A figure formed by three rays emanating from one point O and not lying in the same plane, and three parts of planes enclosed between these rays, is called a trihedral angle (Fig. 352).
Point O is called the vertex of the angle, rays a, b, c are its edges, parts of the planes. The faces are plane angles, also called plane angles of a given trihedral angle. The angles between the flat faces are called the dihedral angles of a given trihedral angle.
Theorem 1. In a trihedral angle, each plane angle is less than the sum of the other two.
Proof. It is enough to prove the theorem for the largest of the plane angles. Let the largest plane angle of the trihedral angle in Fig. 353. Let's construct an angle in the plane equal to the angle its side b passes inside the angle (the largest of the plane angles!).
Let us put on the lines c and b any equal segments Let us draw an arbitrary plane through the points, intersecting the rays a and b at points N and M, respectively.
Triangles are equal as having equal angles concluded between equal parties. Let us show that the angle with vertex O in is greater than the angle with the same vertex in . Indeed, these angles are contained between pairs equal sides, the third side is larger in the triangle
This shows that the sum of two plane angles is greater than the third plane angle, which is what needed to be proven.
Theorem 2. The sum of the plane angles of a trihedral angle is less than four right angles.
Proof. Let's take three points A, B and C on the edges of the trihedral angle and draw a cutting plane through them, as shown in Fig. 354. The sum of the angles of triangle ABC is equal to Therefore, the sum of the six angles OAC, OAB, OCA, OCB, OBC, OVA is greater than according to the previous theorem. But the sum of the angles of three triangles OAB, OBC, OCA in the faces of a trihedral angle is equal to . Thus, the share of flat angles of a trihedral angle remains less than four straight lines: . This sum can be arbitrarily small (“trihedral spire”) or arbitrarily close to if we reduce the height of the SABC pyramid in Fig. 355, preserving its base, then the sum of plane angles at vertex S will tend to
The sum of the dihedral angles of a trihedral angle also has limits. It is clear that each of the dihedral angles and therefore their sum is less than . For the same pyramid in Fig. 355 this sum approaches its limit as the height of the pyramid decreases. It can also be shown that this sum always, although it can differ from as little as desired.
Thus, for plane and dihedral angles of a trihedral angle, the following inequalities hold:
There is a significant similarity between the geometry of a triangle on a plane and the geometry of a trihedral angle. In this case, an analogy can be drawn between the angles of a triangle and the dihedral angles of a trihedral angle, on the one hand, and between the sides of a triangle and the flat angles of a trihedral angle, on the other. For example, with the indicated replacement of concepts, the theorem on the equality of triangles remains valid. Let us present the corresponding formulations in parallel:
However, two trihedral angles whose corresponding dihedral angles are equal are congruent. Meanwhile, two triangles whose angles are respectively equal are similar, but not necessarily equal. For trihedral angles, as well as for triangles, the task of solving a trihedral angle is posed, that is, the task of finding some of its elements from other given ones. Let's give an example of such a task.
Task. The plane angles of a trihedral angle are given. Find its dihedral angles.
Solution. Let us lay down a segment on the edge a and draw a normal section ABC of the dihedral angle a. From right triangle OAV we find We also have
For BC we find by the cosine theorem applied to the triangle BAC (for brevity we denote plane angles simply as ab, ac, bc, dihedral angles - a, b, c)
Now we apply the cosine theorem to triangle BOC:
From here we find
and similarly
Using these formulas, you can find dihedral angles, knowing plane angles. Let us note, without proof, the remarkable relation
called the theorem of sines.
An explanation of the deep analogy between the geometry of a trihedral angle and the geometry of a triangle is not difficult to obtain if we carry out the following construction. Let us place the center of a sphere of unit radius at the vertex of the trihedral angle O (Fig. 357).
Then the edges will intersect the surface of the sphere at three points A, B, C, and the edges of the angle will cut arcs on the sphere large circles AC, AB, BC. A figure ABC is formed on the sphere, called a spherical triangle. The arcs (“sides” of a triangle) are measured by the plane angles of the trihedral angle, the angles at the vertices are the plane angles of the dihedral angles. Therefore, the solution of trihedral angles is nothing other than the solution of spherical triangles, which is the subject of spherical trigonometry. Relations (243.1) and (243.2) are among the basic relations of spherical trigonometry. Spherical trigonometry It has important for astronomy. Thus, the theory of trihedral angles is the theory of spherical triangles and therefore is in many ways similar to the theory of a triangle on a plane. The difference between these theories is that: 1) in a spherical triangle, both angles and sides are measured in angular measure, therefore, for example, in the theorem of sines it is not the sides that appear, but the sines of the sides AB, AC, BC;
Polyhedral angle part of space limited by one polyhedral cavity conical surface, the direction of which is a flat polygon without self-intersections. The faces of this surface are called the faces of the mosaic, and the top is called the top of the mosaic. M. u. is called regular if all its linear angles and all its dihedral angles are equal. Meroy M. u. is the area limited by the spherical polygon obtained by the intersection of the faces of the polygon, a sphere with a radius equal to one, and with the center at the vertex of M. y. See also Solid angle.
Big Soviet encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what a “polyhedral angle” is in other dictionaries:
See solid angle... Big encyclopedic Dictionary
See solid angle. * * * POLYHEDAL ANGLE POLYHEDAL ANGLE, see Solid angle (see SOLID ANGLE) ... encyclopedic Dictionary
Part of space limited by one cavity of a polyhedral conic. surface directing to a swarm of flat polygon without self-intersections. The faces of this surface are called. the edges of the M. u., the top of the apex of the M. u. A polyhedral angle is called correct... Mathematical Encyclopedia
See Solid angle... Natural science. encyclopedic Dictionary
polyhedral angle- math. A part of space bounded by several planes passing through one point (vertex of an angle) ... Dictionary of many expressions
MULTIFACETED, multifaceted, multifaceted (book). 1. Having several faces or sides. Multifaceted stone. Polyhedral angle (a part of space limited by several planes intersecting at one point; mat.). 2. transfer... ... Dictionary Ushakova
- (mat.). If we draw straight lines OA and 0B from point O on a given plane, we obtain angle AOB (Fig. 1). Crap. 1. Point 0 called the vertex of the angle, and straight lines OA and 0B as the sides of the angle. Suppose that two angles ΒΟΑ and Β 1 Ο 1 Α 1 are given. Let us impose them so that... ...
- (mat.). If we draw straight lines OA and 0B from point O on a given plane, we obtain angle AOB (Fig. 1). Crap. 1. Point 0 called the vertex of the angle, and straight lines OA and 0B as the sides of the angle. Suppose that two angles ΒΟΑ and Β1Ο1Α1 are given. Let's superimpose them so that the vertices O... Encyclopedic Dictionary F.A. Brockhaus and I.A. Efron
This term has other meanings, see Angle (meanings). Angle ∠ Dimension ° SI units Radian ... Wikipedia
Flat, geometric figure, formed by two rays (sides of the surface) emanating from one point (the vertex of the surface). Every U. having a vertex at the center O of some circle (central U.), defines on the circle an arc AB bounded by... ... Great Soviet Encyclopedia
Slide 1
The figure formed by the specified surface and one of the two parts of space limited by it is called a polyhedral angle. The common vertex S is called the vertex of a polyhedral angle. The rays SA1, ..., SAn are called the edges of the polyhedral angle, and the plane angles themselves A1SA2, A2SA3, ..., An-1SAn, AnSA1 are called the faces of the polyhedral angle. A polyhedral angle is denoted by the letters SA1...An, indicating the vertex and points on its edges. A surface formed by a finite set of plane angles A1SA2, A2SA3, ..., An-1SAn, AnSA1 with a common vertex S, in which adjacent angles have no common points, except points of a common ray, and non-adjacent angles do not have common points, in addition to a common vertex, will be called a polyhedral surface.
Slide 2
Depending on the number of faces, polyhedral angles are trihedral, tetrahedral, pentagonal, etc.
Slide 3
TRIHEDAL ANGLES
Theorem. Every plane angle of a trihedral angle is less than the sum of its two other plane angles. Proof: Consider the trihedral angle SABC. Let the largest of its plane angles be angle ASC. Then the inequalities ASB ASC are satisfied
Slide 4
Property. The sum of the plane angles of a trihedral angle is less than 360°. Similarly, for trihedral angles with vertices B and C, the following inequalities hold: ABC
Slide 5
CONVEX POLYHEDAL ANGLES
A polyhedral angle is called convex if it is convex figure, i.e., together with any two of its points, it entirely contains the segment connecting them. The figure shows examples of convex and non-convex polyhedral angles. Property: The sum of all plane angles of a convex polyhedral angle is less than 360°. The proof is similar to the proof of the corresponding property for a trihedral angle.
Slide 6
Vertical polyhedral angles
The figures show examples of trihedral, tetrahedral and pentahedral vertical angles. Theorem. Vertical angles are equal.
Slide 7
Measuring polyhedral angles
Since the degree value of a developed dihedral angle is measured by the degree value of the corresponding linear angle and is equal to 180°, then we will assume that the degree value of the entire space, which consists of two unfolded dihedral angles, is equal to 360°. The size of a polyhedral angle, expressed in degrees, shows how much space a given polyhedral angle occupies. For example, a trihedral angle of a cube occupies one eighth of the space and, therefore, its degree value is 360°: 8 = 45°. Triangular angle in the right n-gonal prism equal to half dihedral angle at the side edge. Considering that this dihedral angle equal, we find that the trihedral angle of the prism is equal.
Slide 8
Measuring triangular angles*
Let us derive a formula expressing the magnitude of a trihedral angle in terms of its dihedral angles. Let us describe a unit sphere near the vertex S of the trihedral angle and denote the points of intersection of the edges of the trihedral angle with this sphere as A, B, C. The planes of the faces of the trihedral angle divide this sphere into six pairwise equal spherical digons corresponding to the dihedral angles of this trihedral angle. Spherical triangle ABC and the symmetrical spherical triangle A"B"C" are the intersection of three digons. Therefore, twice the sum of the dihedral angles is equal to 360o plus quadruple the trihedral angle, or SA +SB + SC = 180o + 2SABC.
Slide 9
Measuring polyhedral angles*
Let SA1…An be a convex n-faceted angle. Dividing it into trihedral angles, drawing diagonals A1A3, ..., A1An-1 and applying the resulting formula to them, we will have: SA1 + ... + SAn = 180о(n – 2) + 2SA1…An. Polyhedral angles can also be measured by numbers. Indeed, three hundred and sixty degrees of all space corresponds to the number 2π. Moving from degrees to numbers in the resulting formula, we will have: SA1+ …+SAn = π(n – 2) + 2SA1…An.
Slide 10
Exercise 1
Can there be a trihedral angle with flat angles: a) 30°, 60°, 20°; b) 45°, 45°, 90°; c) 30°, 45°, 60°? No answer; b) no; c) yes.
Slide 11
Exercise 2
Give examples of polyhedra whose faces, intersecting at the vertices, form only: a) trihedral angles; b) tetrahedral angles; c) pentagonal angles. Answer: a) Tetrahedron, cube, dodecahedron; b) octahedron; c) icosahedron.
Slide 12
Exercise 3
The two plane angles of a trihedral angle are 70° and 80°. What are the boundaries of the third plane angle? Answer: 10o
Slide 13
Exercise 4
The plane angles of a trihedral angle are 45°, 45° and 60°. Find the angle between the planes of plane angles of 45°. Answer: 90o.
Slide 14
Exercise 5
In a trihedral angle, two plane angles are equal to 45°; the dihedral angle between them is right. Find the third plane angle. Answer: 60o.
Slide 15
Exercise 6
The plane angles of a trihedral angle are 60°, 60° and 90°. Equal segments OA, OB, OC are laid on its edges from the vertex. Find the dihedral angle between the 90° angle plane and the ABC plane. Answer: 90o.
Slide 16
Exercise 7
Each plane angle of a trihedral angle is 60°. On one of its edges a segment equal to 3 cm is laid off from the top, and a perpendicular is dropped from its end to the opposite face. Find the length of this perpendicular. Answer: see
Slide 17
Exercise 8
Find locus internal points trihedral angle equidistant from its faces. Answer: A ray whose vertex is the vertex of a trihedral angle, lying on the line of intersection of the planes dividing the dihedral angles in half.
Slide 18
Exercise 9
Find the locus of the interior points of a trihedral angle equidistant from its edges. Answer: A ray whose vertex is the vertex of a trihedral angle, lying on the line of intersection of planes passing through the bisectors of plane angles and perpendicular to the planes these angles.
Slide 19
Exercise 10
For the dihedral angles of the tetrahedron we have: , whence 70o30". For the trihedral angles of the tetrahedron we have: 15o45". Answer: 15o45". Find the approximate values of the trihedral angles of the tetrahedron.
Slide 20
Exercise 11
Find approximate values of the tetrahedral angles of the octahedron. For the dihedral angles of the octahedron we have: , whence 109о30". For the tetrahedral angles of the octahedron we have: 38о56". Answer: 38o56".
Slide 21
Exercise 12
Find approximate values of the pentahedral angles of the icosahedron. For the dihedral angles of the icosahedron we have: , whence 138о11". For the pentahedral angles of the icosahedron we have: 75о28". Answer: 75o28".
Slide 22
Exercise 13
For the dihedral angles of the dodecahedron we have: , whence 116o34". For the trihedral angles of the dodecahedron we have: 84o51". Answer: 84o51". Find the approximate values of the trihedral angles of the dodecahedron.
Slide 23
Exercise 14
In a regular quadrangular pyramid SABCD, the side of the base is 2 cm, the height is 1 cm. Find the tetrahedral angle at the apex of this pyramid. Solution: The given pyramids divide the cube into six equal pyramids with the vertices in the center of the cube. Consequently, the 4-sided angle at the top of the pyramid is one sixth of the angle of 360°, i.e. equal to 60o. Answer: 60o.
Slide 24
Exercise 15
In the right triangular pyramid lateral ribs equal to 1, vertex angles 90°. Find the trihedral angle at the vertex of this pyramid. Solution: The indicated pyramids divide the octahedron into eight equal pyramids with the vertices at the center O of the octahedron. Therefore, the 3-sided angle at the top of the pyramid is one eighth of the 360° angle, i.e. equal to 45o. Answer: 45o.
Slide 25
Exercise 16
In a regular triangular pyramid, the lateral edges are equal to 1, and the height Find the trihedral angle at the vertex of this pyramid. Solution: The indicated pyramids are broken regular tetrahedron by four equal pyramids with vertices in the center of the Otetrahedron. Consequently, the 3-sided angle at the top of the pyramid is one-fourth of an angle of 360°, i.e. equal to 90o. Answer: 90o.
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TEXT TRANSCRIPT OF THE LESSON:
In planimetry, one of the objects of study is an angle.
An angle is a geometric figure consisting of a point - the vertex of the angle and two rays emanating from this point.
Two angles, one side of which is common and the other two are a continuation of one another, are called adjacent in planimetry.
A compass can be thought of as a model of a plane angle.
Let us recall the concept of a dihedral angle.
This is a figure formed by a straight line a and two half-planes c common border And, not belonging to the same plane in geometry is called a dihedral angle. Half-planes are the faces of a dihedral angle. Straight line a is an edge of a dihedral angle.
The roof of the house clearly demonstrates the dihedral angle.
But the roof of the house in figure two is made in the form of a figure formed from six flat angles with a common vertex so that the angles are taken at in a certain order and each pair of adjacent angles, including the first and last, has common side. What is this roof shape called?
In geometry, a figure made up of angles
And the angles from which this angle is made are called plane angles. The sides of plane angles are called the edges of a polyhedral angle. Point O is called the vertex of the angle.
Examples of polyhedral angles can be found in the tetrahedron and parallelepiped.
The faces of the tetrahedron DBA, ABC, DBC form the polyhedral angle BADC. More often it is called a trihedral angle.
In a parallelepiped, the faces AA1D1D, ABCD, AA1B1B form the trihedral angle AA1DB.
Well, the roof of the house is made in the shape of a hexagonal angle. It consists of six flat angles.
A number of properties are true for a polyhedral angle. Let us formulate them and prove them. It says here that the statement
First, for any convex polyhedral angle there is a plane intersecting all its edges.
For proof, consider the polyhedral angle OA1A2 A3…An.
By condition, it is convex. An angle is called convex if it lies on one side of the plane of each of its plane angles.
Since, by condition, this angle is convex, then points O, A1, A2, A3, An lie on one side of the plane OA1A2
Let's carry out midline KM of the triangle OA1A2 and select from the edges OA3, OA4, OAn the edge that forms the smallest dihedral angle with the OKM plane. Let this be edge OAi.(оа total)
Let us consider the half-plane α with the boundary CM, dividing the dihedral angle OKMAi into two dihedral angles. All vertices from A to An lie on one side of the plane α, and point O on the other side. Therefore, the plane α intersects all the edges of the polyhedral angle. The statement has been proven.
Convex polyhedral angles have another important property.
The sum of the plane angles of a convex polyhedral angle is less than 360°.
Consider a convex polyhedral angle with a vertex at point O. By virtue of the proven statement, there is a plane that intersects all its edges.
Let us draw such a plane α, let it intersect the edges of the angle at points A1, A2, A3 and so on An.
The plane α from the outer region of the plane angle will cut off the triangle. The sum of the angles is 180°. We obtain that the sum of all plane angles from A1OA2 to AnOA1 is equal to the expression, we transform this expression, we rearrange the terms, we obtain
IN this expression the sums indicated in brackets are the sums of the plane angles of a trihedral angle, and as is known they are greater than the third plane angle.
This inequality can be written for all trihedral angles forming a given polyhedral angle.
Consequently, we obtain the following continuation of the equality
The answer proves that the sum of the plane angles of a convex polyhedral angle is less than 360 degrees.
20. Multi-level study of polyhedral angles, properties of plane angles of a trihedral angle and a polyhedral angle.
A basic level of:
Atanasyan
Considers only the Dihedral angle.
Pogorelov
First, he considers the dihedral angle and then immediately the trihedral and polyhedral angles.
Let's consider three rays a, b, c, emanating from the same point and lying in the same plane. A trihedral angle (abc) is a figure made up of three flat angles (ab), (bc) and (ac) (Fig. 400). These angles are called the faces of a trihedral angle, and their sides are called edges. The common vertex of plane angles is called the vertex of a trihedral angle. The dihedral angles formed by the faces of a trihedral angle are called dihedral angles of a trihedral angle.
The concept of a polyhedral angle is introduced similarly (Fig. 401).
Fig. 400 and Fig. 401
P 
profile level(A.D. Aleksndrov, A.L. Werner, V.I. Ryzhikh):
Leaving the definition and study of arbitrary polyhedral angles until § 31, we will now consider the simplest of them - trihedral angles. If in stereometry dihedral angles can be considered analogues of plane angles, then trihedral angles can be considered as analogues of plane triangles, and in the following paragraphs we will see how they are naturally related to spherical triangles.
You can construct (and therefore constructively define) a trihedral angle like this. Take any three rays a, b, c, having general beginning O and not lying in the same plane (Fig. 150). These rays are the sides of three convex plane angles: angle α with sides b, c, angle β with sides a, c, and angle γ with sides a, b. The union of these three angles α, β, γ is called the trihedral angle Oabc (or, in short, the trihedral angle O). Rays a, b, c are called the edges of the trihedral angle Oabc, and the plane angles α, β, γ are its faces. Point O is called the vertex of a trihedral angle.
3 remark. It would be possible to define a trihedral angle with a non-convex face (Fig. 151), but we will not consider such trihedral angles. 
For each edge of a trihedral angle, a corresponding dihedral angle is determined, one whose edge contains the corresponding edge of the trihedral angle, and whose faces contain the faces of the trihedral angle adjacent to this edge.
The values of the dihedral angles of the trihedral angle Oabc at the edges a, b, c will be denoted respectively by a^, b^, c^ (caps directly above the letters).
Three faces α, β, γ of the trihedral angle Oabc and its three dihedral angles at ribs a, b, с, as well as the quantities α, β, γ and а^, b^, с^ we will call elements of a trihedral angle. (Remember that the elements of a plane triangle are its sides and its angles.)
Our task is to express some elements of a trihedral angle through its other elements, that is, to construct a “trigonometry” of trihedral angles.
1) Let's start by deriving an analogue of the cosine theorem. First, consider a trihedral angle Oabc, which has at least two faces, for example α and β, sharp corners. Let's take point C on its edge c and draw from it in faces α and β perpendiculars CB and CA to edge c until they intersect with edges a and b at points A and B (Fig. 152). Let us express the distance AB from triangles OAB and CAB using the cosine theorem.
AB 2 =AC 2 +BC 2 -2AC*BC*Cos(c^) and AB 2 =OA 2 +OB 2 -2AO*BO*Cosγ.
Subtracting the first from the second equality, we get:
OA 2 -AC 2 +OB 2 -BC 2 +2AC*BC*Cos(c^)-2AO*VO*Cosγ=0 (1). Because triangles OSV and OCA are right-angled, then AC 2 -AC 2 =OS 2 and OB 2 -VS 2 =OS 2 (2)
Therefore, from (1) and (2) it follows that OA*OB*Cosγ=OC 2 +AC*BC*Cos(c^)
those. 
But 
,
,
,
. That's why
(3) – an analogue of the cosine theorem for trihedral angles - cosine formula.
Both faces α and β are obtuse angles.
One of the angles α and β, for example α, is acute, and the other, β, is obtuse.
At least 1 of the angles α or β is straight.
Signs of equality of trihedral angles similar to the signs of equality of triangles. But there is a difference: for example, two trihedral angles are equal if their dihedral angles are correspondingly equal. Remember that two plane triangles whose corresponding angles are equal are similar. And for trihedral angles, a similar condition leads not to similarity, but to equality.
Trihedral angles have a remarkable property which is called duality. If in any theorem about the trihedral angle Oabc we replace values a, b, from to π-α, π-β, π-γand, conversely, replace α, β, γ with π-a^, π-b^, π-c^, then we again obtain a true statement about trihedral angles, dual to the original one theorem. True, if such a replacement is made in the theorem of sines, then we again come to the theorem of sines (it is dual to itself). But if we do this in the cosine theorem (3), we get a new formula
cosc^= -cosa^ cosb^+sina^ sin b^ cosγ.
Why such duality occurs will become clear if for a trihedral angle we construct a trihedral angle dual to it, the edges of which are perpendicular to the faces of the original angle (see section 33.3 and Fig. 356).
Some of the simplest surfaces are polyhedral angles. They are made up of ordinary angles (we will now often call such angles flat angles), just as a closed broken line is made up of segments. Namely, the following definition is given:
A polyhedral angle is called a figure formed by plane angles such that the following conditions are met:
1) No two angles have common points except their common vertex or whole side.
2) For each of these angles, each of its sides is common with one and only one other such angle.
3) From each corner you can go to each corner along the corners that have common sides.
4) No two angles with a common side lie in the same plane (Fig. 324).
Under this condition, the plane angles forming a polyhedral angle are called its faces, and their sides are called its edges.
Under this definition A dihedral angle is also suitable. It is composed of two unfolded flat angles. Its vertex can be considered any point on its edge, and this point splits the edge into two edges that meet at the vertex. But due to this uncertainty in the position of the vertex, the dihedral angle is excluded from the number of polyhedral angles.
P 
The concept of a polyhedral angle is important, in particular, in the study of polyhedra - in the theory of polyhedra. The structure of a polyhedron is characterized by what faces it is made of and how they converge at the vertices, i.e., what polyhedral angles are there.
Consider the polyhedral angles of different polyhedra.
Note that the faces of polyhedral angles can also be non-convex angles.