Lines, lines, planes, and surfaces are parallel to determine theorems and properties

1. Parallel lines.

Lines and lines are parallel) theorem: In the same plane, two straight lines that never intersect are called parallel lines (lines and lines are parallel).

Properties: Two straight lines that are not parallel must intersect, and parallelism is represented by the symbol " ". In the same plane, there is only one straight line parallel to the straight line after passing a point outside the straight line.

2. The line and surface are parallel.

Decision theorem: Theorem 1: A straight line outside the plane is parallel to a straight line in this plane, then the straight line is parallel to this plane.

Theorem 2: A straight line outside the plane is perpendicular to the perpendicular line of this plane, then the straight line is parallel to this plane.

Property: Property 1: A straight line is parallel to a plane, then the intersection of any plane of the line and this plane is parallel to the straight line.

Properties: If a straight line is parallel to a plane, the line is perpendicular to the plane.

3. The faces are parallel.

Decision Theorem: Theorem 1: If two planes are perpendicular to the same straight line, then the two planes are parallel.

Theorem 2: If there are two intersecting lines in one plane that are parallel to the other, then the two planes are parallel.

Theorem 3: If there are two intersecting lines in one plane that are parallel to two intersecting lines in the other plane, then the two planes are parallel.

Properties: Property 1: Two planes are parallel, and any line in one plane is parallel to the other.

Property 2: Two parallel planes, respectively intersecting the third plane, parallel to the intersecting lines.

Property 3: A straight line that is parallel to two planes and perpendicular to one plane must be perpendicular to the other. (Inverse theorem of decision theorem 1).

Extended Information: A Simple Method for Determining Parallel Lines:

Within the same plane, two straight lines are truncated by a third straight line, if the isotopic angle.

equal, then these two straight lines are parallel. It can also be simply said:

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

Within the same plane, two straight lines are truncated by a third straight line, if the inner wrong angle.

equal, then these two straight lines are parallel. It can also be simply said:

2.The inner wrong angles are equal, and the two straight lines are parallel.

In the same plane, two straight lines are truncated by a third straight line, if at the same side of the inner angle.

If a line outside the plane is parallel to a line inside the plane, then the line is parallel to the plane.

If a straight line is parallel to a plane, then the plane passing through the line intersects this plane, then the line is parallel to the intersectional line.

If there are two intersecting lines in a plane that are parallel to the other, then the two planes are parallel.

If two planes are parallel, then the lines in one of the planes are parallel to the other.

If there are two intersecting lines in one plane and two intersecting lines in another plane that are parallel respectively, then the two planes are parallel.

If two parallel planes intersect a third plane at the same time, the intersection lines are parallelBeg.

The line-line perpendicular determination theorem.

If a straight line is perpendicular to any line in a plane, then the line is said to be perpendicular to the plane.

Line-plane perpendicular theorem.

definitions (counter-evidence);

Decision theorem: b, a Gong field Dodu Xin thief goose aldehyde enamel mutu ba ;

Line-plane perpendicular property theorem).

a a (theorem of parallel properties of faces);

β=l，a⊥l，a

a (theorem of perpendicular properties of faces).

The perpendicular determination theorem for faces.

If one plane passes through one perpendicular line of the other plane, then the two planes are perpendicular to each other.

The line surface is vertical, and the face surface is vertical).

First, the lines are parallel.

1. Two straight lines with equal isotopic angles are parallel: in the same plane, two straight lines are truncated by the third straight line, and if the internal wrong angles are equal, then these two straight lines are parallel. It can also be simply said:

2. Two straight lines are parallel when the inner wrong angle is equal: in the same plane, the two straight lines are truncated by the third straight line, and if the inner angles are complementary to the side, then the two straight lines are parallel. It can also be simply said:

3. The two straight lines are parallel to the complementary inner angles of the same side.

Second, the line and surface are parallel.

1. Definition of utilization: prove that there is no common point between the straight line and the plane;

2. Using the decision theorem: from the parallel between the straight line and the straight line, the straight line is parallel to the plane;

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

3. Parallel faces.

1. If two planes are perpendicular to the same straight line, then the two planes are parallel.

2. If there are two intersecting lines in one plane that are parallel to the other, then the two planes are parallel.

3. If there are two intersecting lines in one plane that are parallel to two intersecting lines in the other plane, then the two planes are parallel.

Extended information: The distances between parallel planes are equal everywhere.

Known to: ab , dc , and a, d , b, c

Verification: ab=cd

Proof: Connect AD and BC

From the property theorem of perpendicular to the line surface, ab cd is known, then ab and cd constitute the plane abcd

Planar ABCD = AD, planar ABCD = BC, and

AD BC (Theorem 2).

The quadrilateral ABCD is a parallelogram.

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

The calculation problem of the parallel property theorem of line and surface is mainly the application of the property theorem to synthesize some calculation problems, such as the midpoint information can be transformed into a 1:2 relationship; From the numerical proportion, the conditions required for the application of the line-surface or surface-surface property theorem are deduced. It aims to expand the comprehensiveness of information transformation.

2 questions to help you master quickly!

1. The determination theorem of line and surface parallelism:

If an out-of-plane line is parallel to a line in the plane, the out-of-plane line is parallel to the plane;

2. The property theorem of parallel lines and planes:

If two intersecting lines in a plane are parallel to a known plane, then the two planes are parallel;

3. Use: The determination theorem of line and surface parallelism is mainly to prove that lines and planes are parallel by line and line parallelism;

The property theorem of line-plane parallelism proves that faces are parallel by means of line-plane parallelism;

4. Understanding of theorems:

As the name suggests, the determination theorem of line and surface parallelism is how to judge that lines and surfaces are parallel, that is, through what conditions (lines and lines are parallel) that lines and surfaces can be obtained;

The property theorem of line-surface parallelism, that is, what conclusions can be deduced by the parallel of lines and planes (surface parallelism).

The property theorem of line-surface parallelism: a straight line outside the plane is parallel to a straight line in the plane, then the straight line is parallel to the plane; A line outside the plane perpendicular to the perpendicular line of the plane is parallel to the plane.

A straight line that has no common point (does not intersect) with a plane is said to be parallel to the plane. From the straight line is parallel to the straight line, a straight line is parallel to a plane, and a straight line is parallel to a plane, then the intersection of any plane of the straight line and this plane is parallel to the straight line.