Dihedral angle, perpendicular to the plane. Dihedral angle

CHAPTER ONE LINES AND PLANES

V. DIHEDRAL ANGLES, A RIGHT ANGLE WITH A PLANE,
ANGLE OF TWO CROSSING RIGHTS, POLYHEDRAL ANGLES

dihedral angles

38. Definitions. The part of a plane lying on one side of a line lying in that plane is called half-plane. The figure formed by two half-planes (P and Q, Fig. 26) emanating from one straight line (AB) is called dihedral angle. The straight line AB is called edge, and the half-planes P and Q - parties or faces dihedral angle.

Such an angle is usually denoted by two letters placed at its edge (dihedral angle AB). But if there are no dihedral angles at one edge, then each of them is denoted by four letters, of which two middle ones are at the edge, and two extreme ones are at the faces (for example, the dihedral angle SCDR) (Fig. 27).

If from an arbitrary point D we draw the edges AB (Fig. 28) on each face along the perpendicular to the edge, then the angle CDE formed by them is called linear angle dihedral angle.

The value of a linear angle does not depend on the position of its vertex on the edge. Thus, the linear angles CDE and C 1 D 1 E 1 are equal because their sides are respectively parallel and equally directed.

The plane of a linear angle is perpendicular to the edge because it contains two lines perpendicular to it. Therefore, to obtain a linear angle, it is sufficient to intersect the faces of a given dihedral angle with a plane perpendicular to the edge, and consider the angle obtained in this plane.

39. Equality and inequality of dihedral angles. Two dihedral angles are considered equal if they can be combined when nested; otherwise, one of the dihedral angles is considered to be smaller, which will form part of the other angle.

Like angles in planimetry, dihedral angles can be adjacent, vertical etc.

If two adjacent dihedral angles are equal to each other, then each of them is called right dihedral angle.

Theorems. 1) Equal dihedral angles correspond to equal linear angles.

2) A larger dihedral angle corresponds to a larger linear angle.

Let PABQ, and P 1 A 1 B 1 Q 1 (Fig. 29) be two dihedral angles. Embed the angle A 1 B 1 into the angle AB so that the edge A 1 B 1 coincides with the edge AB and the face P 1 with the face P.

Then if these dihedral angles are equal, then face Q 1 will coincide with face Q; if the angle A 1 B 1 is less than the angle AB, then the face Q 1 will take some position inside the dihedral angle, for example Q 2 .

Noticing this, we take some point B on a common edge and draw a plane R through it, perpendicular to the edge. From the intersection of this plane with the faces of dihedral angles, linear angles are obtained. It is clear that if the dihedral angles coincide, then they will have the same linear angle CBD; if the dihedral angles do not coincide, if, for example, the face Q 1 takes position Q 2, then the larger dihedral angle will have a larger linear angle (namely: / CBD > / C2BD).

40. Inverse theorems. 1) Equal linear angles correspond to equal dihedral angles.

2) A larger linear angle corresponds to a larger dihedral angle .

These theorems are easily proven by contradiction.

41. Consequences. 1) A right dihedral angle corresponds to a right linear angle, and vice versa.

Let (Fig. 30) the dihedral angle PABQ be a right one. This means that it is equal to the adjacent angle QABP 1 . But in this case, the linear angles CDE and CDE 1 are also equal; and since they are adjacent, each of them must be straight. Conversely, if the adjacent linear angles CDE and CDE 1 are equal, then the adjacent dihedral angles are also equal, i.e., each of them must be right.

2) All right dihedral angles are equal, because they have equal linear angles .

Similarly, it is easy to prove that:

3) Vertical dihedral angles are equal.

4) Dihedral angles with correspondingly parallel and equally (or oppositely) directed faces are equal.

5) If we take as a unit of dihedral angles such a dihedral angle that corresponds to a unit of linear angles, then we can say that a dihedral angle is measured by its linear angle.

The purpose of the lesson: introduction of the concept of a dihedral angle and its linear angle.

Tasks:

Educational: to consider tasks for the application of these concepts, to form a constructive skill of finding the angle between planes;

Developing: development of creative thinking of students, personal self-development of students, development of students' speech;

Educational: education of the culture of mental work, communicative culture, reflective culture.

Lesson type: a lesson in learning new knowledge

Teaching methods: explanatory and illustrative

Equipment: computer, interactive whiteboard.

Literature:

Geometry. Grades 10-11: textbook. for 10-11 cells. general education institutions: basic and profile. levels / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev and others] - 18th ed. - M. : Education, 2009. - 255 p.

Lesson plan:

Organizational moment (2 min)

Updating knowledge (5 min)

Learning new material (12 min)

Consolidation of the studied material (21 min)

Homework (2 min)

Summing up (3 min)

During the classes:

1. Organizational moment.

Includes a greeting by the teacher of the class, preparation of the room for the lesson, checking absentees.

2. Actualization of basic knowledge.

Teacher: In the last lesson, you wrote an independent work. In general, the work was well written. Now let's repeat a little. What is called an angle on a plane?

Student: An angle in a plane is a figure formed by two rays emanating from one point.

Teacher: What is the angle between lines in space called?

Student: The angle between two intersecting lines in space is the smallest of the angles formed by the rays of these lines with the vertex at the point of their intersection.

Student: The angle between intersecting lines is the angle between intersecting lines, respectively, parallel to the data.

Teacher: What is the angle between a line and a plane called?

Student: Angle between line and planeAny angle between a straight line and its projection onto this plane is called.

3. Study of new material.

Teacher: In stereometry, along with such angles, another type of angles is considered - dihedral angles. You probably already guessed what the topic of today's lesson is, so open your notebooks, write down today's date and the topic of the lesson.

Writing on the board and in notebooks:

10.12.14.

Dihedral angle.

Teacher : To introduce the concept of a dihedral angle, it should be recalled that any straight line drawn in a given plane divides this plane into two half-planes(Fig. 1a)

Teacher : Let's imagine that we have bent the plane along a straight line so that two half-planes with the boundary turned out to be no longer lying in the same plane (Fig. 1, b). The resulting figure is the dihedral angle. A dihedral angle is a figure formed by a straight line and two half-planes with a common boundary that do not belong to the same plane. The half-planes forming a dihedral angle are called its faces. A dihedral angle has two faces, hence the name - dihedral angle. The straight line - the common boundary of the half-planes - is called the edge of the dihedral angle. Write the definition in your notebook.

A dihedral angle is a figure formed by a straight line and two half-planes with a common boundary that do not belong to the same plane.

Teacher : In everyday life, we often encounter objects that have the shape of a dihedral angle. Give examples.

Student : Half open folder.

Student : The wall of the room together with the floor.

Student : Gable roofs of buildings.

Teacher : Correctly. And there are many such examples.

Teacher : As you know, angles on a plane are measured in degrees. You probably have a question, but how are dihedral angles measured? This is done in the following way.We mark some point on the edge of the dihedral angle, and in each face from this point we draw a ray perpendicular to the edge. The angle formed by these rays is called the linear angle of the dihedral angle. Make a drawing in your notebooks.

Writing on the board and in notebooks.

O ∈ a, AO ⊥ a, VO ⊥ a, SABD- dihedral angle,∠ AOBis the linear angle of the dihedral angle.

Teacher : All linear angles of a dihedral angle are equal. Make yourself something like this.

Teacher : Let's prove it. Consider two linear angles AOB andPQR. Rays OA andQPlie on the same face and are perpendicularOQ, which means they are aligned. Similarly, the rays OB andQRco-directed. Means,∠ AOB= ∠ PQR(like angles with codirectional sides).

Teacher : Well, now the answer to our question is how the dihedral angle is measured.The degree measure of a dihedral angle is the degree measure of its linear angle. Redraw the drawings of an acute, right, and obtuse dihedral angle from the textbook on page 48.

4. Consolidation of the studied material.

Teacher : Make drawings for tasks.

№ 1 . Given: ΔABC, AC = BC, AB lies in the planeα, CD ⊥ α, C∉ a. Construct Linear Angle of Dihedral AngleCABD.

Student : Solution:CM ⊥ AB, DC ⊥ AB.∠ cmd - desired.

№ 2. Given: ΔABC, ∠ C= 90°, BC lies planeα, AO⊥ α, A∈ α.

Construct Linear Angle of Dihedral AngleAVSO.

Student : Solution:AB ⊥ BC, JSC⊥ Sun means OS⊥ Sun.∠ ACO - desired.

№ 3 . Given: ΔABC, ∠ C \u003d 90 °, AB lies in the planeα, CD⊥ α, C∉ a. Buildlinear dihedral angleDABC.

Student : Solution: CK ⊥ AB, DC ⊥ AB,DK ⊥ AB means∠ DKC - desired.

№ 4 . Given:DABC- tetrahedron,DO⊥ ABC.Construct the linear angle of the dihedral angleABCD.

Student : Solution:DM ⊥ sun,DO ⊥ BC means OM⊥ sun;∠ OMD - desired.

5. Summing up.

Teacher: What new did you learn at the lesson today?

Students : What is called dihedral angle, linear angle, how dihedral angle is measured.

Teacher : What did you repeat?

Students : What is called an angle on a plane; angle between lines.

6. Homework.

Writing on the board and in the diaries: item 22, no. 167, no. 170.

TEXT EXPLANATION OF THE LESSON:

In planimetry, the main objects are lines, segments, rays and points. Rays emanating from one point form one of their geometric shapes - an angle.

We know that a linear angle is measured in degrees and radians.

In stereometry, a plane is added to objects. The figure formed by a straight line a and two half-planes with a common boundary a that do not belong to the same plane in geometry is called a dihedral angle. Half planes are the faces of a dihedral angle. The straight line a is the edge of the dihedral angle.

A dihedral angle, like a linear angle, can be named, measured, built. This is what we are going to find out in this lesson.

Find the dihedral angle on the ABCD tetrahedron model.

A dihedral angle with an edge AB is called CABD, where C and D points belong to different faces of the angle and the edge AB is called in the middle

Around us there are a lot of objects with elements in the form of a dihedral angle.

In many cities, special benches for reconciliation have been installed in parks. The bench is made in the form of two inclined planes converging towards the center.

In the construction of houses, the so-called gable roof is often used. The roof of this house is made in the form of a dihedral angle of 90 degrees.

The dihedral angle is also measured in degrees or radians, but how to measure it.

It is interesting to note that the roofs of the houses lie on the rafters. And the crate of the rafters forms two roof slopes at a given angle.

Let's transfer the image to the drawing. In the drawing, to find a dihedral angle, point B is marked on its edge. From this point, two beams BA and BC are drawn perpendicular to the edge of the angle. The angle ABC formed by these rays is called the linear angle of the dihedral angle.

The degree measure of a dihedral angle is equal to the degree measure of its linear angle.

Let's measure the angle AOB.

The degree measure of a given dihedral angle is sixty degrees.

Linear angles for a dihedral angle can be drawn in an infinite number, it is important to know that they are all equal.

Consider two linear angles AOB and A1O1B1. The rays OA and O1A1 lie in the same face and are perpendicular to the straight line OO1, so they are co-directed. Rays OB and O1B1 are also co-directed. Therefore, the angle AOB is equal to the angle A1O1B1 as angles with codirectional sides.

So a dihedral angle is characterized by a linear angle, and linear angles are acute, obtuse and right. Consider models of dihedral angles.

An obtuse angle is one whose linear angle is between 90 and 180 degrees.

A right angle if its linear angle is 90 degrees.

An acute angle, if its linear angle is between 0 and 90 degrees.

Let us prove one of the important properties of a linear angle.

The plane of a linear angle is perpendicular to the edge of the dihedral angle.

Let the angle AOB be the linear angle of the given dihedral angle. By construction, the rays AO and OB are perpendicular to the straight line a.

The plane AOB passes through two intersecting lines AO and OB according to the theorem: A plane passes through two intersecting lines and, moreover, only one.

The line a is perpendicular to two intersecting lines lying in this plane, which means that, by the sign of the perpendicularity of the line and the plane, the line a is perpendicular to the plane AOB.

To solve problems, it is important to be able to build a linear angle of a given dihedral angle. Construct the linear angle of the dihedral angle with the edge AB for the tetrahedron ABCD.

We are talking about a dihedral angle, which is formed, firstly, by the edge AB, one facet ABD, the second facet ABC.

Here is one way to build.

Let's draw a perpendicular from point D to the plane ABC, mark the point M as the base of the perpendicular. Recall that in a tetrahedron the base of the perpendicular coincides with the center of the inscribed circle in the base of the tetrahedron.

Draw a slope from point D perpendicular to edge AB, mark point N as the base of the slope.

In the triangle DMN, the segment NM will be the projections of the oblique DN onto the plane ABC. According to the three perpendiculars theorem, the edge AB will be perpendicular to the projection NM.

This means that the sides of the angle DNM are perpendicular to the edge AB, which means that the constructed angle DNM is the required linear angle.

Consider an example of solving the problem of calculating the dihedral angle.

Isosceles triangle ABC and regular triangle ADB do not lie in the same plane. The segment CD is perpendicular to the plane ADB. Find the dihedral angle DABC if AC=CB=2cm, AB=4cm.

The dihedral angle DABC is equal to its linear angle. Let's build this corner.

Let's draw an oblique CM perpendicular to the edge AB, since the triangle ACB is isosceles, then the point M will coincide with the midpoint of the edge AB.

The line CD is perpendicular to the plane ADB, which means it is perpendicular to the line DM lying in this plane. And the segment MD is the projection of the oblique SM onto the plane ADB.

The line AB is perpendicular to the oblique CM by construction, which means that by the three perpendiculars theorem it is perpendicular to the projection MD.

So, two perpendiculars CM and DM are found to the edge AB. So they form a linear angle СMD of a dihedral angle DABC. And it remains for us to find it from the right triangle СDM.

Since the segment SM is the median and the height of the isosceles triangle ASV, then according to the Pythagorean theorem, the leg of the SM is 4 cm.

From a right triangle DMB, according to the Pythagorean theorem, the leg DM is equal to two roots of three.

The cosine of an angle from a right triangle is equal to the ratio of the adjacent leg MD to the hypotenuse CM and is equal to three roots of three by two. So the angle CMD is 30 degrees.

Your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please read our privacy policy and let us know if you have any questions.

Collection and use of personal information

Personal information refers to data that can be used to identify or contact a specific person.

You may be asked to provide your personal information at any time when you contact us.

The following are some examples of the types of personal information we may collect and how we may use such information.

What personal information we collect:

• When you submit an application on the site, we may collect various information, including your name, phone number, email address, etc.

How we use your personal information:

• The personal information we collect allows us to contact you and inform you about unique offers, promotions and other events and upcoming events.
• From time to time, we may use your personal information to send you important notices and communications.
• We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
• If you enter a prize draw, contest or similar incentive, we may use the information you provide to administer such programs.

Disclosure to third parties

We do not disclose information received from you to third parties.

Exceptions:

• In the event that it is necessary - in accordance with the law, judicial order, in legal proceedings, and / or based on public requests or requests from state bodies in the territory of the Russian Federation - disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public interest purposes.
• In the event of a reorganization, merger or sale, we may transfer the personal information we collect to the relevant third party successor.

Protection of personal information

We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as from unauthorized access, disclosure, alteration and destruction.

Maintaining your privacy at the company level

To ensure that your personal information is secure, we communicate privacy and security practices to our employees and strictly enforce privacy practices.

The angle between two different planes can be determined for any relative position of the planes.

The trivial case is if the planes are parallel. Then the angle between them is considered equal to zero.

Non-trivial case if the planes intersect. This case is the subject of further discussion. First we need the concept of a dihedral angle.

9.1 Dihedral angle

A dihedral angle is two half-planes with a common straight line (which is called an edge of a dihedral angle). On fig. 50 shows a dihedral angle formed by half-planes and; the edge of this dihedral angle is the line a common to the given half-planes.

Rice. 50. Dihedral angle

The dihedral angle can be measured in degrees or radians in a word, enter the angular value of the dihedral angle. This is done in the following way.

On the edge of the dihedral angle formed by the half-planes and, we take an arbitrary point M. Let's draw the rays MA and MB, lying respectively in these half-planes and perpendicular to the edge (Fig. 51).

Rice. 51. Linear angle dihedral angle

The resulting angle AMB is the linear angle of the dihedral angle. The angle " = \AMB is precisely the angular value of our dihedral angle.

Definition. The angular magnitude of a dihedral angle is the magnitude of the linear angle of a given dihedral angle.

All linear angles of a dihedral angle are equal to each other (after all, they are obtained from each other by a parallel shift). Therefore, this definition is correct: the value "does not depend on the specific choice of the point M on the edge of the dihedral angle.

9.2 Determining the angle between planes

When two planes intersect, four dihedral angles are obtained. If they all have the same value (90 each), then the planes are called perpendicular; the angle between the planes is then 90 .

If not all dihedral angles are the same (that is, there are two acute and two obtuse), then the angle between the planes is the value of the acute dihedral angle (Fig. 52).

Rice. 52. Angle between planes

9.3 Examples of problem solving

Let's consider three tasks. The first is simple, the second and third are approximately at the level of C2 on the exam in mathematics.

Task 1. Find the angle between two faces of a regular tetrahedron.

Solution. Let ABCD be a regular tetrahedron. Let's draw the medians AM and DM of the corresponding faces, as well as the height of the tetrahedron DH (Fig. 53).

Rice. 53. To problem 1

Being medians, AM and DM are also the heights of equilateral triangles ABC and DBC. Therefore, the angle " = \AMD is the linear angle of the dihedral angle formed by the faces ABC and DBC. We find it from the triangle DHM:

Answer: arccos 1 3 .

Problem 2. In a regular quadrangular pyramid SABCD (with vertex S), the side edge is equal to the side of the base. Point K is the midpoint of edge SA. Find the angle between planes

Solution. Line BC is parallel to AD and thus parallel to plane ADS. Therefore, the plane KBC intersects the plane ADS along the straight line KL parallel to BC (Fig. 54).

Rice. 54. To problem 2

In this case, KL will also be parallel to the line AD; hence KL is the midline of triangle ADS, and point L is the midpoint of DS.

Draw the height of the pyramid SO. Let N be the midpoint of DO. Then LN is the midline of the triangle DOS, and therefore LN k SO. So LN is perpendicular to the plane ABC.

From the point N we drop the perpendicular NM to the line BC. The straight line NM will be the projection of the oblique LM onto the plane ABC. The three-perpendicular theorem then implies that LM is also perpendicular to BC.

Thus, the angle " = \LMN is a linear angle of the dihedral angle formed by the half-planes KBC and ABC. We will look for this angle from the right triangle LMN.

Let the edge of the pyramid be a. First, find the height of the pyramid:

Solution. Let L be the intersection point of lines A1 K and AB. Then the plane A1 KC intersects the plane ABC along the straight line CL (Fig.55).

A C

Rice. 55. Problem 3

Triangles A1 B1 K and KBL are equal in leg and acute angle. Therefore, the other legs are also equal: A1 B1 = BL.

Consider triangle ACL. In it BA = BC = BL. The CBL angle is 120 ; so \BCL = 30 . Also, \BCA = 60 . Therefore \ACL = \BCA + \BCL = 90 .

So LC? AC. But the line AC is the projection of the line A1 C onto the plane ABC. By the three perpendiculars theorem, we then conclude that LC ? A1C.

Thus, the angle A1 CA is the linear angle of the dihedral angle formed by the half-planes A1 KC and ABC. This is the required angle. From the isosceles right triangle A1 AC we see that it is equal to 45 .