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.
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Figure 1
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Parallel planes.
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Theorem 11: If each of two planes is parallel to a third plane, then the two planes are parallel to each other (Figure
2 ).
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Figure 2
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Two planes parallel to a third plane.
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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 .
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Figure 3
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Perpendicular planes.
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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.
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Figure 4
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Two intersecting planes that are perpendicular to the same plane.
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In Figure
5 , plane
B ⊥ plane
A, plane
C ⊥ plane
A, and plane
B // plane
C.
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Figure 5
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Two parallel planes that are perpendicular to the same plane.
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Fundamental Ideas
Parallel Lines
