Judgment method

(1) defines that two straight lines have no common points on the same plane, and they are called parallel.

(2) Axiom Two straight lines parallel to the same straight line are parallel to each other.

(3) Theorem is equal to complementary angle, or internal angle is equal, or internal angle is complementary, and two lines are parallel.

④ Properties X2 Inverse Theorem, X4, X6 and Vertical Relation Properties

Main attributes

In X 1 theorem space, if two sides of two angles are parallel, then the two angles are equal or complementary. (Equiangular Theorem)

X2 Theorem Three parallel lines cut two straight lines, and the corresponding line segments are proportional.

Parallelism of line and plane

(1) The straight line is in the plane.

Judgment method

(1) a straight line and a plane have countless things in common, said the straight line in the plane.

Axiom If two points on a straight line are on the same plane, then the straight line is on this plane.

(3) Axiom Any two points determine a straight line, and three non-straight lines determine a plane; Two intersecting lines and two parallel lines define a plane.

④ Attribute X3 and vertical relation attribute

Main attributes

X3 Theorem: A straight line passing through a point on a plane is parallel to a parallel line on this plane, so this straight line is on this plane.

(2) The straight line is not in the plane.

Judgment method

① Define that there is no common point between a straight line and a plane, and say that the straight line is parallel to the plane.

② A straight line out of the theorem plane is parallel to a straight line in the plane, then the straight line is parallel to the plane.

③ Attributes X5, X7 and vertical relation attributes

Main attributes

X4 Theorem 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.

Two parallel straight lines out of the plane of X5 theorem, one parallel to this plane and the other parallel to this plane.

Plane parallelism

Judgment method

(1) define two planes have nothing in common, called parallel.

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

③ Theorem Two intersecting straight lines in a plane are parallel to another plane, then the two planes are parallel.

④ Theorem Two intersecting straight lines in one plane are parallel to two intersecting straight lines in another plane, then the two planes are parallel.

⑤ Properties X8 Inverse Theorem, X9 and Vertical Relation Properties

Main attributes

X6 Theorem If two parallel planes intersect the third plane at the same time, their intersection lines are parallel.

X7 Theorem If two planes are parallel, then any straight line in one plane is parallel to the other plane.

The parallel lines between two parallel planes of X8 theorem are equal. Inverse Theorem Two planes are parallel if the parallel lines between them are equal.

The X9 conclusion passes through a point outside the plane and only one plane is parallel to the known plane. (Existence and uniqueness)