Final answer:
To determine the relationship between lines, compare slopes. Lines with the same slope are parallel. Lines with slopes that are negative reciprocals are perpendicular. If neither, lines are oblique.
Step-by-step explanation:
To determine if lines are parallel, perpendicular, or oblique, we need to compare their slopes (the 'm' term in the equation of a line y = mx + b). If two lines have the same slope, they are parallel. If their slopes are opposite reciprocals of each other (meaning one is the negative inverse of the other), the lines are perpendicular. If the slopes are neither the same nor opposite reciprocals, the lines are oblique. Coinciding lines are essentially the same line, thus having the same slope and y-intercept.
In the given scenarios:
- For a, all three lines are parallel to the x-axis, implying they have a slope of zero. Therefore, they are parallel.
- For b, if all three lines are mutually perpendicular to each other, it means that each line has a slope that is the negative reciprocal of the other.
- For c and d, lines are increasing or decreasing, and the steepness varies, indicating that the slopes are different and not negative reciprocals of each other, therefore the lines are oblique.
The example in FIGURE A1 shows a line with a slope of 3, meaning for each increase of 1 in x, y increases by 3. This fact is crucial in determining the relationship between lines.