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EdrawMindspace geometry
MindMap Gallery space geometry
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space geometry
The mind map of space geometry includes the positional relationship between space points and lines and planes, space straight lines, parallel planes, perpendicularity of space straight lines and planes, and space vectors. Let's look at it together.
Edited at 2023-06-20 11:33:48- bacteria
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
- Plant asexual reproduction
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
- Reproductive development of animals
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
space geometry
- bacteria
This is a mind map about bacteria, and its main contents include: overview, morphology, types, structure, reproduction, distribution, application, and expansion. The summary is comprehensive and meticulous, suitable as review materials.
- Plant asexual reproduction
This is a mind map about plant asexual reproduction, and its main contents include: concept, spore reproduction, vegetative reproduction, tissue culture, and buds. The summary is comprehensive and meticulous, suitable as review materials.
- Reproductive development of animals
This is a mind map about the reproductive development of animals, and its main contents include: insects, frogs, birds, sexual reproduction, and asexual reproduction. The summary is comprehensive and meticulous, suitable as review materials.
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space geometry
The positional relationship between space points and lines and planes
Knowledge point 1: Basic properties and applications of planes
Basic properties and corollaries of planes
Axiom 1
Content: There is only one plane passing through 3 points that are not on a straight line.
Function: Provides the basis for determining a plane
Axiom 2
Content: If two points on a straight line are in a plane, then the straight line is in this plane
Function: used to determine whether the straight line is in the plane
Axiom 3
Content: If two non-overlapping planes have a common point, then they have one and only one common straight line passing through the point.
Function: 1. Used to find the intersection line of two planes; 2. Used to prove that the point is on the line
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
Methods to prove that three points are collinear, three lines have common points, and points and lines are coplanar
Prove that three points in space are collinear
That is, first determine that two points are on the intersection of the two planes, and then prove that the third point is both in the first plane and the second plane.
Prove the common point problem of three lines in space
1. Find the intersection point of the two lines. 2. Prove that the third straight line passes the intersection point.
Prove that several points of space are coplanar
You can take 3 points to determine a plane, and then prove that all other points are within this plane.
Prove that several straight lines in space are coplanar
You can first take two straight lines to determine a plane, and then prove that the other straight lines are also in this plane.
Knowledge point 2: The positional relationship between two straight lines in space
Straight lines in different planes: two straight lines that are different in any plane
There are exactly 3 straight lines in space
Coplanar straight lines
intersect
parallel
Different plane straight line
How to prove that two straight lines are out of planes
Proof by contradiction: first assume that the two straight lines are coplanar, then proceed from the assumption, and derive a contradiction through reasoning, thereby denying the assumption and confirming that the two straight lines are out of planes.
Utilize the conclusion: A straight line passing through a point outside the plane and a point in the plane is a straight line out of plane with a straight line in the plane passing through that point.
Basic Fact 4 and Congruent Angles Theorem
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.
Knowledge point 3: Angle formed by straight lines with different planes
How to find the angle formed by straight lines on opposite sides
straight line segment translation method
Use existing parallel lines in the figure to translate
Use the endpoint or midpoint of a special point line segment to perform parallel line translation
Complementary expansion translation
Straight lines on opposite sides are perpendicular: If the angle formed by two straight lines on opposite sides is a right angle
The range of angles formed by out-of-plane straight lines is 0°<a≤90
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°
Knowledge point 4: The positional relationship between straight lines and planes in space
Positional relationship and representation of straight lines and planes
straight line in plane
Intersection of a straight line and a plane
straight line parallel to plane
Knowledge point 5: The positional relationship between planes in space
Two planes are parallel
Two planes intersect
Parallelism of straight lines and planes in space
Knowledge point 1: Judgment and properties of parallel lines and surfaces
Determining whether a straight line is parallel to a 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
Parallel properties of straight lines and planes
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.
How to prove that a straight line is parallel to a plane
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
Knowledge point 2: Judgment and nature of face-to-face parallelism
Judgment of parallelism between faces
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.
The property of face-to-face parallelism
Theorem: Two planes are parallel. If another plane intersects these two planes, then the two intersection lines are parallel.
4 ways to determine whether surfaces 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
Verticality of a straight line plane in space
Knowledge point 1: Determination and properties of straight lines and planes perpendicular to each other
A straight line is perpendicular to a plane: If a straight line is perpendicular to any straight line in a plane, then the straight line is perpendicular to the plane.
Determining whether a straight line is perpendicular to a plane
Theorem: If a straight line is perpendicular to two intersecting straight lines in a plane, then the straight line is perpendicular to the plane.
Parallel properties of straight lines and planes
Theorem: Two straight lines perpendicular to the same plane are parallel
How to prove that a straight line is perpendicular to a plane
angle between a straight line and a plane
Knowledge point 2: Determination and properties of plane and plane perpendicularity
plane perpendicular to plane
space vector