Theorems Relating Lines and Planes

Theorems Relating Lines and Planes

Consider the following theorems relating lines and planes.  A diagram is supplied for each theorem that represents one possible depiction of the situation.

1. 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 them.
   
2. Through a given point there passes one and only one plane perpendicular to a given line.
   
3. Through a given point there passes one and only one line perpendicular to a given plane.
   
4. Two lines perpendicular to the same plane are coplanar.
   
5. Two planes are perpendicular to each other if and only if one plane contains a line perpendicular to the second plane.
   
6. 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.
   
7. If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane.
   
8. If a plane intersects two parallel planes, then the intersection is two parallel lines.
   
9. If two planes are perpendicular to the same line, they are parallel.
   
10. The angle where two planes meet is called a dihedral angle. Woodworkers and construction workers deal with dihedral angles. For example, creating a rafter for a hip roof requires an understanding of dihedral angles.
    

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