Question

Consider two unique parallel lines. What aspects of these two lines are the same? What aspects of these two lines would have to be different

Answer 1

I assume you mean two distinct (noncoincident) parallel lines in the plane.

Same:
- Direction/orientation: they point the same way (their direction vectors are scalar multiples).
- Slope (in Cartesian coordinates): the slopes are equal (or both undefined if vertical).
- Angle with any given transversal or with the axes: the same.
- Normal direction: the same line-normal direction (up to sign).
- Shape and dimension: both are straight 1-D lines, infinite, zero curvature.
- Form of equation: their linear equation coefficients for x and y are proportional (e.g. ax + by + c1 = 0 and ax + by + c2 = 0).
- Perpendicular separation is constant along their length.

Different:
- Position: they consist of different sets of points (they are not the same line).
- Intercept constants: different y-intercepts or different constant term c in ax + by + c = 0.
- Perpendicular distance between them is nonzero.
- They do not intersect (no common points).

Example: y = 2x + 1 and y = 2x − 3 — same slope 2, different intercepts and separated by a fixed nonzero distance.