(1) Definition of a straight line parallel to a plane: If a straight line and a plane have nothing in common, we say that the straight line is parallel to the plane.

(2) Theorem for judging whether a straight line is parallel to a plane: If a straight line out of the plane is parallel to a straight line in the plane, the straight line is parallel to the plane.

Note: This theorem is the most commonly used theorem to prove that a straight line is parallel to a plane, that is, to prove that a straight line is parallel to a plane means that the straight line is not in this plane, which means that there is a straight line in a known plane that is parallel to a known straight line.

2. Determination of parallelism of two planes

(1) Definition of two planes being parallel: two planes are parallel if they have no common points.

(2) The judging theorem of plane parallelism: if two intersecting straight lines in one plane are parallel to another plane, the two planes are parallel.

Note: Another expression of this theorem is "If two intersecting straight lines in one plane and two intersecting straight lines in another plane are parallel respectively, then the two planes are parallel".

(3) Two planes parallel to the same plane are parallel to each other.

3. The property that the straight line is parallel to the plane

( 1)

Theorem of the property that a straight line is parallel to a plane: If a straight line is parallel to a plane, the intersection line between any plane passing through the straight line and the plane is parallel to the straight line.

Note: If a straight line is parallel to a plane, it is parallel to countless straight lines in the plane, but it cannot be misunderstood as "If a straight line is parallel to a plane, it is parallel to any straight line in the plane".

(2) The property that the straight line is parallel to the plane: if the straight line passing through a point in the plane is parallel to the straight line parallel to the plane, the straight line is in the plane.

4. The nature of the plane parallel to the plane

(1) If two planes are parallel, then any straight line on one plane is parallel to the other plane.

This conclusion can be used as a theorem to determine the parallelism between a straight line and a plane.

(2) Two-plane parallelism theorem: If two parallel planes intersect the third plane at the same time, their intersection lines are parallel.

(3) The parallel lines sandwiched between two parallel planes are equal.