Parallel, Perpendicular, & Skew Lines
Three relationships define how lines interact in space: they can be parallel (never meet), perpendicular (meet at 90°), or skew (non-coplanar, non-intersecting).
Introduction
Before studying how angles form when lines cross, you need to classify the lines themselves. This lesson introduces the vocabulary that drives every theorem in Units 3–5.
Past Knowledge
Points, lines, planes (1.1). Angle basics (1.3).
Today's Goal
Identify and classify parallel, perpendicular, and skew lines and planes.
Future Success
The Parallel Lines Postulate and angle theorems (3.1.3–3.1.6) build on this vocabulary.
Key Concepts
Three Line Relationships
| Type | Symbol | Definition |
|---|---|---|
| Parallel | Coplanar lines that never intersect | |
| Perpendicular | Lines that intersect at 90° angles | |
| Skew | — | Non-coplanar, non-intersecting lines |
Interactive Diagram — Desmos Geometry
Explore two parallel lines crossed by a transversal. You can drag the points to change the angle — notice how the lines stay parallel.
Parallel Planes
Two planes that never intersect are parallel planes. Example: the floor and ceiling of a room.
Notation
Write for parallel lines and for perpendicular lines. Arrows on diagrams indicate parallel; small squares indicate right angles.
Worked Examples
Classifying from a Diagram
A rectangular box has edges AB (top front), CD (bottom back), and EF (top right side). Classify each pair of edges.
AB and CD: Skew — not in the same plane, never intersect.
AB and the edge below it: Parallel — same plane, same direction, never meet.
AB and EF: Perpendicular — intersect at a right angle at a shared vertex.
Identifying Parallel Planes
In a rectangular prism, name a pair of parallel planes and a pair of perpendicular planes.
Parallel: Top face and bottom face — they never intersect.
Perpendicular: Front face and bottom face — they meet along an edge at 90°.
True or False Reasoning
“If two lines do not intersect, they must be parallel.” True or false?
False. They could be skew — non-coplanar lines don't intersect but aren't parallel. Parallel lines must be coplanar.
Common Pitfalls
Forgetting Skew Lines
Students often think non-intersecting = parallel. In 3D space, lines can be non-intersecting and non-parallel — those are skew lines.
Confusing Perpendicular with Intersecting
All perpendicular lines intersect, but not all intersecting lines are perpendicular. The angle must be exactly .
Real-Life Applications
Architecture & City Planning
City streets are designed with parallel and perpendicular relationships — a grid layout uses parallel streets crossed by perpendicular avenues. Skew lines appear in overpasses and multi-level interchanges where roads occupy different planes.
Practice Quiz
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