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Parallelism and perpendicularity of straight lines and planes

Criteria of parallelism of straight line and plane. Criteria of parallelism of planes.

Criteria of perpendicularity of straight line and plane. Straight line inclined to plane.

Theorem about three perpendiculars. Criteria of parallelism of straight lines in a space.

Criteria of perpendicularity of planes. Theorem about common perpendicular

to two crossing straight lines.

Criteria of parallelism of straight line and plane:

 

1)  If a straight line, lying out of plane, is parallel to some straight line, lying in the  

     plane, then it is parallel to this plane.

 

2)  If a straight line and a plane are perpendicular to the same straight line, then they  

     are parallel.

 

Criteria of parallelism of planes:

 

1)  If two intersecting straight lines of the same plane are parallel correspondingly to  

     two intersecting straight lines of other plane, then these planes are parallel.

 

2)  If two planes are perpendicular to the same straight line, then they are parallel.

 

Criteria of perpendicularity of straight line and plane:

 

1)  If a straight line is perpendicular to two interesting straight lines, lying in a plane,  

     then it is perpendicular to this plane.

 

2)  If a plane is perpendicular to one of parallel straight lines, then it is perpendicular  

     to the other.

 

Straight line inclined to plane. A straight line, intersecting a plane and not perpendicular to it, is called a straight line inclined to plane.

 

Theorem about three perpendiculars. A straight line, that lies in a plane and is perpendicular to a projection of straight line inclined to this plane, is perpendicular to this straight line inclined to plane.

 

Criteria of parallelism of straight lines in a space.

 

1)  If two straight lines are perpendicular to the same plane, then they are parallel.

 

2)  If a straight line lies in one of interesting planes and it is parallel to another plane,  

     then it is parallel to line of intersection of these planes.

 

Criterion of perpendicularity of planes:  if a plane goes through a straight line that is perpendicular to another plane, then these planes are perpendicular.

 

Theorem about common perpendicular to two crossing straight lines.  Any two crossing straight lines have the only common perpendicular.

 

 

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