Mutual arrangement of lines and planes V space
Two straight
two planes
Line and plane
Mutual arrangement of lines in space
do not have a common point
do not have a common point
have a common point
lie in the same plane
lie in the same plane
do not lie in the same plane
interbreed
are parallel
intersect
V
V
A
A
A
A
V
Given cube ABCDA 1B1C1D1
B 1
C 1
Specify:
- Edges that lie on lines parallel to edge AA 1
- Edges that lie on lines intersecting edge AA1
- Straight lines that cross with straight line AA1
A 1
D 1
B
C
A
D
Given pyramid ABCD Specify:
1. planes in which the straight lines PE, MK, DB, AB, EC lie;
2.points of intersection of the line DK with the plane ABC, the line CE with the plane ADB;
3. points lying in the planes ADB and DBC;
4. straight lines along which the planes ABC and DCB, ABD and CDA, PDC and ABC intersect.
Mutual arrangement of a straight line and a plane in space
have many points in common
have a common point
Do not have common points
The line lies in the plane
A straight line intersects a plane
Line and plane are parallel
A
A
A
A
A
A ‖
S
Given pyramid ABCS
Specify:
1. Straight lines that lie in the BSC plane
2. Straight lines intersecting the plane ABC
A
WITH
Let's check:
ABOUT
TO
1. SB,SC,BC,SK
2. SA, SB, SC, SK, SO
IN
Mutual arrangement of planes in space
There are common points
No common points
planes are parallel
planes intersect
With
‖
slide 1
slide 2
Lesson Objectives: Introduce the definition of skew lines. Introduce formulations and prove the sign and property of skew lines.
slide 3
Location of lines in space: α α a b a b a ∩ b a || b Lie in the same plane!
slide 4
??? Given a cube ABCDA1B1C1D1 Are the lines AA1 and DD1 parallel; AA1 and CC1? Why? AA1 || dd1 like opposite sides square, lie in the same plane and do not intersect. AA1 || DD1; DD1 || CC1 →AA1 || CC1 by the three parallel lines theorem. 2. Are AA1 and DC in parallel? Do they intersect? Two lines are said to intersect if they do not lie in the same plane.
slide 5
Sign of intersecting lines. If one of two lines lies in a certain plane, and the other line intersects this plane at a point not lying on the first line, then these lines are skew. a b
slide 6
Sign of intersecting lines. Given: AB α, CD ∩ α = C, C AB. a b Proof: Assume that CD and AB lie in the same plane. Let this be the plane β. Prove that AB Crosses with CD A B C D α coincides with β The planes coincide, which cannot be, because line CD intersects α. There is no plane to which AB and CD belong, and therefore, by the definition of skew lines, AB is skew to CD. Ch.t.d.
Slide 7
Consolidation of the studied theorem: Determine the relative position of the lines AB1 and DC. 2. Indicate the relative position of the line DC and the plane AA1B1B 3. Is the line AB1 parallel to the plane DD1C1C?
Slide 8
Theorem: Through each of the two intersecting lines there passes a plane parallel to the other plane, and moreover, only one. Given: AB crosses with CD. Construct α: AB α, CD || a. A B C D Through point A draw a line AE, AE || CD. E 2. The lines AB and AE intersect and form the plane α. AB α, CD || a. α is the only plane. Prove that α is unique. 3. Proof: α is unique by the corollary of the axioms. Any other plane to which AB belongs intersects AE and hence the line CD.
Slide 9
Task. Construct a plane α passing through the point K and parallel to the intersecting lines a and b. Construction: Through the point K draw a straight line a1 || A. 2. Through the point K draw a line b1 || b. a b К а1 b1 3. Draw the plane α through the intersecting lines. α is the desired plane.
- 1. Parallel lines
- 2. Intersecting lines
- 3. Crossing straight lines
- 1) Parallel lines are straight lines that lie in the same plane and either coincide or do not intersect.
- 2) Signs of Parallelism:
- I. Two lines parallel to a third are parallel.
- II. If interior cross-lying angles are equal, then the lines are parallel.
- III. If the sum of one-sided interior angles is 180°, then the lines are parallel.
- IV. If the corresponding angles are equal, then the lines are parallel.
- Two lines are said to intersect if they have a common point.
- Lines are called intersecting if one of the lines lies in a plane, and the other intersects this plane at a point not belonging to the first line.
- 1) Parallel planes
- 2) Intersecting planes
- Planes that do not have common points are called parallel.
- Planes are said to intersect if they have common points
- A line and a plane are said to be parallel if they do not intersect and have no common points.
- A plane and a line are said to intersect if they have a common point of intersection.
- A line intersecting a plane is said to be perpendicular to that plane if it is perpendicular to every line that lies in the given plane and passes through the point of intersection.
Answer the questions:
Yes
- Can a line and a plane have no common points?
- Is it true that if two lines do not intersect, then they are parallel?
- planes α And β are parallel, the line m lies in the plane α . Is it true that the line m is parallel to the plane β ?
- Is it true that if the line a is parallel to one of the two parallel planes, the line a has one common point with the other plane?
- Is it true that planes are parallel if a line lying in one plane is parallel to another plane?
No
Yes
No
No
Problem solving
points E, F,M,N - the middle of the ribs.
1). Prove: EF ll MN ;
2). Determine the relative position of the lines DC And AB
Given: α || β
AO = 5,
OB = 4,
OA 1 = 3,
A 1 IN 1 = 6.
Find: AB and OB 1
A 1
B 1
Parallelepiped ABCDA 1 B 1 C 1 D 1
№ 6
B 1
C 1
The section passes through the points M , N and P lying on the edges BC , AD and AA 1 respectively.
A 1
D 1
Tetrahedron DABC
№ 2
The section passes through the point M lying on the edge DA parallel to the face ABC .
Find: the cross-sectional area of a tetrahedron with an edge equal to 3 cm, if the point M is the midpoint of the edge YES.
Determine the relative position of the lines.
B 1
C 1
D 1
A 1
B 1
C 1
A 1
D 1
C 1
B 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
Determine the relative position of lines and planes.
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
Determine the relative position of the planes.
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
B 1
C 1
D 1
A 1
- Crossbreed.
- Intersect.
- Parallel.
- Crossbreed.
- Intersect.
- Parallel.
- Intersect.
- Intersect.
- Parallel.
- Parallel.
- Intersect.
- Parallel.
- Homework:
- 1. prep. to the test pp. 35-36 "Test yourself"
Mutual arrangement of lines in space Three cases of mutual arrangement of two lines in space are possible: - lines intersect, i.e. have only one common point - the lines are parallel, i.e. lie in the same plane and do not intersect - the lines intersect, i.e. do not lie in the same plane
A 2 If two points of a line lie in a plane, then all points of the line lie in this plane. The property expressed in axiom A 2 is used to check the "evenness" of the drawing ruler. For this purpose, the ruler is applied with an edge to the flat surface of the table. If the edge of the ruler is even (rectilinear), then it is adjacent to the surface of the table with all its points. If the edge is uneven, then in some places a gap is formed between them and the table surface.
A3 If two planes have a common point, then they have a common straight line on which all common points of these planes lie. In this case, the planes are said to intersect in a straight line. A clear illustration of Axiom A3 is the intersection of two adjacent walls, the wall and the ceiling of the classroom.
Parallelism of a line and a plane If two points of a line lie in a given plane, then according to A2 the entire line lies in this plane. It follows that there are three cases of mutual arrangement of a straight line and a plane in space: a) the straight line lies on the plane b) the straight line and the plane have one common point, that is, they intersect c) the straight line and the plane do not have a single common point
Parallelism of planes So, we know that if two planes have a common point, then they intersect in a straight line (axiom A3). It follows from this that two planes either intersect in a straight line, or do not intersect, i.e., do not have a single common point. Definition Two planes are said to be parallel if they do not intersect. The idea of parallel planes is given by the floor and ceiling of the room, two opposite walls, the surface of the table and the plane of the floor.
Theorem If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel. Proof Consider two planes and β. The lines a and b intersecting at the point M lie in the plane, and the lines a 1 and b 1 lie in the plane β, and a a 1 and b 1. Let us prove that β. First of all, we note that on the basis of parallelism of a straight line and a plane, a β and b β. Assume that the planes and β are not parallel. Then they intersect along some line c. We have obtained that the plane passes through the line a, parallel to the plane β, and intersects the plane β in a straight line. From this it follows (by property 1 0) that the lines a and c are parallel. But the plane also passes through the line b parallel to the plane β. Therefore b s. Thus, two lines a and b pass through point M and are parallel to line c. But this is impossible, since, according to the theorem on parallel lines, only one line passes through the point M, parallel to the line c. Hence, our assumption is wrong and, therefore, β. The theorem has been proven.
