Summary of solid geometry knowledge points

1. Determination of plane straight line

(1) Using axiom 1: If two points on a straight line are in a plane, then the straight line is in a plane.

(2) If two planes are perpendicular to each other, a straight line perpendicular to the second plane through a point in the first plane is in the first plane, that is, if α⊥β,A∈α, AB⊥β, then ABα.

(3) All lines perpendicular to the known straight line passing through a point are in the plane perpendicular to the known straight line, that is, if A∈a,a⊥b, A∈α,b⊥α, then aα.

(4) The straight lines passing through a point outside the plane and parallel to the plane are all in the plane passing through the point and parallel to the plane, that is, if Pα, P∈β, β ∈ α, P∈a, a∧α, then aβ.

(5) If a straight line is parallel to a plane, then the straight line passing through this plane and parallel to this straight line must be in this plane, that is, if A∧α, A∈α, A∈b, b∑A, then bα.

2. Existence and uniqueness theorem

(1) At a point outside the straight line, only one straight line is parallel to this straight line;

(2) There is only one straight line perpendicular to the known plane;

(3) There is only one plane parallel to this plane at a point outside the plane;

(4) There is only one straight line perpendicular to two straight lines in different planes;

(5) There is only one plane perpendicular to the known straight line;

(6) Only one plane intersects a diagonal of the plane and is perpendicular to the plane;

(7) Only one plane passes through one of the two straight lines and is parallel to the other;

(8) Only one plane passes through one of two mutually perpendicular straight lines and is perpendicular to the other.

3. Projection and related attributes

The projection of the (1) point on the plane leads to a vertical line from the point to the plane. The vertical foot is called the projection of this point on this plane, and the projection of this point is still a point.

(2) The projection of a straight line on the plane leads a vertical line from two points on the straight line to the plane, and a straight line passing through two vertical feet is called the projection of the straight line on this plane.

The projection of a straight line perpendicular to the projection plane is a point; The projection of a line that is not perpendicular to the projection plane is a straight line.

(3) Projection of a figure on a plane The set of projections of all points on a plane figure is called the projection of a plane figure on a plane.

When the plane of the figure is perpendicular to the projection plane, the projection is a line segment;

When the plane of the figure is not perpendicular to the projection plane, the projection is still a figure.

(4) Related properties of projection

In the vertical and diagonal lines drawn from a point outside the plane to the plane:

(i) The diagonal lines of two equal projections are equal, and the diagonal lines with longer projections are also longer;

(ii) Equal diagonal lines have equal projections, and longer diagonal lines have longer projections;

(iii) The vertical line segment is shorter than any diagonal line segment.

4. Various angles in space

Equiangular Theorem and Its Inference

Theorem If two sides of an angle and two sides of another angle are parallel and in the same direction, the two angles are equal.

It is inferred that if two intersecting straight lines are parallel to the other two intersecting straight lines, the acute angles (or right angles) formed by the two groups of straight lines are equal.

The angle formed by straight lines on different planes.

(1) Definition: A and B are two straight lines in different planes. When they pass through any point O in the space and lead to straight lines A'∨A and B'∨B respectively, the acute angle (or right angle) formed by A' and B' is called the angle formed by straight lines A and B. 。

(2) Value range: 0 < θ ≤ 90.

(3) Solutions

(1) According to the definition, through translation, find the angle θ formed by straight lines on different planes;

② Solve the triangle containing θ and find the angle θ.

5. Angle between straight line and plane

(1) There are three angles between the definition and the plane:

(1) The acute angle formed by the diagonal of the angle formed by the vertical plane and its projection on the plane is called the angle formed by this straight line and this plane.

(ii) If the angle formed by the vertical line and the plane is perpendicular to the plane, the angle formed by them is a right angle.

(iii) If a straight line is parallel to or in a plane, they form an angle of 0.

(2) The value range is 0 ≤ θ≤ 90.

(3) Solutions

(1) as the projection of the oblique line on the plane, and find the angle θ formed by the oblique line and the plane.

② Solve the triangle containing θ and find its size.

③ Minimum angle theorem

The angle between the diagonal line and the plane is the smallest of all angles between the diagonal line and the straight line passing through the diagonal foot in the plane. In other words, the angle formed by the diagonal line and the plane is not greater than the angle formed by the diagonal line and any straight line in the plane.

6. dihedral angle and plane angle dihedral angle

(1) Half-plane line divides a plane into two parts, and each part is called a half-plane.

(2) The figure formed by two half planes starting from the straight line of dihedral angle is called dihedral angle. This straight line is called the edge of dihedral angle, and these two planes are called the faces of dihedral angle, that is, dihedral angle consists of half plane, edge and half plane.

If two planes intersect, four dihedral angles are formed with the intersection line of the two planes as the edge.

The size of dihedral angle is measured by its plane angle. It is generally believed that the range of plane angle θ of dihedral angle is