GEOMETRY
Lesson plans for 10th grade

Subject. Properties of a straight line and a plane perpendicular to each other

Purpose of the lesson: to develop students' knowledge about the properties of perpendicular lines and planes.

Equipment: stereometric set, diagram “Properties of straight lines and planes perpendicular to each other” (p. 116).

During the classes

I. Checking homework

1. Group discussion solutions to problem No. 10.

2. Mathematical dictation.

An image of a cube is given: option 1 - fig. 151, option 2 - fig. 152.

Using the image, write down:

1) a plane that passes through the point M of the straight line AM and is perpendicular to it; (2 points)

2) a straight line that is perpendicular to plane ABC and passes through point D; (2 points)

3) a straight line that is perpendicular to the ABC plane and passes through point N; (2 points)

4) a plane that is perpendicular to straight line BD; (2 points)

5) straight lines, perpendicular to the AMC plane; (2 points)

6) planes that are perpendicular to the straight line DC. (2 points)

Option 1. 1) (MNK); 2) KD; 3) BN; 4) (AFM); 5) BD and KN; 6) (ADK) and (BCL).

Option 2. 1) (MNK); 2) DL; 3) CN; 4) (AFM); 5) BD i KL; 6) (BCN) and (ADM).

II. Perception and awareness of new material

Properties of a straight line and a plane perpendicular to each other

Theorem 1.

If a plane is perpendicular to one of two parallel lines, then it is also perpendicular to the second.

Finishing

Let a1 || a2 and a1α. Let us prove that αа2 (Fig. 153). Points A1 and A2 are the points of intersection of a1 and a2 with the plane α.

In the plane α, through point A2 we draw an arbitrary straight line x2, and through point A1 - a straight line x1 such that x1 || x2. Since a1 || a2, x1 || x2 and a1x1, then by Theorem 3.1 a2x2. Since x2 is chosen arbitrarily in the α plane, then a2α.

Theorem 2.

If two lines are perpendicular to the same plane, then the lines are parallel.

Finishing

Let aα, b α . Let us prove that a || b (Fig. 154). Let's assume that ab . Then through point C of line b we draw b 1 parallel to a. And since α , then b1α by the proven theorem, and by condition bα . If points A and B are the points of intersection of lines b 1 and b with the plane α, then it follows from the assumption that in the triangle A = B = 90°, which cannot be. Therefore, a || b.

Problem solving

1. Determine the type of quadrilateral AA 1B 1B if:

a) AA1α; AA1 || BB1; Aα, Bα; AA 1 ≠ BB1 (Fig. 155);

b) AA1α; BB1α; α, Bα (Fig. 156);

c) α; α; AA1α; BB1α; AA1 = BB1 (Fig. 156).

2. Problem No. 12 from the textbook (p. 35).

3. Problem No. 13 from the textbook (p. 35).

4. Problem No. 16 from the textbook (p. 35).

Theorem 3.

If a line is perpendicular to one of two parallel planes, then it is also perpendicular to the second.

Finishing

Let α || β, aα. Let us prove that α β . (Fig. 157). Let points A and B be the points of intersection of straight line a with planes α and β. In the β plane, draw an arbitrary straight line b through point B. Through line b and point A we draw a plane γ, which intersects α along the line c, and with || b. And since α, then ac (by definition of a line perpendicular to the plane). So ac, b || c and a, b, c lie in γ, then ab. Considering that b is an arbitrary straight line of the plane β, we have aβ.

Theorem 4.

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

Finishing

Let α and β a, let us prove that α || β (Fig. 158). Let points A and B be the points of intersection of straight line a with planes α and β. Let us assume that α β .

Problem solving

Let's take point C on the line of intersection of planes α and β. Ca, because otherwise two different planes α and β would pass through point C, perpendicular to straight line a, which is impossible. Let us draw a plane γ through point C and straight line a; this plane intersects α and β along straight lines AC and BC, respectively. And since α, then aAC, similar to aBC. Consequently, in the plane α, two different straight lines AC and BC pass through point C, perpendicular to straight line a, which is impossible. Therefore α || β.

1. Let ABCD be a rectangle, BSAB, AMAB (Fig. 159). How are the AMD and BSC planes located?

2. B1β; AA1α, AA1β; B B1 || AA1; AA1 = 12 cm, A1B = 13 cm (Fig. 160). Find AB.

Lesson 3.2.1

Perpendicularity of lines.

Perpendicular and oblique.

Theorem of three perpendiculars. Definition:

Two lines in space are called perpendicular (mutually perpendicular) if the angle between them is 90 degrees.. .

Designation Consider the straight lines A And.

Lines can intersect, cross, or be parallel. In order to construct an angle between them, you need to select a point and draw a straight line a` through it, parallel to the straight line A, and a line b` parallel to the line And.

Lines a` and b` intersect. The angle between them is the angle between the lines Consider the straight lines And b. If the angle is 90°, then straight a and b perpendicular.

Lemma: If one of two lines is perpendicular to the third line, then the other line is also perpendicular to this line.

Proof:

Let's take an arbitrary point M. Through the point M draw a line a` parallel to the line Consider the straight lines and a line c` parallel to the line c. Then the angle AMS equals 90°.

Straight And parallel to the line Consider the straight lines by condition, line a` is parallel to line Consider the straight lines by construction. This means that the straight lines a` and And parallel.

We have, straight and And parallel, straight With and parallel in construction. So, the angle between the lines And And With - this is the angle between lines a` and b`, that is, the angle AMS, equal to 90°. So it's straight And And With are perpendicular, which is what needed to be proven.

Perpendicularity of a line and a plane.

Definition: A line is called perpendicular to a plane if it is perpendicular to any line lying in this plane.

Property: If a line is perpendicular to a plane, then it intersects this plane.

(If a┴ α, then a∩ α.)

Reminder. A straight line and a plane either intersect at one point, or are parallel, or the straight line lies in the plane.

Properties of perpendicular lines and planes:

Theorem: If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to this plane.

In the first lesson, we studied the Lemma - if one of the parallel lines intersects the plane, then the other parallel line intersects the plane. Straight Consider the straight lines intersects at an angle of 90 0, that is, perpendicular, then the other parallel line is perpendicular

Theorem: If two lines are perpendicular to a plane, then they are parallel.

Sign of perpendicularity of a line and a plane

Theorem: If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to the plane

Theorem: Through any point in space there passes a straight line perpendicular to a given plane and, moreover, only one.

We constantly see that perpendiculars to the same plane are parallel. For example, vertical segments are parallel to each other. These segments can be represented by parallel pillars or masts, trunks of slender pine trees in a ship's forest, columns of museum buildings (Fig. 84) or vertical supports of a bridge, etc.

Rice. 84

This elegant geometry is expressed in a theorem that we will now prove.

8.1 Parallelism of lines perpendicular to one plane

Proof. Let two straight lines a and b be perpendicular to plane a and intersect it, respectively, at points A and B (Fig. 85). Let us draw a plane p through line a and point B and show that line b also lies in the plane β.

Rice. 85

In plane a, take a segment MN perpendicular to segment AB and having point A as its midpoint. Since AM = AN and AB ⊥ MN, then BM = BN.

Take any point C ≠ B on line b and draw segments CA, CM, CN. Since b ⊥ a, then the triangles CBM and CBN are right-angled. They are equal because they have a common side CB and equal sides BM and BN. Therefore CM = CN, i.e. triangle CMN is isosceles. Its median CA is also its height, i.e. CA ⊥ MN.

So, three lines passing through point A - AC, AB and a - are perpendicular to the line MN. According to the theorem on the plane of perpendiculars (Section 7.2), they lie in the same plane - the plane β, which passes through the lines AB and a.

Since the straight line AC lies in the plane β, then point C ∈ β. This means that straight line b lies in the β plane (like straight line a). But in the plane β, lines a and b are perpendicular to the same line AB (since a ⊥ α, then b ⊥ α and line AB lies in α). Therefore b||a.

The proven theorem is a sign of parallelism of lines in space.

8.2 Parallel to perpendicular

In this section we will prove the theorem, converse of the theorem on the parallelism of perpendiculars.

Proof. Let two straight lines a and b be parallel and a perpendicular to plane a (Fig. 86). The straight line b intersects the plane α at some point B (by Lemma 3.3). There are two possibilities:

1. b ⊥ α;
2. b is not perpendicular to α.

Rice. 86

Let's assume the second one is running. Then we draw a line with ⊥ α through point B (problem section 7.3). By the theorem on the parallelism of perpendiculars with||α. It turns out that through point B there are two straight lines parallel to straight line A, which is impossible.

So b ⊥ α.

The parallel to perpendicular theorem is another sign of the perpendicularity of a line and a plane.

Questions for self-control

1. What signs of parallel lines have you learned?
2. What signs of perpendicularity of a line and a plane do you now know?

Perpendicularity in space can have:

1. Two straight lines

3. Two planes

Let's look at these three cases in turn: all the definitions and statements of theorems related to them. And then we will discuss the very important theorem about three perpendiculars.

Perpendicularity of two lines.

Theorem of three perpendiculars.

You can say: they discovered America too for me! But remember that in space everything is not quite the same as on a plane.

On a plane, only the following lines (intersecting) can be perpendicular:

But two straight lines can be perpendicular in space even if they do not intersect. Look:

a straight line is perpendicular to a straight line, although it does not intersect with it. How so? Let us recall the definition of the angle between straight lines: to find the angle between intersecting lines and, you need to draw a straight line through an arbitrary point on line a. And then the angle between and (by definition!) will be equal to angle between and.

Do you remember? Well, in our case, if the straight lines and turn out to be perpendicular, then we must consider the straight lines and to be perpendicular.

For complete clarity, let's look at example. Let there be a cube. And you are asked to find the angle between the lines and. These lines do not intersect - they intersect. To find the angle between and, let's draw.

Due to the fact that it is a parallelogram (and even a rectangle!), it turns out that. And due to the fact that it is a square, it turns out that. Well, that means.

Perpendicularity of a line and a plane.

Theorem of three perpendiculars.

Here's a picture:

a straight line is perpendicular to a plane if it is perpendicular to all, all straight lines in this plane: and, and, and, and even! And a billion other direct ones!

Yes, but how then can you generally check perpendicularity in a straight line and in a plane? So life is not enough! But fortunately for us, mathematicians saved us from the nightmare of infinity by inventing sign of perpendicularity of a line and a plane.

We formulate:

Rate how great it is:

if there are only two straight lines (and) in the plane to which the straight line is perpendicular, then this straight line will immediately turn out to be perpendicular to the plane, that is, to all straight lines in this plane (including some straight line standing on the side). This is a very important theorem, so we will also draw its meaning in the form of a diagram.

And let's look again example.

Let us be given a regular tetrahedron.

Task: prove that. You will say: these are two straight lines! What does the perpendicularity of a straight line and a plane have to do with it?!

But look:

let's mark the middle of the edge and draw and. These are the medians in and. Triangles are regular and...

Here it is, a miracle: it turns out that, since and. And further, to all straight lines in the plane, which means and. They proved it. And the most important point was precisely the use of the sign of perpendicularity of a line and a plane.

When the planes are perpendicular

Definition:

That is (for more details, see the topic “ dihedral angle"") two planes (and) are perpendicular if it turns out that the angle between the two perpendiculars (and) to the line of intersection of these planes is equal. And there is a theorem that connects the concept of perpendicular planes with the concept of perpendicularity in the space of a line and a plane.

This theorem is called

Criterion for the perpendicularity of planes.

Let's formulate:

As always, the decoding of the words “then and only then” looks like this:

• If, then passes through the perpendicular to.

• If it passes through the perpendicular to, then.

(naturally, here we are planes).

This theorem is one of the most important in stereometry, but, unfortunately, also one of the most difficult to apply.

So you need to be very careful!

So, the wording:

And again deciphering the words “then and only then.” The theorem states two things at once (look at the picture):

let's try to apply this theorem to solve the problem.

Task: a regular hexagonal pyramid is given. Find the angle between the lines and.

Solution:

Due to the fact that in a regular pyramid the vertex, when projected, falls into the center of the base, it turns out that the straight line is a projection of the straight line.

But we know that in regular hexagon. We apply the theorem of three perpendiculars:

And we write the answer: .

PERPENDICULARITY OF STRAIGHT LINES IN SPACE. BRIEFLY ABOUT THE MAIN THINGS

Perpendicularity of two lines.

Two lines in space are perpendicular if there is an angle between them.

Perpendicularity of a line and a plane.

A line is perpendicular to a plane if it is perpendicular to all lines in that plane.

Perpendicularity of planes.

Planes are perpendicular if the dihedral angle between them is equal.

Criterion for the perpendicularity of planes.

Two planes are perpendicular if and only if one of them passes through the perpendicular to the other plane.

Three Perpendicular Theorem:

Well, the topic is over. If you are reading these lines, it means you are very cool.

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