Two beams at different levels never touch. Their directions may even cross in a photograph. Are they parallel? In a plane, saying that two distinct lines do not meet was enough. In space, we need another condition.
Two distinct lines in space are parallel if they lie in a common plane and do not intersect. If no common plane exists, they are skew. A flat drawing cannot decide this alone: we need information about the spatial figure. We return to axioms and proof, parallel lines and parallelograms.
A plane continues beyond the drawn edge
Points, lines and planes are basic objects of solid geometry. A plane is flat, unbounded and has no thickness. A drawn parallelogram shows only a portion of it, rather than four real edges. Likewise, a cube’s edge is a segment, while line AB extends past both ends of edge AB.
In the Euclidean model we use these starting assumptions:
- Three noncollinear points determine one and only one plane.
- If two distinct points on a line lie in a plane, the entire line lies in it.
- Two distinct planes with a common point intersect in a line through that point.
- For every plane there are points on it and outside it.
These are axioms of the model rather than measurements of our illustration. A tabletop gives intuition, but its boundary and thickness do not belong to an ideal plane. AMSI, points, lines and planes.
Specifying a unique plane
Three collinear points are insufficient: infinitely many planes can rotate about their line. A plane can instead be specified by three noncollinear points, a line and a point outside it, two intersecting lines, or two distinct parallel lines.
For example, choose A,B on line AB and C outside it. The first axiom gives a unique plane through A,B,C; the second makes it contain all of AB. This short proof explains the method rather than adding an unrelated rule.
Three cases for distinct lines
Use cube ABCDA₁B₁C₁D₁. Lower vertices A,B,C,D run around one face in order; each upper vertex is above the matching lower one. The faces are squares, so opposite sides of a face are parallel.
Parallel AB and CD
Intersection at B
Skew AB and CC₁
Lines and planes α, β
AB and CD lie in the lower-face plane and do not intersect: they are parallel. AB and BC share B: they intersect. AB and CC₁ have no common plane: they are skew. Two names for the same line describe coincidence, separate from these cases for distinct lines. Purdue, relationships between lines.
Why can AB and CC₁ not lie in a common plane? Such a plane would contain AB and C, so it would be the lower plane ABC. It would then contain C₁ too, although C₁ is outside the lower plane: a contradiction. Uniqueness of the plane does the work; the picture alone is insufficient.
A line and a plane: no point, one point or all points
Call the upper plane α and the lower β. AB lies in β. AA₁ meets β only at A. AB has no common points with α, so AB is parallel to α. In our terminology a line contained in a plane is not called parallel to that plane.
Criterion: if l is not contained in α and is parallel to a line m in α, then l∥α. In the cube, AB∥A₁B₁ and A₁B₁ lies in α. “l is not contained in α” is essential: parallel lines also exist within a plane, sharing all their points with it.
AB∥A_(1)B_(1), AB∥α, AB⊂β.
Two planes and two pairs of directions
Two distinct planes either have no common points and are parallel, or intersect in a line. Coincidence means one plane with two names. Our cube’s upper α and lower β are parallel. Front plane ABB₁A₁ meets the lower plane in AB.
To prove distinct planes parallel, it suffices to find two intersecting lines in one respectively parallel to two lines in the other. For β and α, use AB∥A₁B₁ and AD∥A₁D₁. Two intersecting directions determine a plane.
One pair is insufficient: CD in the lower face is parallel to A₁B₁ in the front face, yet these planes intersect in AB. A transversal plane cuts two parallel planes in parallel lines. Otherwise a common point of those lines would belong to both parallel planes, which is impossible.
Why the criterion worksOptional · safe to skip
From parallel lines to a parallel plane
Let l∥m, with m in α and l not contained in α. A plane γ through l and m is distinct from α; their intersection is m. If l met α at X, X would also lie in γ and thus on m. But parallel l,m have no common point. Hence l does not meet α.
What a view cannot tell us
In a flat picture, skew lines can appear to cross. A line can extend beyond the drawn portion of a plane while still lying in that plane. Do not measure true spatial angles or edge lengths from this drawing. Establish common points and planes from the conditions first, then use the illustration to explain those relationships.