Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school
Brian was a geometry teacher through the Teach for America program and started the geometry program at his school
In Geometry, a plane is any flat, two-dimensional surface. Two planes that do not intersect are said to be parallel. Parallel planes are found in shapes like cubes, which actually has three sets of parallel planes. The two planes on opposite sides of a cube are parallel to one another.
In Geometry when we talk about this concept of two things being parallel, we aren't just talking about two parallel lines. We could be talking about well, the obvious the two coplanar lines that's what we're going to see the most. But a line and plane can be parallel to each other and two planes can be parallel to each other.
So let's start off by identifying two coplanar lines in this cube right here. So this cube we'll assume that we have 6 congruent faces and that opposite faces are parallel. So if we start off by saying well two coplanar lines, if I look at this front face, so that's going to be one plane. That's going to be place a, b, c, d. I could say that this segment a, b so I'm going to write segment a, b is parallel to segment c, d. So those will be 2 that are in the same plane that will never intersect. Now what about a line and a plane? How can those be parallel? Well taking that same plane a, b, c, d if I took one edge let's say a, b so I'm going to say line segment a, b line segment a, b intersects this plane a, b, c, d it also intersects this plane a, b, e, f. Which means it could be parallel to this bottom face, so the bottom face is c, d, h and g. So we can say that line segment a, b is parallel to plane c, d, h, g so that line will never intersect that plane they're considered parallel.
And last what about two planes? Well since we have a cube we have 3 pairs of parallel planes, so we could start off with front plane a, b, c, d. So I'm going to say a, b, c, d is parallel to the face that is opposite to it e, f, g, h, so I'm going to say e, f, g, h but we could also consider the other 2 pairs, so we could say this side face a, e, h, d. a, e, h, d is parallel to this other side face b, f, g, c and last we could say our two bottom faces or the top and the bottom face. So we have a, b, f, e is parallel to this bottom face which is c, d, h, g.
So two coplanar lines if we look at our cube, there's lots of them but I only named one pair and that was a, b, and c, d. We said that we could have a line parallel to a plane again there's many and I just chose one. And we can say that two planes can be parallel if they never intersect. And because there's only 3 pairs I decided to write them all up, so don't just think that parallelism applies only to two coplanar lines. It could also apply to a line and a plane and two planes.