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POLYHEDRAL ANGLES The figure formed by the specified surface and one of the two parts of 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. A surface formed by a finite set of plane angles A1SA2, A2SA3, …, An-1SAn, AnSA1 with a common vertex S, in which the adjacent corners have no common points, except for the points of a common ray, and the non-neighboring corners have no common points, except for a common vertex, will be called a polyhedral surface.slide 2
POLYHEDRAL CORNERS Depending on the number of faces, polyhedral angles are trihedral, tetrahedral, pentahedral, etc.
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TRIHEDRAL ANGLES 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
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TRIHEDRAL CORNERS 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
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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.
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Vertical polyhedral angles The figures show examples of trihedral, tetrahedral and pentahedral vertical angles Theorem. Vertical angles are equal.
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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.
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Measurement of trihedral angles* Let us derive a formula expressing the value of a trihedral angle in terms of its dihedral angles. We 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 its symmetrical spherical triangle A"B"C" are the intersection of three digons. Therefore, twice the sum of the dihedral angles is 360o plus quadruple the trihedral angle, or SA + SB + SC = 180o + 2 SABC.
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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) + 2 SA1…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) + 2 SA1…An.
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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.
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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.
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Exercise 3 The two plane angles of a trihedral angle are 70° and 80°. What is the boundary of the third plane angle? Answer: 10o< < 150о.
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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.
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Exercise 5 In a trihedral angle, two flat angles are equal to 45 °; the dihedral angle between them is right. Find the third flat corner. Answer: 60o.
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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.
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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 lowered from its end to the opposite face. Find the length of this perpendicular.
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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 dividing the dihedral angles in half.
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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.
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Slides captions:
Dihedral angles The work was completed by: teacher of mathematics Serebryanskaya L.A.
A dihedral angle is a part of space enclosed between two half-planes that have one common boundary.
The half-planes α and β that form a dihedral angle are called its faces
We choose an arbitrary point C on the edge A D of the dihedral angle and draw a plane α through it perpendicular to the edge AP. This angle is called the linear angle of the dihedral angle.
When two planes intersect, four dihedral angles are formed. The smallest of these dihedral angles is called the angle between these planes.
If the planes are parallel, then the angle between them is 0° by definition. If φ is the angle between two planes, then 0°
Problem In the cube ABCDA 1 B 1 C 1 D 1 find the angle between the planes BC 1 D and BA 1 D . A B C D A 1 B 1 C 1 E 1
Problem Given: cube ABCDA 1 B 1 C 1 D 1 Find: the angle between the planes BC 1 D and BA 1 D Solution: A B C D A 1 B 1 C 1 D 1 O ∆ BDA 1 and ∆ DC 1 B are equal isosceles AO and C 1 O are perpendicular DB \u003d "A 1 O C 1 the desired C 1 O is the diagonal of a square with a side equal to 1.
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On the topic: methodological developments, presentations and notes
One of the main topics in stereometry is the topic “Dihedral angles”. Despite the fact that students easily learn the concepts of a dihedral angle and its linear angle, there are many difficulties in solving ...
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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. Theorem. The sum of all plane angles of a convex polyhedral angle is less than 360°.
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CONVEX POLYTOPES An angle polyhedron 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 a convex and non-convex pyramid. The cube, parallelepiped, triangular prism and pyramid are all convex polyhedra.
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PROPERTY 1 Property 1. In a convex polyhedron, all faces are convex polygons. Indeed, let F be some face of the polyhedron M, and the points A, B belong to the face F. From the convexity condition of the polyhedron M, it follows that the segment AB is entirely contained in the polyhedron M. Since this segment lies in the plane of the polygon F, it will be entirely be contained in this polygon, i.e., F is a convex polygon.
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PROPERTY 2 Indeed, let M be a convex polyhedron. Let us take some interior point S of the polytope M, i.e., a point of it that does not belong to any face of the polytope M. Let's connect the point S with the vertices of the polytope M by segments. Note that, due to the convexity of the polytope M, all these segments are contained in M. Consider pyramids with vertex S whose bases are the faces of the polytope M. These pyramids are entirely contained in M, and together they form the polytope M. Property 2. Any convex polytope can be composed of pyramids with a common vertex, the bases of which form the surface of a polyhedron.
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Exercise 1 In the figure, indicate convex and non-convex flat figures. Answer: a), d) - convex; b), c) are non-convex.
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Exercise 2 Is the intersection of convex figures always a convex figure? Answer: Yes.
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Exercise 3 Is the union of convex figures always a convex figure? Answer: No.
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Exercise 4 Is it possible to make a convex tetrahedral angle with such flat angles: a) 56o, 98o, 139o and 72o; b) 32o, 49o, 78o and 162o; c) 85o, 112o, 34o and 129o; d) 43o, 84o, 125o and 101o. No answer; b) yes; c) no; d) yes.
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Exercise 5 In the figure, indicate convex and non-convex polyhedra. Answer: b), e) - convex; a), c), d) are non-convex.
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Exercise 6 Can a non-convex polygon be a face of a convex polyhedron? Answer: No.
The presentation "Polyhedral angle" is a visual material for presenting educational information to students on the topic. During the presentation, theoretical basis concepts of a polyhedral angle, the basic properties of a polyhedral angle, which must be known to solve problems, are proved. With the help of the manual, it is easier for the teacher to form an idea of the multifaceted angle, the ability to solve problems on the topic. The presentation, among other visual aids, helps to increase the effectiveness of the lesson.
Techniques are used in the presentation to improve presentation educational material. These are animation effects, highlighting, inserting pictures, diagrams. Using animation effects, information is presented sequentially, highlighting important points. With the help of animation, constructions are presented more alive, close to the traditional demonstration using chalkboard to make it easier for students to understand the properties being represented. The use of highlighting tools helps students to remember educational information more easily.
The demonstration begins with a reminder of the educational material from which the study of angles began in the course of mathematics. Definition of an angle as a figure consisting of a point and two rays that emanate from the point. Under the definition, the image of the angle ∠ABC is given, the angle, vertex and points on the rays are indicated. Next, we recall the information about what the adjacent angles ∠LOM and ∠MON are. The figure shows adjacent angles, the angles themselves are indicated, the vertex O and points on the rays - L, M, N. The compass shown on slide 4 serves as the angle model. The compass opening can change, creating angles of various sizes.
With the help of slide 5, students are reminded of the definition of a dihedral angle as a figure made up of two half-planes that do not belong to the same plane, and their common boundary is a straight line. A dihedral angle is shown below the text of the definition. Examples of polyhedral angles are the roofs of houses. Slide 6 shows buildings with a dihedral and polyhedral roof.
Slide 7 shows an image of a polyhedral angle OA 1 A 2 A 3 ... A n . In the figure, the vertex of the corner is indicated, a point is marked on each ray, creating the designation of a polyhedral angle along the vertex and rays. The designation is displayed next to the figure and enclosed in a frame for memorization. The structure of polyhedral angle ОА 1 А 2 А 3 …А n . The following shows the trihedral angle ABCD, in which the flat corners are marked. The trihedral angle AA 1 DB is represented in the cube ABCDA 1 B 1 C 1 D 1, shown in the figure of slide 10. The image highlights a trihedral angle, the forming faces of which are painted in different colors, and flat corners are indicated. The next slide shows the roofs of buildings that have a shape - a hexagonal corner. In the figure, a flat corner and a hexagonal corner are marked.
The existence property of a plane intersecting all edges of a convex polyhedral angle is presented. To understand the essence of the property, you need to know the definition of a convex angle. It is marked next to the property. The definition states that the convex angle is on one side of the plane that contains each of the planar angles. The condition of the polyhedral angle property theorem stipulates that there is a convex polyhedral angle ∠ ОА 1 А 2 А 3 …An. On the rays OA 1 and OA 2 points K and M are marked, the connection of which is middle line triangle Δ OA 1 A 2. The plane passing through the CM and some point A i is located in such a way that all points A 1 , A 2 , A 3 , ... And n are on one side of α, and the vertex of the angle point O lies on the other side of the plane. It follows from this that the plane intersects all edges of a convex polyhedral angle. The theorem has been proven.
The next theorem, presented on slide 4, states that the sum of all plane angles of a polyhedral angle is less than 360°. The theorem is formulated as a property highlighted in a red frame for memorization. The proof of the property is illustrated in the figure, which shows the polyhedral angle ∠ ОА 1 А 2 А 3 …An. On the polyhedral angle marked vertex O, points belonging to the rays, A 1 , A 2 , A 3 ,…An. This is a convex polyhedral angle. The angle is intersected by a plane intersecting the rays at points A 1 , A 2 , A 3 ,…An. The sum of the flat angles of a polyhedral angle is represented by the expression A 1 OA 2 + A 2 OA 3 + ... + A n OA 1. Knowing the sum of the angles of the triangle, each of the flat angles is represented by expressions, for example, A 1 OA 2 \u003d 180 ° - OA 1 A 2 - OA 2 A 1, etc. As a result of the transformation of the expression, we obtain 180 ° n-(OA 1 A n + OA 1 A 2) - ... - (OA n A n-1 + OA n A 1). Considering the validity of the inequality OA 1 A n + OA 1 A 2 > A n A 1 A 2 ..., we calculate 180 ° n-(A n A 1 A 2 + A 1 A 2 A 3 + ... + A n-1 A n A 1 \u003d 180 ° n-180 ° (n-2) \u003d 360 ° The statement is proved.
The Polyhedral Angle presentation is used to improve the effectiveness of a traditional lesson at school. Also, this visual aid can become a teaching tool during distance learning. The material can be useful for students who independently master the topic, as well as for those who need an additional lesson for a deeper understanding of it.
Trihedral and polyhedral angles: A trihedral angle is a figure formed by three planes bounded by three rays emanating from one point and not lying in one plane. Consider some flat polygon and a point lying outside the plane of this polygon. Let's draw from this point the rays passing through the vertices of the polygon. We get a figure called a polyhedral angle.
A trihedral angle is a part of space bounded by three flat angles with a common vertex and pairwise common sides that do not lie in the same plane. The common vertex O of these angles is called the vertex of the trihedral angle. The sides of the corners are called edges, the flat corners at the vertex of a trihedral angle are called its faces. Each of the three pairs of faces of a trihedral angle forms a dihedral angle with flat angles
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The faces of a polyhedron are the polygons that form it. The edges of a polyhedron are the sides of the polygons. The vertices of a polyhedron are the vertices of a polygon. The diagonal of a polyhedron is a segment connecting 2 vertices that do not belong to the same face.
