The specific method process of determining the parallel and perpendicular lines and planes

The line surface is perpendicular: now find two intersecting straight lines on the plane, and then prove that this straight line is perpendicular to these two intersecting straight lines; Line and surface parallelism is to find the straight line parallel to the straight line in the plane, generally find the intersection line between the plane passing through the straight line and the other plane, and then prove that the straight line is parallel to the intersection line.

Answer: 1, line and surface vertical:

Prove that the line l is perpendicular to the plane: the common method is to prove that the line l is perpendicular to the two intersecting lines l1 and l2 in the plane respectively (theorem: if a straight line in space is perpendicular to the other two intersecting lines, then the line is perpendicular to the plane determined by the two intersecting lines); Another way is to prove that the plane on which the line is located (let's say a plane) is perpendicular to the plane, and then prove that the line l is perpendicular to the intersection of the two planes m, so that it can be proved that the line l is perpendicular to the plane (theorem:

Two planes are perpendicular, and if a straight line in one plane is perpendicular to the intersection of the two, then the line is perpendicular to the other plane).

Line and surface parallelism: (When the line surface is parallel, the plane on which the line is located may be intersecting or parallel to the known plane, which is divided into two cases).

1, when two planes intersect, what is commonly used is to prove that this straight line l is parallel to any straight line in the plane (if a straight line l is parallel to a straight line in a plane, and this straight line l is not on this plane, then this straight line is parallel to this plane); Another way is to prove that the straight line l is parallel to the intersection of two planes.

2. When these two planes are parallel, it can be directly concluded that the straight line l is parallel to the plane (theorem: if two planes are parallel, then any straight line in one of the planes is parallel to the other plane).

Perpendicular: The line is perpendicular to any line of the plane, parallel: There is no common point between the line and any line of the plane.

Property theorem: the straight line l is parallel to the plane, and the plane passes through l and intersects the plane with the straight line l', then l l'; Decision theorem: the line l' is on the plane , the line l is not on the plane , and l'l, then l.

Determination theorem, if a straight line outside the plane is parallel to a straight line in this plane, then this straight line is parallel to this plane, and the property theorem, if a straight line is parallel to a plane, and the plane of the straight line intersects with this plane, then the straight line is parallel to the intersection line.

Proof of parallel lines.

Known: a b, a , b, verify: a Counterproof proves that assuming that a and are not parallel to , then they intersect, let the intersection point be a, then a

A B, A is not on B.

If a is passed within c b, then a c = a

and a b, b c, a c, contradict a c = a.

Assuming that the slag section does not stand, a

The vector method proves that the direction vector of a is a, and the direction vector of b is b, for example, the normal vector of the absolute surface is p. ∵b⊂α

b p, i.e., p·b=0

a b, from the fundamental theorem of collinear vectors, we know that there is a real number k such that a=kb

Then p·a=p·kb=kp·b=0

i.e. a PA

The above content reference:Encyclopedia - Lines and planes are parallel

In geometry, two straight lines that never intersect and never coincide in the same plane are called parallel lines. The definition of parallel lines includes three basic characteristics: one is in the same plane, two straight lines, and three are disjoint.

The method of determining parallel lines is as follows: where to bury.

1. The isotopic angle is equal, and the two straight lines are parallel;

2. The internal staggered angles are equal, and the two straight lines are parallel;

3. The inner angles of the same side are complementary, and the two straight lines are parallel;

4. When two straight lines are parallel to the third straight line, the two straight lines are parallel;

5. In the same plane, two straight lines perpendicular to the same straight line are parallel to each other;

6. In the same plane, two straight lines parallel to the same straight line are parallel to each other;

7. Two straight lines that never intersect in the same plane are parallel to each other.

Determination theorem and property theorem for the perpendicularity of lines and surfaces:

1. Line-surface perpendicularity theorem: If a straight line is perpendicular to two intersecting straight lines in the plane, then the straight line is perpendicular to the plane. Note that the keyword "intersection", if the chain is only parallel to the line, it cannot be determined that the line surface is perpendicular.

2. Theorem of the perpendicular properties of lines and planes:

1) If a straight line is perpendicular to a plane, then the line is perpendicular to all the lines in the plane.

2) Passing through a point in space, there is and only one straight line perpendicular to the known plane.

3) If one of the two parallel lines is perpendicular to one plane, then the other line is also perpendicular to this plane of rocks.

4) Two straight lines perpendicular to the same plane are parallel.

5) Corollary: If both lines are parallel to the third line in space, then the two lines are parallel. (This inference implies that the transmissibility of parallel lines is true not only in plane geometry, but also in spatial geometry.) ）

Method for determining the perpendicularity of lines and planes:

1.The judgment of the perpendicular line and surface is determined to be coarse and slag: the straight line is perpendicular to the two intersecting straight lines in the plane.

2.The property that the face is perpendicular: if the two planes are perpendicular to each other, the line perpendicular to the intersection line in one side must be perpendicular to the other plane.

3.The property that the line surface is perpendicular: if one of the two parallel lines is perpendicular to the plane, the other is also perpendicular to the plane.

4.The nature of parallel faces: if one line is perpendicular to one of the two parallel planes, it must be perpendicular to the other plane.

5.Definition: A straight line is perpendicular to any straight line in the plane.

Hello lz. Vertical or parallel can be directly judged by other means, not to be discussed.

Directly from the definition of absolute search, take a little p on the straight line, and do (find) projection p to the plane', if the line and the plane have an intersection point s within the field of view, then PSP'That is, the angle between the line and the surface; If there is no S in the field of view, then find another point R, also do the projection R', and then search for macro knowledge pr and p'r'Angle (find p.)'r'It is better to pass through P or R, or to find PR than through P' or R'）

Pay attention to the special situation of the angle between the surface and the line and the line and the angle. Pay special attention to the planar angles of the dihedral corners.

Take two points of pr on the straight line, find the normal vector n of the plane, and then find the angle between the vector pr and the vector n, the result is its cos value, if it is positive, it is the cos corresponding to the angle between the line and the surface; If it is negative, it is the cos corresponding to the co-angle of the line-surface angle

The method of determining the parallelism of lines and planes is shown in the following figure

Judgment that the straight line is parallel to the Pingsun meeting].

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

Rent refers to the method of judging that a straight line is parallel to a plane].

1) Use the definition of regular collision: prove that there is no common point between the straight line and the plane;

2) Using the decision theorem: from the straight line parallel to the straight line, the straight line is parallel to the plane;

3) Take advantage of the property of parallel surfaces: if two planes are parallel, then the straight lines in one plane must be parallel to the other.

Determination and properties of line-surface and face-to-surface parallelism.

The foundation is strengthened and strengthened.

1.(Text) (2011 Beijing Haidian Period) Knowing that the plane l,m is a straight line in which is different from l, then the following proposition is wrong ( ).

a.If m, then m lbIf m l, then m

c.If m, then m ldIf m l, then m

Answer]d analysis]a conforms to the property theorem that a straight line is parallel to a plane; b conforms to the determination theorem that a straight line is parallel to a plane; c. Conforms to the property that the straight line is perpendicular to the plane; For d, only when is , it can be true.

(2011 Tai'an simulation) Let m and n denote different straight lines, and , denote different planes, then the correct one of the following propositions is ().

a.If m, m n, then n

b.If m, n, m, n, then

c.If m, m n, then n

d.If m , n m, n , then n

Answer]D Analysis] Option A is incorrect, n may still be in the plane, option B is incorrect, the plane may also intersect the plane, option C is incorrect, n may also be in the plane, option d is correct.

2.(text) (2011 Handan period) let m and n be two straight lines, and two planes, then among the following four propositions, the correct proposition is ().

a.If m , n , and m , n , then

b.If m, m n, then n

c.If m , n then m n