Final answer:
To prove that lines B and C are parallel, it is necessary that they have the same slope. Intersecting at a right angle or being perpendicular imply they are not parallel, and different y-intercepts alone do not establish parallelism.
Step-by-step explanation:
To prove that lines B and C are parallel, it must be true that the lines have the same slope. This is because parallel lines never intersect and maintain a constant distance from each other. Option b) stating that the lines intersect at a right angle, and option d) stating that the lines are perpendicular, both imply that the lines would indeed intersect at some point, which would mean that they are not parallel. On the other hand, option c) mentions that the lines have different y-intercepts, but this alone does not guarantee parallelism, as non-parallel lines can also have different y-intercepts.
If two lines are parallel and within the same plane, they will have identical slopes, represented mathematically if the slope of line B is 'm' then the slope of line C will also be 'm'. This is the foundational property of parallel lines in Euclidean geometry.