# Why are two planes parallel to the same line not necessarily parallel?

What is a case in which the statement, "Two planes parallel to the same line are parallel" be false?

• simply because for a given line there are infinite planes parallel with this line. A way to visualize has been pointed out by @IvoTerek – Ant May 10 '15 at 23:26
• @Ant: Sure, but given any line, there are infinite lines parallel to that line too, and yet parallelness is transitive for lines. – user2357112 May 11 '15 at 5:38
• @user2357112 you're right, totally overlooked that :) – Ant May 11 '15 at 9:18
• Pavement. Wall. Yellow line in the middle of the street. – Did May 11 '15 at 13:33
• Two planes that intersect are both parallel to the line of intersection. – Daniel R Hicks May 11 '15 at 17:39

Another example: take a right cylinder. Every tangent plane to the cylinder is parallel to the cylinder's axis.

• I like this one! – Martigan May 11 '15 at 7:28
• This is what I was thinking about, but couldn't have articulated it this nicely. – Cruncher May 12 '15 at 19:37
• I wish that you were my teacher in school. – Ejaz May 12 '15 at 23:11 Think in the floor of your room as the gray plane and think in the wall of your room as the blue plane. the blue line will be the line between the wall in front of the wall blue and the ceiling.

• if and only if, you have a squeare-room. – Luis Felipe May 11 '15 at 11:07
• Judging by the dashed line, the line is not parallel to those planes, as the length of those differs greatly. – Vlasec May 11 '15 at 12:37
• The comment before the draw improves the imagination ;) – Luis Felipe May 11 '15 at 15:12
• This was the example I'd have given. In a "perfect" house, the wall and the floor will both be parallel to the line where they intersect, but they will be perpendicular to each other. – TecBrat May 12 '15 at 4:13
• About this kind of propositions always I seen the room where I am and try to find a relation with things around me – Luis Felipe May 12 '15 at 4:15

A very simple example is two different planes containing the same line: both are parallel to that line and to any other line parallel to it.

• And since all pairs of non-parallel planes intersect, this exists for all of them. – Random832 May 11 '15 at 4:39
• As real world examples, any two planes connected by a hinge, like the keyboard and monitor of a laptop, a door and a wall... – null May 11 '15 at 12:13

In $\mathbb{R}^3$, the planes $y = 1$ and $z=1$ are both parallel to the $x$-axis.

Because in three-dimensional space, the planes that are anchored to a line still have one more degree of freedom. They can rotate around the line. So the two planes that are parallel to the same line could be at an infinite number of angles to each other.

Making lines parallel in 3D is simpler than making planes parallel, because planes have extra dimension. You can think of a plane as an intersection of two lines.

So as you can see there is one more line involved. Omitting that extra line is what makes your statement incomplete. Not necessarily false. It may still be true in one case out of infinity.

If you had two lines crossing each other (not necessarily at the right angle) and you had two planes that were parallel to both of the lines, then you could guarantee that the two planes are parallel, because then you would have anchored both of the plane's dimensions, not just one.

• To make the planes parallel you'd then have to make their both lines parallel. That's not true. You can rotate these intersecting lines around normals of their containing planes, and the planes will still remain parallel. – Ruslan May 13 '15 at 6:41
• Hmmm... You sir are right. – zvolkov May 13 '15 at 13:06
• Fixed by removing the offending statement. – zvolkov May 13 '15 at 21:22

If you take any two intersecting planes they will both be parallel to the line formed by their intersection, but they can't be parallel to each other because they intersect. In the image above both planes are parallel to the line defined by points A and B.

More examples can be found on Google

If both planes are infinite, and not parallel, their intersection will be a line, and any line parallel to that line will also be parallel to both planes. If the planes are parallel they of course won't have an intersection (Thanks @vlasic).

• Provided that they have an intersection :) They might actually be parallel. – Vlasec May 11 '15 at 15:26

All the given solutions are particular case which can be generalized in the following context:

consider that in a three dimensional space, a given plane has infinite parallels planes; the order of such an infinite is one (infinite of the first order, i.e. you can fix a single parameter, for instance the distance from the line and the plane, to identify the plane).

A straight line, in the same three dimensional space, has infinite planes parallel to itself; but the order of such an infinite is two (infinite of the second order: you need two parameters (distance and angle in a given space of coordinates).

So, we simply cannot compare the two situations: the only case in which the straight line and the two planes are parallel, happens when the angle is fixed in an opportune way. So, you remove a degree of freedom and finally obtain infinite^1 plans parallel to the line and each other.

The answer to Why are two planes parallel to the same line not necessarily parallel? is: because for every two planes there is a line parallel to both of them, so if the two planes parallel to the line had to be parallel, every two planes would be parallel, which implies all planes would be parallel. Which is false.