Positional Relationships Between Lines and Planes

Conditions for Determining a Plane

1. Three non-collinear points
   
   
2. A line and a point not on it
   
   
3. Two intersecting lines
   
   
4. Two parallel lines
   
   

Positional Relationships Between Two Different Lines

1. Intersect at a point (lie in the same plane)
   
   
2. Parallel (lie in the same plane)
   
   
3. Skewed (do not lie in the same plane)
   
   

Positional Relationships Between a Line and a Plane

1. Contained in the plane (they intersect)
   
   
2. Intersect at a point (they intersect)
   
   
3. Parallel (do not intersect)
   
   

Positional Relationships Between Two Different Planes

1. Intersect
   Two different planes intersect, and when they do, they share a line called the line of intersection of the planes.
   
   
   
2. Parallel
   When two planes do not intersect, they are parallel. This is represented by the symbol .
   
   
   

Angles Between Two Lines and Perpendicularity Between a Line and a Plane

1. Angle between two skewed lines
   When two lines and are skewed (not in the same plane), the angle between them is defined by taking a point on line and finding the angle between line and a line that is parallel to and passes through .
   
   
   
   
   
2. Perpendicularity between a line and a plane
   If a line intersects a plane at point , and is perpendicular to every line in the plane that passes through point , then the line is perpendicular to the plane , denoted as . In this case, line is called the normal to the plane, and point is called the foot of the perpendicular.
   
   
   
   If line is perpendicular to two non-parallel lines on the plane, then is perpendicular to the plane, .