You may be tempted to think of planes as vehicles to be found up in the sky or at the airport. Well, rest assured, geometry is no fly‐by‐night operation.

**Parallel planes**

**Parallel planes** are two planes that do not intersect. In Figure 1, plane *P* // plane *Q*.

**Figure 1**Parallel planes

Theorem 11: If each of two planes is parallel to a third plane, then the two planes are parallel to each other (Figure 2).

**Figure 2 **Two planes parallel to a third plane

Perpendicular planes

A line *l* is perpendicular to plane *A* if *l* is perpendicular to all of the lines in plane *A* that intersect *l*. (Think of a stick standing straight up on a level surface. The stick is perpendicular to all of the lines drawn on the table that pass through the point where the stick is standing).

A plane *B* is perpendicular to a plane *A* if plane *B* contains a line that is perpendicular to plane *A*. (Think of a book balanced upright on a level surface.) See Figure 3.

**Figure 3 **Perpendicular planes

*Theorem 12:* If two planes are perpendicular to the same plane, then the two planes either intersect or are parallel.

In Figure 4, plane *B* ⊥ plane *A*, plane *C* ⊥ plane *A*, and plane *B* and plane *C* intersect along line *l*.

**Figure 4 **Two intersecting planes that are perpendicular to the same plane

In Figure 5, plane *B* ⊥ plane *A*, plane *C* ⊥ plane *A*, and plane *B* // plane *C*.

**Figure 5 **Two parallel planes that are perpendicular to the same plane