Final answer:
Parallel lines are characterized by having the same slope and being coplanar, which means they are consistently equidistant and never intersect due to having the same tilt within the same plane.
Step-by-step explanation:
To complete the definition of parallel lines, you must know that they have the same slope and are coplanar. This means that two lines are parallel if, for every given point on one line, there is a corresponding point on the other line that is exactly the same distance away. Importantly, this distance is consistent along the entire length of both lines. Therefore, the correct answer to the student's question is both (a) They have the same slope, and (d) They are coplanar.
As for the meaning of slope, it denotes the steepness of the line, or how much the line tilts. A consistent slope between two lines ensures they will never intersect, precisely because they are tilted at the same angle. Being coplanar means that the lines are in the same plane, which is also necessary for lines to be considered parallel in Euclidean geometry. To find a slope, you would typically select two points on a line graph and use the formula (change in y)/(change in x) between them.