Question

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?

Answer 1

Parallel lines do not intersect.

Parallel lines must be coplanar.

Perpendicular lines intersect at one point.

Perpendicular lines intersect so plane is not required in the definition.

Explanation:

Answer 2

Point, line, and plane are the undefined expression that relinquish the starting location for geometry. When we define words, we ordinarily use simpler words, and these simpler words are in turn defined using yet simpler words. This procedure must eventually abort; at some stage, the definition must use a word whose meaning is accepted as intuitively clear. Because that meaning is accepted without definition, we refer to these words as undefined terms. These terms will be used in defining other terms. Although these expressions are not formally defined, a brief intuitive dialogue is needed. A point is the most fundamental object in geometry. It is represented by a dot and named by a capital letter. A point constitute position only. A line (straight line) can be thought of as a connected set of infinitely many points. It extends infinitely far in two opposite directions. A line has boundless length, zero width, and zero height. Any two points on the line name it. The symbol ↔ written on top of two letters is used to denote that line. A plane may be contemplating as an infinite set of points creating a connected flat surface extending infinitely far in all directions. It is usually represented in drawings by a four‐sided figure. A single capital letter is used to designate a plane.