclear which pairs of triangles the student needs to consider for congruence. Without the specific details of triangles A, B, and C or a clear description of the triangles being referenced, it is not possible to accurately determine the congruence of any specific pairs of triangles.
The provided information references geometry concepts such as angles, parallel lines, and vectors but does not provide a clear context for assessing congruence. One general piece of information given is that the sum of angles in any triangle is always 180 degrees. Also mentioned is a situation with a shared segment on a surface AB', but without additional context or a diagram, it's not possible to use this to determine congruence between specified triangles.
In conclusion, to address the student's question fully, it would be essential to have a clear and accurate representation of the triangles in question. Without this, or additional context provided, a definitive answer cannot be given on which pairs show congruent triangles.