Let us consider three rays a, b, c, emanating from the same point and not lying in the same plane. A trihedral angle (abc) is a figure composed of "three flat angles (ab), (bc) and (ac) (Fig. 2). These angles are called the faces of a trihedral angle, and their sides are edges, the common vertex of the flat angles is called 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 defined similarly (Fig. 3).
Polyhedron
In stereometry, figures in space, called bodies, are studied. Visually, a (geometric) body must be imagined as a part of space occupied by a physical body and bounded by a surface.
A polyhedron is a body whose surface consists of a finite number of flat polygons (Fig. 4). A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of such a plane and the surface of a convex polyhedron is called a face. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called the edges of the polyhedron, and the vertices are called the vertices of the polyhedron.
Let us explain what was said on the example of a familiar cube (Fig. 5). The cube is a convex polyhedron. Its surface consists of six squares: ABCD, BEFC, .... They are its faces. The edges of the cube are the sides of these squares: AB, BC, BE,.... The vertices of the cube are the vertices of the squares: A, B, C, D, E, .... The cube has six faces, twelve edges and eight vertices.
The simplest polyhedra - prisms and pyramids, which will be the main object of our study - we will give definitions that, in essence, do not use the concept of a body. They will be defined as geometric figures with indication of all points of space belonging to them. The concept of a geometric body and its surface in general case will be given later.
TEXT EXPLANATION OF THE LESSON:
In planimetry, one of the objects of study is the 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, which are common and the other two are a continuation of one another, are called adjacent in planimetry.
The compass can be viewed as a model of a flat angle.
Recall the concept of a dihedral angle.
This is a 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.
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 corners with a common vertex so that the corners are taken in a certain order and each pair of adjacent corners, including the first and last, has a common side. What is this type of roof called?
In geometry, a figure made up of angles
And the angles that make up this angle are called flat angles. The sides of flat angles are called edges of a polyhedral angle. Point O is called the vertex of the corner.
Examples of polyhedral angles can be found in the tetrahedron and cuboid.
The faces of the tetrahedron DBA, ABC, DBC form a polyhedral angle BADC. More often it is called a trihedral angle.
In a parallelepiped, faces AA1D1D, ABCD, AA1B1B form a trihedral angle AA1DB.
Well, the roof of the house is made in the form of a hexagonal corner. It consists of six flat corners.
A number of properties hold for a polyhedral angle. Let's 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.
Consider for proof the polyhedral angle OA1A2 A3…An.
By definition, it is convex. An angle is called convex if it lies on one side of the plane of each of its flat angles.
Since by the condition this angle is convex, then the points O, A1, A2, A3, An lie on one side of the plane OA1A2
Let us draw the midline KM of the triangle OA1A2 and choose from the edges OA3, OA4, OAn the edge that forms the smallest dihedral angle with the OCM plane. Let this be the edge OAi. (Oa total)
Let us consider the half-plane α with the CM boundary 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 edges of the polyhedral angle. The assertion 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 exists 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 will cut off the triangle from the outer area of the flat angle. The sum of the angles is 180°. We get that the sum of all plane angles from А1ОА2 to АnОА1 is equal to the expression we transform, this expression we regroup the terms, we get
In this expression, the amounts indicated in brackets are the sums of the plane angles of the trihedral angle, and as you know, they are greater than the third plane angle.
This inequality can be written for all trihedral angles forming a given polyhedral angle.
Therefore, we obtain the following continuation of the equality
The answer obtained proves that the sum of the plane angles of a convex polyhedral angle is less than 360 degrees.
№1 Date05.09.14
Subject Geometry
Class 11
Lesson topic: The concept of a polyhedral angle. triangular angle.
Lesson Objectives:
introduce the concepts: “trihedral angles”, “polyhedral angles”, “polyhedron”;
to acquaint students with the elements of trihedral and polyhedral angles, a polyhedron, as well as the definitions of a convex polyhedral angle and the properties of flat angles of a polyhedral angle;
to continue work on the development of spatial representations and spatial imagination, as well as the logical thinking of students.
Lesson type: learning new material
DURING THE CLASSES
1. Organizational moment.
Greeting students, checking the readiness of the class for the lesson, organizing the attention of students, disclosing the general objectives of the lesson and its plan.
2. Formation of new concepts and methods of action.
Tasks: To ensure the perception, comprehension and memorization of the studied material by students. To ensure that students master the methodology for reproducing the studied material, to promote the philosophical understanding of the concepts, laws, rules, formulas being assimilated. To establish the correctness and awareness of the studied material by students, to identify gaps in the primary understanding, to carry out a correction. To ensure that students correlate their subjective experience with the signs of scientific knowledge.
Let three rays be givena, b ands s common start pointO (Fig. 1.1). These three rays do not necessarily lie in the same plane. In figure 1.2, the raysb andwith lie in a planeR, a raya does not lie in this plane.
Raysa, b andwith pairs define three flat angles distinguished by arcs (Fig. 1.3).
Consider a figure consisting of the three angles indicated above and the part of space bounded by these flat angles. This spatial figure is calledtrihedral angle
(Fig. 2).
Raysa, b and with callededges of a trihedral angle, and the corners: = AOC, = AOB,
=
BOC
,
limiting the trihedral angle, - itsfaces.
These corners formtrihedral surface.
DotO
calledvertex of a trihedral angle.
A trihedral angle can be denoted as follows: OABC
Having carefully examined all the polyhedral angles shown in Figure 3, we can conclude that each of the polyhedral angles has the same number of edges and faces:
4 faces and one vertex;
a hexagonal corner has 6 edges, 6 faces and one vertex, etc.
a five-sided corner has 5 edges, 5 faces and one vertex;
Polyhedral angles are convex and non-convex.
Imagine that we took four rays with a common origin, as in Figure 4. In this case, we gotnon-convex polyhedral angle.
Definition 1. A polyhedral angle is called a convex angle,if helies on one side of the plane of each of its faces.
In other words, a convex polyhedral angle can always be placed by any of its faces on some plane. You can see that in the case shown in Figure 4, this is not always possible. The tetrahedral angle shown in Figure 4 is non-convex.
Note that in our tutorial, if we say “polyhedral angle”, we mean that it is convex. If the considered polyhedral angle is non-convex, this will be discussed separately.
Properties of Plane Corners of a Polyhedral Corner
Theorem 1.Each flat angle of a trihedral angle is less than the sum of the other two flat angles.
Theorem 2.The sum of the values of all plane angles of a convex polyhedral angle is less than 360°.
3. Application. Formation of skills and abilities.
Objectives: To ensure that students apply the knowledge and methods of action that they need for SW, to create conditions for students to identify individual ways of applying what they have learned.
6. The homework information stage.
Objectives: To ensure that students understand the purpose, content and methods of doing homework.
§1(1.1, 1.2) p. 4, no. 9.
7. Summing up the lesson.
Objective: To give a qualitative assessment of the work of the class and individual students.
8. Stage of reflection.
Tasks: To initiate students' reflection on self-assessment of their activities. To ensure that students learn the principles of self-regulation and cooperation.
Conversation on:
What did you find interesting in the lesson?
What's not clear?
What should the teacher pay attention to in the next lesson?
How would you rate your work in class?
slide 1
The figure formed by the specified surface and one of the two parts of the space bounded by it is called a polyhedral angle. The common vertex S is called the vertex of the 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. The surface formed by a finite set of plane angles A1SA2, A2SA3, …, An-1SAn, AnSA1 with a common vertex S, in which neighboring angles have no common points, except for points of a common ray, and non-neighboring angles have no common points, except for a common vertex, we will called a polyhedral surface.
slide 2
Depending on the number of faces, polyhedral angles are trihedral, tetrahedral, pentahedral, etc.
slide 3
TRIHEDRAL CORNERS
Theorem. Every flat angle of a trihedral angle is less than the sum of its other two flat angles. Proof. Consider the trihedral angle SABC. Let the largest of its flat angles be the angle ASC. Then the inequalities ASB ASC
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: ABС
slide 5
CONVEX POLYHEDRAL ANGLES
A polyhedral angle is called convex if it is a 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
Measurement of 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°, we will assume that the degree value of the entire space, which consists of two developed dihedral angles, is 360°. The value of a polyhedral angle, expressed in degrees, shows what part of the space the given polyhedral angle occupies. For example, the trihedral angle of a cube occupies one eighth of the space and, therefore, its degree value is 360o:8 = 45o. The trihedral angle in a regular n-gonal prism is equal to half the dihedral angle at the side edge. Considering that this dihedral angle is equal, we obtain that the trihedral angle of the prism is equal.
Slide 8
Measurement of trihedral angles*
We derive a formula expressing the value 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 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 the given trihedral angle. The spherical triangle ABC and the spherical triangle A "B" C symmetric to it are the intersection of three digons. Therefore, the double sum of the dihedral angles is 360o plus the quadruple value of the trihedral angle, or SA + SB + SC = 180o + 2SABC.
Slide 9
Measurement of polyhedral angles*
Let SA1…An be a convex n-faced angle. Dividing it into trihedral angles, drawing the 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 the whole space corresponds to the number 2π. Passing 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 corners: 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 corners; c) five-sided corners. Answer: a) Tetrahedron, cube, dodecahedron; b) octahedron; c) icosahedron.
slide 12
Exercise 3
The two planar angles of a trihedral angle are 70° and 80°. What is the boundary 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 planes of flat angles of 45°. Answer: 90o.
Slide 14
Exercise 5
In a trihedral angle, two plane angles are 45° each; the dihedral angle between them is right. Find the third flat corner. Answer: 60o.
slide 15
Exercise 6
The plane angles of a trihedral angle are 60°, 60° and 90°. Equal segments OA, OB, OC are plotted 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 flat 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 lowered from its end to the opposite face. Find the length of this perpendicular. Answer: see
Slide 17
Exercise 8
Find the locus of interior points of a 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 that bisect the dihedral angles.
Slide 18
Exercise 9
Find the locus of 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 of 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 the approximate values of the tetrahedral angles of the octahedron. For the dihedral angles of the octahedron we have: , whence 109o30". For the tetrahedral angles of the octahedron we have: 38o56". Answer: 38o56".
slide 21
Exercise 12
Find the approximate values of the five-sided angles of the icosahedron. For the dihedral angles of the icosahedron we have: , whence 138o11". For the pentahedral angles of the icosahedron we have: 75o28". 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 top of this pyramid. Solution: The indicated pyramids divide the cube into six equal pyramids with vertices in the center of the cube. Therefore, the 4-sided angle at the top of the pyramid is one sixth of the 360° angle, i.e. equal to 60o. Answer: 60o.
slide 24
Exercise 15
In a regular triangular pyramid, the side edges are equal to 1, the angles at the top are 90o. Find the trihedral angle at the top of this pyramid. Solution: The indicated pyramids divide the octahedron into eight equal pyramids with vertices in 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 side edges are equal to 1, and the height Find the trihedral angle at the top of this pyramid. Solution: The indicated pyramids divide the regular tetrahedron into four equal pyramids with vertices in the center of the tetrahedron. Therefore, the 3-sided angle at the top of the pyramid is one fourth of the 360° angle, i.e. is equal to 90o. Answer: 90o.
View all slides
Definitions. Let's take several angles (Fig. 37): ASB, BSC, CSD, which, adjoining one another in series, are located in the same plane around the common vertex S.
Let us rotate the angle plane ASB around the common side SB so that this plane makes some dihedral angle with the plane BSC. Then, without changing the resulting dihedral angle, we rotate it around the straight line SC so that the BSC plane makes some dihedral angle with the CSD plane. Let's continue this sequential rotation around each common side. If in this case the last side of SF is combined with the first side of SA, then a figure is formed (Fig. 38), which is called polyhedral angle. Angles ASB, BSC,... are called flat corners or faces, their sides SA, SB, ... are called ribs, and the common vertex S- summit multifaceted angle.
Each edge is also an edge of some dihedral angle; therefore, in a polyhedral angle, there are as many dihedral angles and as many flat angles as there are all edges in it. The smallest number of faces in a polyhedral angle is three; this angle is called trihedral. There may be four-sided, five-sided, etc. angles.
A polyhedral angle is denoted either by a single letter S placed at the vertex, or by a series of letters SABCDE, of which the first denotes the vertex, and the others denote the edges in the order in which they are located.
A polyhedral angle is called convex if it is all located on one side of the plane of each of its faces, which is indefinitely extended. Such, for example, is the angle shown in drawing 38. On the contrary, the angle in drawing 39 cannot be called convex, since it is located on both sides of the ASB face or the BSC face.
If all faces of a polyhedral angle are intersected by a plane, then a polygon is formed in the section ( abcde ). In a convex polyhedral angle, this polygon is also convex.
We will consider only convex polyhedral angles.
Theorem. In a trihedral angle, each flat angle is less than the sum of the other two flat angles.
Let in the trihedral angle SABC (Fig. 40) the largest of the flat angles be the angle ASC.
Let us plot the angle ASD on this angle, which is equal to the angle ASB, and draw some straight line AC intersecting SD at some point D. Put SB = SD. Connecting B with A and C, we get \(\Delta\)ABC, in which
AD+DC< АВ + ВС.
Triangles ASD and ASB are congruent because they each contain an equal angle between equal sides: hence AD = AB. Therefore, if we discard the equal terms AD and AB in the derived inequality, we get that DC< ВС.
Now we notice that triangles SCD and SCB have two sides of one equal to two sides of the other, and the third sides are not equal; in this case, a larger angle lies opposite the larger of these sides; means,
∠CSD< ∠ CSВ.
Adding the angle ASD to the left side of this inequality, and the angle ASB equal to it to the right side, we obtain the inequality that was required to be proved:
∠ASC< ∠ CSB + ∠ ASB.
We have proved that even the largest flat angle is less than the sum of the other two angles. So the theorem is proven.
Consequence. Subtract from both parts of the last inequality in the angle ASB or in the angle CSB; we get:
∠ASC - ∠ASB< ∠ CSB;
∠ASC - ∠CSB< ∠ ASB.
Considering these inequalities from right to left, and taking into account that the angle ASC as the largest of the three angles is greater than the difference of the other two angles, we conclude that in a trihedral angle, each plane angle is greater than the difference of the other two angles.
Theorem. In a convex polyhedral angle, the sum of all planar angles is less than 4d (360°) .
Let's intersect the faces (Fig. 41) of the convex angle SABCDE with some plane; from this in the section we get a convex n-gon ABCDE.
Applying the theorem proved earlier to each of the trihedral angles whose vertices are at points A, B, C, D and E, paholim:
∠ABC< ∠ABS + ∠SВC, ∠BCD < ∠BCS + ∠SCD и т. д.
Let's add all these inequalities term by term. Then on the left side we get the sum of all angles of the polygon ABCDE, which is equal to 2 dn - 4d , and on the right - the sum of the angles of the triangles ABS, SBC, etc., except for those angles that lie at the vertex S. Denoting the sum of these last angles by the letter X , we get after addition:
2dn - 4d < 2dn - x .
Since in differences 2 dn - 4d and 2 dn - x minuends are the same, then for the first difference to be less than the second, it is necessary that the subtrahend 4 d was more than subtracted X ; means 4 d > X , i.e. X < 4d .
The simplest cases of equality of trihedral angles
Theorems. Trihedral angles are equal if they have:
1) by an equal dihedral angle enclosed between two respectively equal and equally spaced plane angles, or
2) along an equal plane angle enclosed between two respectively equal and equally spaced dihedral angles.
1) Let S and S 1 be two trihedral angles (Fig. 42), in which ∠ASB = ∠A 1 S 1 B 1 , ∠ASC = ∠A 1 S 1 C 1 (and these equal angles are equally located) and dihedral the angle AS is equal to the dihedral angle A 1 S 1 .
Let us embed the angle S 1 into the angle S so that the points S 1 and S, the lines S 1 A 1 and SA, and the planes A 1 S 1 B 1 and ASB coincide. Then the edge S 1 B 1 will go along SB (due to the equality of angles A 1 S 1 B 1 and ASB), the plane A 1 S 1 C 1 will go along ASC (due to the equality of dihedral angles), and the edge S 1 C 1 will go along the edge SC (due to the equality of the angles A 1 S 1 C 1 and ASC). Thus, the trihedral angles will be combined by all their edges, i.e. they will be equal.
2) The second criterion, like the first one, is proved by an embedding.
Symmetric polyhedral angles
As you know, vertical angles are equal when it comes to angles formed by straight lines or planes. Let's see if this statement is true for polyhedral angles.
We continue (Fig. 43) all the edges of the angle SABCDE beyond the vertex S, then another polyhedral angle SA 1 B 1 C 1 D 1 E 1 is formed, which can be called vertical with respect to the first corner. It is easy to see that both angles have equal plane and dihedral angles, respectively, but both are in reverse order. Indeed, if we imagine an observer who looks from outside the polyhedral angle at its vertex, then the edges SA, SB, SC, SD, SE will seem to him to be located in a counterclockwise direction, while looking at the angle SA 1 B 1 C 1 D 1 E 1 , he sees the edges SA 1 , SВ 1 , ... located clockwise.
Polyhedral angles with respectively equal plane and dihedral angles, but located in the reverse order, cannot be combined at all when embedding; that means they are not equal. Such angles are called symmetrical(relative to the top S). More about the symmetry of figures in space will be discussed below.
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