Question

contrasting parallel and perpendicular lines
the definition of parallel lines requires the undefined terms line and plane, while the definition of perpendicular lines requires the undefined terms of line and point. what characteristics of these geometric figures create the different requirements?

Explanation:

• For parallel lines: Parallel lines are coplanar (lie in the same plane) and never intersect. The plane is needed because parallel lines must exist within a single plane (in 2D or as coplanar in 3D); without a plane, lines in 3D could be skew (not parallel and not intersecting, but not coplanar). So the plane’s characteristic of containing lines (and defining coplanarity) drives the need for the plane in parallel lines’ definition.
• For perpendicular lines: Perpendicular lines intersect at a right angle at a single point. The point is the intersection point of the two lines. Since perpendicular lines just need to meet at a point (with a 90° angle), the point (as the intersection location) and the line (the figures themselves) are key—no plane is required (they can be in different planes, but in standard definitions, often considered in a plane, but the core requirement is the intersection point). The point’s role as the intersection of two lines (and the right angle condition) means the undefined term point (for the intersection) is needed.

Answer:

• Parallel lines: Require a plane because parallel lines must be coplanar (in the same plane) to never intersect (skew lines in 3D aren’t parallel). The plane defines the “flat” space where they remain equidistant and non - intersecting.
• Perpendicular lines: Require a point (the intersection point) because they intersect at a right angle at a single point. The point is the location of their intersection, and the line is the figure; no plane is required in the core definition (though they often lie in a plane, the key is the intersection point).