Axiom 1

Axiom 2

Axiom 3

Corollary 1: There is only one plane passing through a straight line and a point outside the straight line.

Corollary 2: After two intersecting straight lines, there is only one plane

Corollary 3: passing through two parallel lines, there is only one plane

Prove that three points in space are collinear

Prove the common point problem of three lines in space

Prove that several points of space are coplanar

Prove that several straight lines in space are coplanar

There are exactly 3 straight lines in space

How to prove that two straight lines are out of planes

Two straight lines parallel to the same straight line are parallel, that is, parallel transitivity

If the two sides of two angles in space are parallel, then the two angles are equal or complementary.

straight line segment translation method

When straight lines a and b are parallel to each other, a and b are coplanar. It is stipulated that the angle formed by them is 0, so the value range of the angle formed by the two straight lines in space is 0°≤a≤90°

straight line in plane

Intersection of a straight line and a plane

straight line parallel to plane

Theorem: If a straight line outside a plane is parallel to a straight line in this plane, then the straight line is parallel to this plane

Theorem: A straight line is parallel to a plane. If a plane passing through the straight line intersects with the plane, then the straight line is parallel to the intersection line.

Using definitions, prove that a straight line and a plane have five common points

Use the determination theorem of straight lines and planes: prove that a straight line outside the plane is parallel to a straight line inside the plane

Using the definition of planes and planes: if two planes are parallel, then all straight lines in one plane are parallel to the other plane

Theorem: If two intersecting straight lines on a plane are parallel to another plane, then the two planes are parallel

Corollary: If two intersecting straight lines in one plane are parallel to two straight lines in another plane, then the two planes are parallel.

Theorem: Two planes are parallel. If another plane intersects these two planes, then the two intersection lines are parallel.

Utilize the definition: prove that the two faces have no common points

Using the Determination Theorem of Surface Parallelism

Use two planes perpendicular to the same straight line to be parallel

Utilizing plane-parallel transitivity

Theorem: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.

Theorem: Two straight lines perpendicular to the same plane are parallel