Subject: geometry.

Class: 10

Teacher: Prikhodko Svetlana Ivanovna

Topic : « Parallelism of a straight line and a plane "(2 lessons of 40 minutes each)

Lesson equipment: multimedia projector, board, cards with tasks for independent work, textbook "Geometry. 10-11 grades" / L.S. Atanasyan, V.F. Butuzov, etc.

Target: introduce the concepts of parallelism of a straight line and a plane; to study the sign of parallelism of a straight line and a plane; generalize and systematize knowledge about the relative position of a straight line and a plane.

Tasks:

Create conditions for control (self-control, mutual control);

Develop spatial representations when constructing parallel lines, straight lines and planes;

To form the ability to prove the sign of parallelism of a straight line and a plane;

To develop the ability to use theoretical material in solving problems.

DURING THE CLASSES

organizational stage.

The teacher greets the students, formulates the goals and objectives of the lesson, reports the lesson plan.

Knowledge update.

Frontal work using a multimedia projector.

slide 1.

Slide 2.

3. Learning new material. (Front work.)

Slide 3.

A visual representation of a straight line parallel to a plane is given by:

Power lines and ground plane;

The line of intersection of the ceiling and walls and the plane of the floor.

slide 4.

Consider the theorem (a sign of parallelism of a straight line and a plane).

If a line not lying in a given plane is parallel to some line lying in this plane, then it is parallel to the given plane.

a Given: the line in lies in the plane α.

a║c

Prove: a║α

(Proof of the theorem should be done by students on their own, discussed, offered to prove at the blackboard, written in a notebook. If you find it difficult, you can press the button clue to proof.)

4. Consolidation of the studied material.

Orally (Front work)

Slide 5.

A task: Given a trapezoid ABCD (AB and CD bases). Point K does not belong to the plane of the trapezoid. Prove that the line DC is parallel to the plane (ABK).

By depicting: 1) a trapezoid;

2) depict a plane a;

3) depict segments VC and KS;

4) write down: given, prove.

We discuss and write down the solution to the problem.

slide 6.

We solve the problem orally.

5. Learning new things. (Work in groups of 4 people.)

Consider two statements that are used in solving problems.

Slide 7.

(Students prove by working in groups.)

Discussion of the work of groups.(During the work of the group (5-7 min.), students write down their evidence in a notebook.) The representative of the group writes down the evidence on the board. Summing up the work of the group.

6. Consolidation of the studied material.

slide 8.

slide 9.

Some words have been erased and dots added. In the course of the solution, instead of the ellipsis, the complete solution of the problem appears.

Task number 23 (textbook).

(On a regular board).

M Given: ABCD is a rectangle, point M does not lie in

plane ABC.

B C Prove: CD ║ (AVM).

BUT D

7
. Solving problems to consolidate the studied material. (Assignment with mutual verification - in pairs).

slide 10.

8. Work with the textbook.

Task number 27.(Student at the blackboard.)

9. Summing up.

Conversation with students

Describe the relationship between a line and a plane.

Which line is said to be parallel to the given plane?

Name the sign of parallelism of a straight line and a plane.

What can be said about a straight line parallel to a plane if some plane passes through it and intersects the first plane?

Continue the phrase: if one of two parallel lines is parallel to a given plane, then ...

10. Independent work(according to card options).

Option 1

Option 2

The segment AB does not intersect the plane α.

Through the ends of this segment - points A, B

and its middle (point M) are drawn

parallel lines that intersect

plane α at points A 1 , B 1 , M 1 .

Prove that the points A 1 ,B 1 ,M 1 lie

on one straight line.

2) Find AA 1 if BB 1 =12cm, MM 1 =8cm.

A plane α is drawn through the end A of the segment AB.

Through point M (midpoint AB) and point B

parallel lines are drawn that intersect

plane α at points M 1 and B 1, respectively.

1) Prove that points A, B 1 , M 1 lie

on one straight line.

2) Find BB 1 if MM 1 \u003d 4 cm.

Optional: No. 31 (textbook.)

11. Homework: theory §1 (theorems with proof), No. 29,30.

After the students have studied the topic "Parallelism of lines in space", it's time to consider the parallelism of a line with respect to a plane. This topic is also important. The theorems that will be studied in this presentation will be useful for solving various kinds of problems in stereometry. If you skip this topic, it will be difficult to understand other topics and practical tasks.

What are the straight lines with respect to the plane? Firstly, they may intersect them, secondly, they may not have any common points, and thirdly, the line may lie directly on the plane. These three cases are discussed on the first slide of this eLearning resource. There are also illustrations to them, which demonstrate all cases.

In which of these cases will the line and plane be parallel? The next slide is devoted to determining the parallelism of a straight line with respect to a plane. It is allocated in a special block and it will be easy to remember.

Since it will be necessary to use this concept quite often, the notation is given on the next page. It says that the line A is parallel to the plane alpha.

If some line is parallel to another line that lies on a plane, then the first line will be parallel directly to the plane. This is the first theorem in this presentation. To avoid any ambiguities, a simple proof is given that can be easily disassembled with a teacher or tutor. The theorem is proved by contradiction, which is a frequently used technique in many cases. Students should have gotten used to it and understood it by now.

We have a straight path and some plane that is parallel to it. If through a given line to draw an intersecting plane with an existing plane, then the line of intersection and the original line will be parallel. This statement requires proof, because it is not an axiom. The proof is not voluminous and will not make any difficulty in understanding.

If it is known that there are two parallel lines, one of which is in turn parallel to the plane, then these lines must either be parallel to each other, or one of them must lie on the plane.

You can view and analyze the presentation during the lesson with the teacher. If he correctly comments on everything, then the students will understand this lesson and remember it for a long time, there will be no problems when doing homework, writing independent and test papers.

, Competition "Presentation for the lesson"

Class: 10

Presentations for the lesson

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Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested this work please download the full version.

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Lesson type: lesson of repetition, generalization and systematization of knowledge.

The purpose of the lesson: repetition and generalization of theoretical knowledge on the topic; solving problems related to this topic, basic and advanced levels of complexity.

Methods and pedagogical techniques: conversation with elements of discussion on solving tasks; problem solving; differentiated teaching method

During the classes

1. Organizing time. Greetings. Setting the goal of the lesson.

2. Actualization of students' knowledge.

1. Theoretical survey. We use a table.

Mutual arrangement of lines in space

1.1. one student talks about the relative position of two lines in space;

1.2. the second student recalls the definition of parallel lines, intersecting lines, skew lines;

1.3. the third doctrine proves the sign of parallelism of a straight line and a plane;

1.4. the fourth student repeats the definition of parallel planes, a sign of parallel planes.

2.1. We solve problems according to the finished drawings. Presentation I. (4 slides)

Before slide IV, we repeat the theorem on angles with codirectional sides.

3. Problem solving.

3.1. As the presentation is shown, the solution to the problems is discussed orally, written down on the board and in notebooks.

Presentation II. (5 slides)

3.2. Independent problem solving.

I level

II level

3. Summing up.

Using slide 6, check the implementation of the solution to the problem of level I.

4. Homework.

In a regular tetrahedron DABC, a section parallel to the plane DBC is drawn through the midpoint of height DH. Find the cross-sectional area if the edge of the tetrahedron is

Triangle MRH is given. The plane parallel to the straight line MK intersects MP at the point M 1 , PK – at the point K 1 . Find if .

Triangle ABK is given, point M does not belong to the plane of the triangle; E, D are the points of intersection of medians of triangles MBK and ABM; AK=14cm. Prove that ADEK is a trapezoid. Find segment DE.

Literature.

1. L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev, L.S. Kiseleva, E.G. Poznyak. Geometry: a textbook for grades 10–11.
2. V.A.Yarovenko. Lesson developments in geometry: Grade 10.
3. A. Zambrzhitsky. Parallelism of a straight line and a plane: a system of lessons.
4. A.V. Beloshinskaya. Mathematics: Thematic planning of exam preparation lessons.
5. A.P. Ershova, V.V. Goloborodko, A.S. Ershov. Independent and test papers in geometry for grade 10.
6. THEM. Smirnova, V.A. Smirnov. Geometry. Distances and angles in space.
7. E.V.Potoskuev. Solving problems on stereometry. Workshop. Preparation for the exam.

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Slides captions:

Parallelism of lines and planes in space MBOU Secondary School No. 63 SHIPILOVA E.S.

cases relative position lines in space lines are parallel lines intersect lines intersect Parallel lines in space lines do not intersect

α d a b c Definition: Two lines in space are called parallel if they lie in the same plane and do not intersect. Parallelism of lines a and b is denoted as follows: a || b In the figure, lines a and b are parallel, but lines a and c, a and d are not parallel.

Parallelism of three lines Lemma: If one of the two parallel lines intersects given plane, then another line intersects this plane. a b a M

Theorem: If two lines are parallel to a third, then they are parallel. α a b c

Ways of specifying a plane ● A ● C ● B α a ● M α b a ● O α a b α

Intersecting lines Two lines are called intersecting if they do not lie in the same plane a b

α Theorem: If one of the 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 D C Assume that the lines AB and C D lie in some plane β .

Parallelism of a straight line and a plane Cases of mutual arrangement of a straight line and a plane in space a straight line lies in a plane a straight line and a plane intersect (have one common point) a straight line and a plane do not have a single common point α A B α a M a α

Definition: A line and a plane are called parallel if they have no common points. Theorem: If a line not lying in a given plane is parallel to some line lying in this plane, then it is parallel to the given plane. Prove the theorem by contradiction?

Material models of the relation of parallelism of a straight line and a plane Each edge cuboid parallel to the planes of its two faces. And the straight line drawn in the face of the bar with the help of a thickness gauge - to the planes of three faces. Masons lay the wall under a plumb line, the cord of which is parallel to the planes of the wall. If the submarine is moving in a straight line at the same depth, then it is parallel to the surface of the sea.

Prove two more statements that are often used in solving problems. If a plane passes through a given point parallel to another plane and intersects this plane, then the line of intersection of the planes is parallel to the given line. If one of two parallel lines is parallel to a given plane, then the other line is either also parallel to the given plane or lies in this plane.

Parallelism of planes Cases of mutual arrangement of planes in space planes parallel to planes intersect β α α β

Definition: Two planes are said to be parallel if they do not intersect. Theorem: If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel. Prove a theorem? α a b β c d M

Parallel planes In parallel planes, floors of floors of multi-storey buildings, glass of double windows, upper edges of stair steps are placed. Parallel layers of plywood, saws sawing a log into boards, opposite faces of a brick, channel, I-beam, etc.

Properties of Parallel Planes If two parallel planes are intersected by a third, then the lines of their intersection are parallel. Segments of parallel lines enclosed between parallel planes are equal. Prove the properties (p. 21) ?

Now for a little test! Is the statement true: if two lines have no common points, then they are parallel? The point M does not lie on the line a. How many lines not intersecting line a pass through point M? How many of these lines are parallel to line a? Lines a and c are parallel, and lines a and b intersect. Can lines b and c intersect. Can lines b and c be parallel? The line a is parallel to the plane α. Is it true that this line does not intersect any line lying in the plane α? The line a is parallel to the plane α. How many lines lying in the plane α are parallel to the line a? Are these lines parallel to each other, lying in the plane α? Can two non-parallel segments enclosed between parallel planes be equal? The two sides of the parallelogram are parallel to the plane α. Are the plane α and the plane of the parallelogram parallel?

Let's check the answers! - ∞ , 1 +,- + ∞ , + - +