Parallel Postulate, Parallel Lines, and Skew Lines

The Parallel Postulate

• Classical Euclidean statement (Euclid’s 5th postulate)

  • Given any line $l$ and a point $P$ not on $l$, there exists exactly one line in the same plane that passes through $P$ and is parallel to $l$.

  • Symbolic form: $\forall\, l,\; P \notin l,\; \exists!\, m \; (P \in m \land m \parallel l).$

• Key implications

  • “Exactly one” means uniqueness; even the slightest rotation of that candidate line will eventually cause an intersection with $l$.

  • Guarantees the familiar grid-like structure in Euclidean geometry (rectangles, coordinate planes, etc.).

• Visual/Conceptual aid

  • Imagine sliding a ruler (the candidate line) through point $P$. Only one tilt leaves the ruler never touching $l$.

Parallel Lines in the Same Plane

• Definition recap

  • Two lines are parallel iff they lie in the same plane and never intersect.

• Practical consequences

  • Parallel railroad tracks, edges of a rectangle, opposite sides of a sheet of paper—all coplanar examples.

• Non-examples

  • If two lines live in distinct planes, they cannot be classified as “parallel” under the Euclidean definition, even if they never meet.

Skew Lines

• Formal definition

  • Two lines are skew if they

  1. Do not intersect and

  2. Are non-coplanar (no single plane contains them both).

• Distinguishing features

  • Unlike parallel lines, skew lines exist only in three or more dimensions.

• Why perpendicular or parallel lines can’t be skew

  • Perpendicular lines intersect by definition.

  • Parallel lines require coplanarity; skew lines violate that.

Planes, Parallel Planes, and Embedded Lines

• A plane can be thought of as an infinite collection of parallel “grid” lines running in two independent directions.

• When two planes are parallel (never meet), every point of one plane is a fixed distance from the other.

• Example in the transcript

  • Planes $r$ and $w$ are parallel.

  • Pick one line $DE$ on plane $r$ and one line $FG$ on plane $w$. Because

  • $DE \cap FG = \varnothing$ (they never meet) and

  • no single plane contains both (planes $r$ and $w$ are distinct),

  • $DE$ and $FG$ are skew.

Three-Dimensional Diagram Discussion

• Provided setup: two planes $a$ and $b$ with several lines.

• Observations made

  • Lines $p$ and $m$ are marked as parallel (coplanar within the same plane, no intersection).

  • Search task: identify other pairs as parallel, perpendicular, or skew by checking coplanarity and intersection behavior.

Quick Reference Checklist

• To classify a pair of lines:

  1. Intersect?

  • Yes ⇒ either perpendicular (if angle $=90^\circ$) or just intersecting.

  1. Common plane?

  • Yes, No intersection ⇒ parallel.

  • No, No intersection ⇒ skew.

• Remember: In Euclidean geometry, parallel implicitly means coplanar.

perpendicular bisector theorm: the points on the perpendicular bisector of a segment are equidstant from the endpoints