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You may be tempted come think of planes together vehicles come be discovered up in the skies or at the airport. Well, remainder assured, geometry is no fly‐by‐night operation.

You are watching: Two planes perpendicular to a third plane are parallel

Parallel planes

Parallel planes are two airplane that perform not intersect. In figure 1, plane P // aircraft Q.


Figure 1Parallel planes

Theorem 11: If each of two planes is parallel come a 3rd plane, then the two planes are parallel come each other (Figure 2).


Figure 2Two plane parallel come a 3rd plane


Perpendicular planes

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

A aircraft B is perpendicular to a aircraft A if airplane B includes a line the is perpendicular to aircraft A. (Think of a book well balanced upright ~ above a level surface.) See number 3.

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Figure 3Perpendicular planes

Theorem 12: If 2 planes space perpendicular come the same plane, then the two planes either crossing or are parallel.

In figure 4, plane B ⊥ plane A, aircraft C ⊥ aircraft A, and aircraft B and plane C intersect follow me line l.


Figure 4Two intersecting plane that space perpendicular come the very same plane

In number 5, airplane B ⊥ plane A, airplane C ⊥ airplane A, and aircraft B // aircraft C.