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The intersection of a line and a plane can be the empty set, a point, or a line.

Consider the following theorems relating lines and planes. The diagrams supplied for each theorem represent one possible depiction of the situation.


theorem If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by the two intersecting lines.

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theorem Through a given point there passes one and only one plane perpendicular to a given line.

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theorem Through a given point there passes one and only one line perpendicular to a given plane.

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theorem If two planes are perpendicular to the same line, the planes are parallel.

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theorem Two lines perpendicular to the same plane are coplanar.

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theorem If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the given plane.

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theorem Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.

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theorem If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.

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theorem If a plane intersects two parallel planes, then the intersection is two parallel lines.

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The angle where two planes meet is called a dihedral angle.

Example: Carpenters and construction workers deal with dihedral angles when planning the construction of the trusses in a roof.

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