What does the dihedral angle at the base mean. Dihedral angle. Complete illustrated guide (2019)

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, the edges AB (Fig. 28) are drawn 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 concept of a dihedral angle

To introduce the concept of a dihedral angle, first we recall one of the axioms of stereometry.

Any plane can be divided into two half-planes of the line $a$ lying in this plane. In this case, the points lying in the same half-plane are on the same side of the straight line $a$, and the points lying in different half-planes are on opposite sides of the straight line $a$ (Fig. 1).

Picture 1.

The principle of constructing a dihedral angle is based on this axiom.

Definition 1

The figure is called dihedral angle if it consists of a line and two half-planes of this line that do not belong to the same plane.

In this case, the half-planes of the dihedral angle are called faces, and the straight line separating the half-planes - dihedral edge(Fig. 1).

Figure 2. Dihedral angle

Degree measure of a dihedral angle

Definition 2

We choose an arbitrary point $A$ on the edge. The angle between two lines lying in different half-planes, perpendicular to the edge and intersecting at the point $A$ is called linear angle dihedral angle(Fig. 3).

Figure 3

Obviously, every dihedral angle has an infinite number of linear angles.

Theorem 1

All linear angles of one dihedral angle are equal to each other.

Proof.

Consider two linear angles $AOB$ and $A_1(OB)_1$ (Fig. 4).

Figure 4

Since the rays $OA$ and $(OA)_1$ lie in the same half-plane $\alpha $ and are perpendicular to one straight line, they are codirectional. Since the rays $OB$ and $(OB)_1$ lie in the same half-plane $\beta $ and are perpendicular to one straight line, they are codirectional. Hence

\[\angle AOB=\angle A_1(OB)_1\]

Due to the arbitrariness of the choice of linear angles. All linear angles of one dihedral angle are equal to each other.

The theorem has been proven.

Definition 3

The degree measure of a dihedral angle is the degree measure of a linear angle of a dihedral angle.

Task examples

Example 1

Let us be given two non-perpendicular planes $\alpha $ and $\beta $ which intersect along the line $m$. The point $A$ belongs to the plane $\beta $. $AB$ is the perpendicular to the line $m$. $AC$ is perpendicular to the plane $\alpha $ (point $C$ belongs to $\alpha $). Prove that the angle $ABC$ is a linear angle of the dihedral angle.

Proof.

Let's draw a picture according to the condition of the problem (Fig. 5).

Figure 5

To prove this, we recall the following theorem

Theorem 2: A straight line passing through the base of an inclined one, perpendicular to it, is perpendicular to its projection.

Since $AC$ is a perpendicular to the $\alpha $ plane, then the point $C$ is the projection of the point $A$ onto the $\alpha $ plane. Hence $BC$ is the projection of the oblique $AB$. By Theorem 2, $BC$ is perpendicular to an edge of a dihedral angle.

Then, the angle $ABC$ satisfies all the requirements for defining the linear angle of a dihedral angle.

Example 2

The dihedral angle is $30^\circ$. On one of the faces lies the point $A$, which is at a distance of $4$ cm from the other face. Find the distance from the point $A$ to the edge of the dihedral angle.

Decision.

Let's look at Figure 5.

By assumption, we have $AC=4\ cm$.

By definition of the degree measure of a dihedral angle, we have that the angle $ABC$ is equal to $30^\circ$.

Triangle $ABC$ is a right triangle. By definition of the sine of an acute angle

\[\frac(AC)(AB)=sin(30)^0\] \[\frac(5)(AB)=\frac(1)(2)\] \

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

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

Student : Decision: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.

Dihedral angle. Linear angle of a dihedral angle. A dihedral angle is a figure formed by two half-planes that do not belong to the same plane and have a common boundary - a straight line a. The half-planes that form a dihedral angle are called its faces, and the common boundary of these half-planes is called the edge of the dihedral angle. The linear angle of a dihedral angle is the angle whose sides are the rays along which the faces of the dihedral angle intersect with a plane perpendicular to the edge of the dihedral angle. Each dihedral angle has as many linear angles as desired: through each point of an edge one can draw a plane perpendicular to this edge; the rays along which this plane intersects the faces of the dihedral angle, and form linear angles.

All linear angles of a dihedral angle are equal to each other. Let us prove that if the dihedral angles formed by the plane of the base of the pyramid KABC and the planes of its side faces are equal, then the base of the perpendicular drawn from the vertex K is the center of the circle inscribed in the triangle ABC.

Proof. First of all, we construct linear angles of equal dihedral angles. By definition, the plane of a linear angle must be perpendicular to the edge of a dihedral angle. Therefore, the edge of the dihedral angle must be perpendicular to the sides of the linear angle. If KO is perpendicular to the plane of the base, then we can draw OP perpendicular to AC, OR perpendicular to CB, OQ to perpendicular AB, and then connect points P, Q, R With point K. Thus, we will construct a projection of oblique RK, QK, RK so that the edges AC, CB, AB are perpendicular to these projections. Consequently, these edges are also perpendicular to the inclined ones. And therefore the planes of the triangles ROK, QOK, ROK are perpendicular to the corresponding edges of the dihedral angle and form those equal linear angles, which are mentioned in the condition. Right-angled triangles ROK, QOK, ROK are equal (since they have a common leg OK and the angles opposite to this leg are equal). Therefore, OR = OR = OQ. If we draw a circle with center O and radius OP, then the sides of the triangle ABC are perpendicular to the radii OP, OR and OQ and therefore are tangent to this circle.

Plane perpendicularity. Planes alpha and beta are called perpendicular if the linear angle of one of the dihedral angles formed at their intersection is 90". Signs of perpendicularity of two planes If one of the two planes passes through a line perpendicular to the other plane, then these planes are perpendicular.

The figure shows a rectangular parallelepiped. Its bases are rectangles ABCD and A1B1C1D1. And the side edges AA1 BB1, CC1, DD1 are perpendicular to the bases. It follows that AA1 is perpendicular to AB, i.e., the side face is a rectangle. Thus, it is possible to substantiate the properties of a cuboid: In a cuboid, all six faces are rectangles. In a cuboid, all six faces are rectangles. All dihedral angles of a cuboid are right angles. All dihedral angles of a cuboid are right angles.

Theorem The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions. Let us turn again to the figure, And we will prove that AC12 \u003d AB2 + AD2 + AA12 Since the edge CC1 is perpendicular to the base ABCD, then the angle AC1 is right. From the right triangle ACC1, according to the Pythagorean theorem, we obtain AC12=AC2+CC12. But AC is the diagonal of the rectangle ABCD, so AC2 = AB2+AD2. Also, CC1 = AA1. Therefore, AC12=AB2+AD2+AA12 The theorem is proved.

This lesson is intended for self-study of the topic "Dihedral angle". During this lesson, students will be introduced to one of the most important geometric shapes, the dihedral angle. Also in the lesson, we have to learn how to determine the linear angle of the geometric figure under consideration and what is the dihedral angle at the base of the figure.

Let's repeat what an angle on a plane is and how it is measured.

Rice. 1. Plane

Consider the plane α (Fig. 1). From a point O two beams come out OV and OA.

Definition. The figure formed by two rays emanating from the same point is called an angle.

Angle is measured in degrees and radians.

Let's remember what a radian is.

Rice. 2. Radian

If we have a central angle whose arc length is equal to the radius, then such a central angle is called a 1 radian angle. , ∠ AOB= 1 rad (Fig. 2).

Relation between radians and degrees.

glad.

We get it, happy. (). Then,

Definition. dihedral angle called a figure formed by a straight line a and two half-planes with a common boundary a not belonging to the same plane.

Rice. 3. Half planes

Consider two half-planes α and β (Fig. 3). Their common border is a. This figure is called a dihedral angle.

Terminology

The half-planes α and β are the faces of the dihedral angle.

Straight a is the edge of a dihedral angle.

On a common edge a dihedral angle choose an arbitrary point O(Fig. 4). In the half-plane α from the point O restore the perpendicular OA to a straight line a. From the same point O in the second half-plane β we construct the perpendicular OV to the rib a. Got a corner AOB, which is called the linear angle of the dihedral angle.

Rice. 4. Dihedral angle measurement

Let us prove the equality of all linear angles for a given dihedral angle.

Let we have a dihedral angle (Fig. 5). Pick a point O and point About 1 on a straight line a. Let's construct a linear angle corresponding to the point O, i.e. we draw two perpendiculars OA and OV in the planes α and β, respectively, to the edge a. We get the angle AOB is the linear angle of the dihedral angle.

Rice. 5. Illustration of the proof

From a point About 1 draw two perpendiculars OA 1 and OB 1 to the rib a in the planes α and β, respectively, and we obtain the second linear angle A 1 O 1 B 1.

Rays O 1 A 1 and OA co-directional, since they lie in the same half-plane and are parallel to each other as two perpendiculars to the same line a.

Likewise, rays About 1 in 1 and OV aligned, which means ∠ AOB =∠ A 1 O 1 B 1 as angles with codirectional sides, which was to be proved.

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

Prove: a ⊥ AOW.

Rice. 6. Illustration of the proof

Proof:

OA ⊥ a by construction, OV ⊥ a by construction (Fig. 6).

We get that the line a perpendicular to two intersecting lines OA and OV out of plane AOB, which means straight a perpendicular to the plane OAB, which was to be proved.

A dihedral angle is measured by its linear angle. This means that as many degrees of radians are contained in a linear angle, as many degrees of radians are contained in its dihedral angle. In accordance with this, the following types of dihedral angles are distinguished.

Sharp (Fig. 6)

A dihedral angle is acute if its linear angle is acute, i.e. .

Straight (Fig. 7)

Dihedral angle is right when its linear angle is 90 ° - Obtuse (Fig. 8)

A dihedral angle is obtuse when its linear angle is obtuse, i.e. .

Rice. 7. Right angle

Rice. 8. Obtuse angle

Examples of constructing linear angles in real figures

ABCD- tetrahedron.

1. Construct a linear angle of a dihedral angle with an edge AB.

Rice. 9. Illustration for the problem

Building:

We are talking about a dihedral angle, which is formed by an edge AB and faces ABD and ABC(Fig. 9).

Let's draw a straight line DH perpendicular to the plane ABC, H is the base of the perpendicular. Let's draw an oblique DM perpendicular to the line AB,M- inclined base. By the three perpendiculars theorem, we conclude that the projection of the oblique NM also perpendicular to the line AB.

That is, from the point M restored two perpendiculars to the edge AB on two sides ABD and ABC. We got a linear angle DMN.

notice, that AB, the edge of the dihedral angle, perpendicular to the plane of the linear angle, i.e., the plane DMN. Problem solved.

Comment. A dihedral angle can be denoted as follows: DABC, where

AB- edge, and points D and With lie on different sides of the corner.

2. Construct a linear angle of a dihedral angle with an edge AC.

Let's draw a perpendicular DH to the plane ABC and oblique DN perpendicular to the line AS. By the three perpendiculars theorem, we get that HN- oblique projection DN to the plane ABC, also perpendicular to the line AS.DNH- linear angle of a dihedral angle with a rib AC.

in a tetrahedron DABC all edges are equal. Dot M- middle of the rib AC. Prove that the angle DMV- linear angle of dihedral angle YOUD, i.e., a dihedral angle with an edge AC. One of its edges is ACD, second - DIA(Fig. 10).

Rice. 10. Illustration for the problem

Decision:

Triangle ADC- equilateral, DM is the median and hence the height. Means, DM ⊥ AS. Likewise, the triangle AATC- equilateral, ATM is the median, and hence the height. Means, VM ⊥ AS.

So from the point M ribs AC dihedral angle restored two perpendiculars DM and VM to this edge in the faces of the dihedral angle.

So ∠ DMAT is the linear angle of the dihedral angle, which was to be proved.

So, we have defined the dihedral angle, the linear angle of the dihedral angle.

In the next lesson, we will consider the perpendicularity of lines and planes, then we will learn what a dihedral angle is at the base of the figures.

References on the topic "Dihedral angle", "Dihedral angle at the base of geometric figures"

1. Geometry. Grade 10-11: a textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p .: ill.
2. Geometry. Grade 10: a textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M.: Bustard, 2008. - 233 p.: ill.

1. Yaklass.ru ().
2. e-science.ru ().
3. Webmath.exponenta.ru().
4. Tutoronline.ru ().

Homework on the topic "Dihedral angle", determining the dihedral angle at the base of the figures

Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M.: Mnemozina, 2008. - 288 p.: ill.

Tasks 2, 3 p. 67.

What is the linear angle of a dihedral angle? How to build it?

ABCD- tetrahedron. Construct a linear angle of a dihedral angle with an edge:

a) ATD b) DWITH.

ABCDA 1 B 1 C 1 D 1 - cube Plot Linear Angle of Dihedral Angle A 1 ABC with a rib AB. Determine its degree measure.