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.

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Lesson Objectives: introduce the concept of a dihedral angle and its linear angle;

consider tasks for the application of these concepts;

to form a constructive skill of finding the angle between planes;

consider tasks for the application of these concepts.

During the classes

I. Organizational moment.

Inform the topic of the lesson, form the objectives of the lesson.

II. Actualization of students' knowledge (slide 2, 3).

1. Preparation for the study of new material.

What is called an angle on a plane?

What is the angle between lines in space called?

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

Formulate the three perpendiculars theorem

III. Learning new material.

• The concept of a dihedral angle.

The figure formed by two half-planes passing through the line MN is called a dihedral angle (slide 4).

Half-planes are faces, straight line MN is an edge of a dihedral angle.

What objects in everyday life have the shape of a dihedral angle? (Slide 5)

• The angle between the planes ACH and CHD is the dihedral angle ACND, where CH is an edge. Points A and D lie on the faces of this angle. Angle AFD is the linear angle of the dihedral angle ACHD (slide 6).

• Algorithm for constructing a linear angle (slide 7).

1 way. On the edge, take any point O and draw perpendiculars to this point (PO DE, KO DE) and get the angle ROCK - linear.

2 way. Take a point K in one half-plane and drop two perpendiculars from it to the other half-plane and an edge (KO and KR), then by the inverse TTP theorem PODE

• All linear angles of a dihedral angle are equal (slide 8). Proof: rays OA and O 1 A 1 are co-directed, rays OB and O 1 B 1 are also co-directed, angles BOA and B 1 O 1 A 1 are equal as angles with co-directed sides.

• The degree measure of a dihedral angle is the degree measure of its linear angle (slide 9).

IV. Consolidation of the studied material.

• Problem solving (orally according to ready-made drawings). (Slides 10-12)

1. RAVS - pyramid; the angle ACB is 90°, the straight line PB is perpendicular to the plane ABC. Prove that angle PCB is a linear angle of a dihedral angle with

2. RAVS - pyramid; AB \u003d BC, D is the midpoint of the segment AC, the straight line PB is perpendicular to the plane ABC. Prove that angle PDB is a linear angle of a dihedral angle with edge AC.

3. PABCD - pyramid; line PB is perpendicular to plane ABC, BC is perpendicular to DC. Prove that angle PKB is a linear angle of a dihedral angle with edge CD.

• Tasks for constructing a linear angle (slides 13-14).

1. Construct a linear angle of a dihedral angle with an edge AC, if in the pyramid RABC the face ABC is a regular triangle, O is the intersection point of the medians, the straight line RO is perpendicular to the plane ABC

2. Rhombus ABCD is given. The straight line PC is perpendicular to the plane ABCD.

Construct a linear angle of a dihedral angle with edge BD and a linear angle of a dihedral angle with edge AD.

• Computational task. (Slide 15)

In the parallelogram ABCD, the angle ADC is 120 0, AD = 8 cm,

DC = 6 cm, straight line PC is perpendicular to the plane ABC, PC = 9 cm.

Find the value of the dihedral angle with the edge AD and the area of ​​the parallelogram.

V. Homework (slide 16).

P. 22, No. 168, 171.

Used Books:

1. Geometry 10-11 L.S. Atanasyan.
2. The system of tasks on the topic “Dihedral angles” by M.V. Sevostyanova (Murmansk), journal Mathematics at school 198 ...

Between perpendiculars to the edge of a dihedral angle, restored in both faces from the same point.

Mathematical encyclopedia. - M.: Soviet Encyclopedia. I. M. Vinogradov. 1977-1985.

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"Dihedral angle" - Find the distance from point B to the plane. Angle C is acute. Triangle ABC is an obtuse triangle. Angle C is obtuse. The distance from a point to a line. In a tetrahedron DABC all edges are equal. Angle between slopes. The distance between the bases of the inclined. The linear angles of a dihedral angle are equal. Algorithm for constructing a linear angle.

"Dihedral angle geometry" - angle RSV - linear for a dihedral angle with edge AC. Find (see) the edge and faces of the dihedral angle. The model can be both three-dimensional and folding. Section of a dihedral angle by a plane perpendicular to the edge. Facets. the line CP is perpendicular to the edge CA (by the three perpendiculars theorem). angle RKV - linear for a dihedral angle with RSAV.

"Trihedral angle" - Signs of equality of trihedral angles. Given: Оabc – trihedral angle; ?(b; c) = ?; ?(a; c) = ?; ?(a; b) = ?. Lesson 6 1) To calculate the angle between a straight line and a plane, the formula is applicable: Formula of three cosines. . Given a trihedral angle Oabc. triangular angle. Theorem. In a regular triangular pyramid, the flat angle at the apex is less than 120°.

"Trihedral and polyhedral angles" - Trihedral angles of the dodecahedron. Trihedral and tetrahedral angles of the rhombic dodecahedron. Tetrahedral corners of the octahedron. Trihedral corners of a tetrahedron. Measurement of polyhedral angles. Task. Multifaceted corners. Five-sided angles of the icosahedron. Vertical polyhedral angles. Trihedral corner of the pyramid. Let SA1…An be a convex n-faced angle.

"The angle between the line and the plane" - In the regular 6th prism A ... F1, whose edges are equal to 1, find the angle between the line AC1 and the plane ADE1. In the correct 6th prism A…F1, whose edges are equal to 1, find the angle between the line AA1 and the plane ACE1. The angle between a line and a plane. In the regular 6th prism A…F1, whose edges are equal to 1, find the angle between the line AB1 and the plane ADE1.

"Polyhedral angle" - Convex polyhedral angles. Multifaceted corners. Depending on the number of faces, polyhedral angles are trihedral, tetrahedral, pentahedral, etc. B) icosahedron. The two planar angles of a trihedral angle are 70° and 80°. Hence, ? ASB+? BSC+? ASC< 360° . Сумма плоских углов трехгранного угла меньше 360°.

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Slides captions:

DOUBLE ANGLE Mathematics teacher GOU secondary school №10 Eremenko M.A.

The main objectives of the lesson: Introduce the concept of a dihedral angle and its linear angle Consider tasks for the application of these concepts

Definition: A dihedral angle is a figure formed by two half-planes with a common boundary line.

The value of a dihedral angle is the value of its linear angle. AF ⊥ CD BF ⊥ CD AFB is the linear angle of the dihedral angle ACD B

Let us prove that all linear angles of a dihedral angle are equal to each other. Consider two linear angles AOB and A 1 OB 1 . Rays OA and OA 1 lie on the same face and are perpendicular to OO 1, so they are co-directed. Rays OB and OB 1 are also co-directed. Therefore, ∠ AOB = ∠ A 1 OB 1 (as angles with codirectional sides).

Examples of dihedral angles:

Definition: The angle between two intersecting planes is the smallest of the dihedral angles formed by these planes.

Task 1: In the cube A ... D 1 find the angle between the planes ABC and CDD 1 . Answer: 90o.

Task 2: In the cube A ... D 1 find the angle between the planes ABC and CDA 1 . Answer: 45o.

Task 3: In the cube A ... D 1 find the angle between the planes ABC and BDD 1 . Answer: 90o.

Task 4: In the cube A ... D 1 find the angle between the planes ACC 1 and BDD 1 . Answer: 90o.

Task 5: In the cube A ... D 1 find the angle between the planes BC 1 D and BA 1 D . Solution: Let O be the midpoint of B D. A 1 OC 1 is the linear angle of the dihedral angle A 1 B D C 1 .

Problem 6: In the tetrahedron DABC all edges are equal, point M is the midpoint of edge AC. Prove that ∠ DMB is a linear angle of dihedral angle BACD .

Solution: Triangles ABC and ADC are regular, so BM ⊥ AC and DM ⊥ AC and hence ∠ DMB is a linear angle of dihedral angle DACB .

Task 7: From the vertex B of the triangle ABC, the side AC of which lies in the plane α, a perpendicular BB 1 is drawn to this plane. Find the distance from point B to the line AC and to the plane αif AB=2, ∠BAC=150 0 and the dihedral angle BACB 1 is 45 0 .

Solution: ABC is an obtuse triangle with an obtuse angle A, so the base of height BK lies on the extension of side AC. VC is the distance from point B to AC. BB 1 - distance from point B to plane α

2) Since AS ⊥VK, then AS⊥KV 1 (by the theorem converse to the three perpendiculars theorem). Therefore, ∠VKV 1 is the linear angle of the dihedral angle BACB 1 and ∠VKV 1 =45 0 . 3) ∆VAK: ∠A=30 0 , VK=VA sin 30 0 , VK =1. ∆VKV 1: VV 1 \u003d VK sin 45 0, VV 1 \u003d