MAOU "Lyceum of Innovative Technologies"
Polyhedral angles. Convex polyhedra
Prepared by 10B class student: Alexey Burykin
Checked by: Dubinskaya I.A.
Khabarovsk
Polyhedral angle
Polyhedral angle is a figure formed by plane angles such that the following conditions are met:
1) no two angles have common points, except for 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 corners that have a common side;
4) no two angles with common side do not lie in the same plane.
- Angles ASB, BSC,... are called flat angles or edges, their sides SA, SB, ... are called ribs, and the common vertex S- top polyhedral angle.
Theorem 1.
In a trihedral angle, each plane angle is less than the sum of the other two plane angles.
Consequence
- / ASC- / ASB/CSB; / ASC- / CSB/ASB.
In a trihedral angle, each plane angle is greater than the difference of the other two angles .
Theorem2.
- The sum of the values of all three plane angles of a trihedral angle is less than 360° .
180°, which means that α + β + γ " width="640" Proof
Let us denote
then from the triangles ASC, ASB, BSC we have
Now the inequality takes the form
180° - α + 180° - β + 180° - γ 180°,
whence it follows that
α + β + γ
The simplest cases of equality of trihedral angles
- 1) along an equal dihedral angle enclosed between two correspondingly equal and identically spaced plane angles , or 2) along an equal plane angle enclosed between two correspondingly equal and identically located dihedral angles .
Convex polyhedral angle
- A polyhedral angle is called convex if it is entirely located on one side of the plane of each of its faces, which is extended indefinitely.
Polyhedron.
Polyhedron, in three-dimensional space - a collection finite number flat polygons, such that each side of any of the polygons is simultaneously the side of another, called adjacent to the first.
Convex polyhedra
Polyhedron called convex, if it lies entirely on one side of the plane of any of its faces; then its edges are also convex.
Convex polyhedron cuts the space into two parts - external and internal. Its inner part is a convex body. Conversely, if the surface of a convex body is polyhedral, then the corresponding polyhedron is convex.
Theorem. The sum of all the plane angles of a convex polyhedral angle is less than 360 degrees.
Property 1. In a convex polyhedron, all faces are convex polygons.
Property2. Any convex polyhedron can be composed of pyramids with a common vertex, the base of which forms the surface of the polyhedron.
A dihedral angle is a figure formed by two half-planes with a common straight line limiting them. Half-planes are called faces, and the straight line limiting them is called an edge of a dihedral angle.
Figure 142 shows a dihedral angle with edge a and faces a and (3.
A plane perpendicular to the edge of a dihedral angle intersects its faces along two half-lines. The angle formed by these half-lines is called the linear angle of the dihedral angle. The measure of a dihedral angle is taken to be the measure of its corresponding linear angle. If through point A of edge a of a dihedral angle we draw a plane y perpendicular to this edge, then it will intersect planes a and (3 along half-lines (Fig. 142); the linear angle of a given dihedral angle. The degree measure of this linear angle is degree measure dihedral angle. The measure of the dihedral angle does not depend on the choice of the linear angle.
A trihedral angle is a figure made up of three flat angles (Fig. 143). 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 and their extensions are called the dihedral angles of a trihedral angle.
The concept of a polyhedral angle is defined similarly as a figure made up of flat angles (Fig. 144). For a polyhedral angle, the concepts of faces, edges, and dihedral angles are defined in the same way as for a trihedral angle.
A polyhedron is a body whose surface consists of a finite number of flat polygons (Fig. 145).
A polyhedron is called convex if it is located on one side of the plane of each polygon on its surface (Fig. 145, a, b). a common part such a plane and the surface of a convex polyhedron is called a face. The faces of a convex polyhedron are convex polygons. The sides of the faces are called the edges of the polyhedron, and the vertices are called the vertices of the polyhedron.
Polyhedral angles A polyhedral angle is a spatial analogue of a polygon on a plane. Recall that a polygon on a plane is a figure formed by a simple closed broken line of this plane and the internal region limited by it.
Definition of polyhedral angle A surface formed by a finite set of plane angles A 1 SA 2, A 2 SA 3, ..., An-1 SAn, An. SA 1 with a common vertex S, in which neighboring angles do not have common points, except for points of a common ray, and non-neighboring corners do not have common points, except for a common vertex, will be called a polyhedral surface. 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 SA 1, ..., SAn are called the edges of the polyhedral angle, and the plane angles themselves A 1 SA 2, A 2 SA 3, ..., An-1 SAn, An. SA 1 – faces of a polyhedral angle. A polyhedral angle is denoted by the letters SA 1...An, indicating the vertex and points on its edges.
Types of polyhedral angles Depending on the number of faces, polyhedral angles are trihedral, tetrahedral, pentagonal, etc.
Exercise 1 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.
Exercise 2 Give examples of polyhedra whose faces, intersecting at the vertices, form only: a) trihedral and tetrahedral angles; b) trihedral and pentagonal angles; c) tetrahedral and pentagonal angles. Answer: a) quadrangular pyramid, triangular bipyramid; b) pentagonal pyramid; c) pentagonal bipyramid.
Triangle inequality For a triangle, the following theorem holds. Theorem (Triangle inequality). Each side of a triangle is less than the sum of the other two sides. Let us prove that for a trihedral angle the following holds: spatial analogue this theorem. 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
Intersection point of bisectors For a triangle, the following theorem holds. Theorem. The bisectors of a triangle intersect at one point - the center of the inscribed circle. Let us prove that for a trihedral angle the following spatial analogue of this theorem holds. Theorem. The bisector planes of the dihedral angles of a trihedral angle intersect along one straight line.
Proof Consider the trihedral angle SABC. The bisector plane SAD of the dihedral angle SA is locus points of this angle equidistant from its faces SAB and SAC. Similarly, the bisector plane SBE of a dihedral angle SB is the locus of the points of this angle equidistant from its faces SAB and SBC. The line of their intersection SO will consist of points equidistant from all faces of the trihedral angle. Consequently, the bisector plane of the dihedral angle SC will pass through it.
Intersection point of perpendicular bisectors For a triangle, the following theorem holds. Theorem. The perpendicular bisectors to the sides of the triangle intersect at one point - the center of the circumcircle. Let us prove that for a trihedral angle the following spatial analogue of this theorem holds. Theorem. Planes passing through the bisectors of the faces of a trihedral angle and perpendicular to these faces intersect along one straight line.
Proof Consider the trihedral angle SABC. The plane passing through the bisector SD of the angle BSC and perpendicular to its plane consists of points equidistant from the edges SB and SC of the trihedral angle SABC. Similarly, the plane passing through the bisector SE of the angle ASC and perpendicular to its plane consists of points equidistant from the edges SA and SC of the trihedral angle SABC. The line of their intersection SO will consist of points equidistant from all the edges of the trihedral angle. Consequently, it will be contained by a plane passing through the bisector of angle ASB and perpendicular to its plane.
Intersection point of medians For a triangle, the following theorem holds. Theorem. The medians of a triangle intersect at one point - the center of the inscribed circle. Let us prove that for a trihedral angle the following spatial analogue of this theorem holds. Theorem. The planes passing through the edges of a trihedral angle and the bisectors of opposite faces intersect along one straight line.
Proof Consider the trihedral angle SABC. We'll put it on his ribs equal segments SA = SB = CS. Bisectors SD, SE, SF of plane angles of a trihedral angle are the medians of triangles SBC, SAB, respectively. Therefore, AD, BE, CF are medians triangle ABC. Let O be the intersection point of the medians. Then straight line SO will be the line of intersection of the planes under consideration.
Point of intersection of altitudes For a triangle, the following theorem holds. Theorem. The altitudes of a triangle or their extensions intersect at one point. Let us prove that for a trihedral angle the following spatial analogue of this theorem holds. Theorem. Planes passing through the edges of a trihedral angle and perpendicular to the planes of opposite faces intersect along one straight line.
Proof Consider the trihedral angle Sabc. Let d, e, f be the lines of intersection of the planes of the faces of a trihedral angle with the planes passing through the edges a, b, c of this angle and perpendicular to the corresponding planes of the faces. Let's choose some point C on the edge c. Let us drop the perpendiculars CD and CE from it onto lines d and e, respectively. Let us denote by A and B the points of intersection of lines CD and CE with lines SB and SA, respectively. Line d is orthogonal projection direct AD to the BSC plane. Since BC is perpendicular to line d, it is also perpendicular to line AD. Similarly, line AC is perpendicular to line BE. Let O be the intersection point of lines AD and BE. Line BC is perpendicular to plane SAD, therefore it is perpendicular to line SO. Similarly, Line AC is perpendicular to plane SBE, hence it is perpendicular to line SO. Thus, line SO is perpendicular to lines BC and AC, therefore, perpendicular to plane ABC, which means it is perpendicular to line AB. On the other hand, line CO is perpendicular to line AB. Thus, line AB is perpendicular to plane SOC. Plane SAB passes through line AB, perpendicular to the plane The SOC is therefore itself perpendicular to this plane. This means that all three planes under consideration intersect along the straight line SO.
Sum of plane angles Theorem. The sum of the plane angles of a trihedral angle is less than 360°. Proof. Let SABC be the given trihedral angle. Consider a trihedral angle with vertex A formed by faces ABS, ACS and angle BAC. Due to the triangle inequality, the inequality BAC holds
Convex polyhedral 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. 
Exercise 5 Two plane angles of a trihedral angle are 70° and 80°. What are the boundaries of the third plane angle? Answer: 10 o
Exercise 6 The plane angles of a trihedral angle are 45°, 45° and 60°. Find the angle between the planes of plane angles of 45°. Answer: 90 o.
Exercise 7 In a trihedral angle, two plane angles are equal to 45°; the dihedral angle between them is right. Find the third plane angle. Answer: 60 o.
Exercise 8 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: 90 o.
Exercise 9 Each plane angle of a trihedral angle is equal to 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
Definitions. Let's take several angles (Fig. 37): ASB, BSC, CSD, which, adjacent sequentially to one another, 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 a certain 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 a certain dihedral angle with the CSD plane. Let's continue this sequential rotation around each common side. If the last side SF coincides with the first side SA, then a figure is formed (Fig. 38), which is called polyhedral angle. Angles ASB, BSC,... are called flat angles or edges, their sides SA, SB, ... are called ribs, and the common vertex S- top polyhedral angle.
Each edge is also an edge of a certain dihedral angle; therefore, in a polyhedral angle there are as many dihedral angles and as many plane angles as there are all the edges in it. Smallest number there are three faces in a polyhedral angle; this angle is called triangular. There may be tetrahedral, pentagonal, 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 - the edges in the order of their location.
A polyhedral angle is called convex if it is entirely located on one side of the plane of each of its faces, which is extended indefinitely. This is, for example, 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 edge or the BCC edge.
If we intersect all the faces of a polyhedral angle with 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 plane angle is less than the sum of the other two plane angles.
Let the largest of the plane angles in the trihedral angle SABC (Fig. 40) be the angle ASC.
Let us plot on this angle the angle ASD, equal to the angle ASB, and draw some straight line AC intersecting SD at some point D. Let us plot SB = SD. By 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: therefore AD = AB. Therefore, if in the derived inequality we discard the equal terms AD and AB, we obtain that DC< ВС.
Now we notice that in triangles SCD and SCB, two sides of one are equal to two sides of the other, but the third sides are not equal; in this case, the larger angle lies opposite the larger of these sides; Means,
∠CSD< ∠ CSВ.
By adding the angle ASD to the left side of this inequality, and the angle ASB equal to it to the right, we obtain the inequality that needed to be proved:
∠ASC< ∠ CSB + ∠ ASB.
We have proven that even the largest plane angle is less than the sum of the other two angles. This means the theorem is proven.
Consequence. Subtract from both sides of the last inequality by angle ASB or angle CSB; we get:
∠ASC - ∠ASB< ∠ CSB;
∠ASC - ∠CSB< ∠ ASB.
Considering these inequalities from right to left and taking into account that angle ASC as the largest of three corners greater than the difference of the other two angles, we come to the conclusion 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 plane angles is less than 4d (360°) .
Let's cross the edges (Fig. 41) convex angle SABCDE by some plane; from this we get a convex cross-section n-gon ABCDE.
Applying the theorem proved earlier to each of the trihedral angles whose vertices are located at points A, B, C, D and E, we pacholym:
∠ABC< ∠ABS + ∠SВC, ∠BCD < ∠BCS + ∠SCD и т. д.
Let us add up 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 triangles ABS, SBC, etc., except for those angles that lie at the vertex S. Denoting the sum of these last angles with the letter X , we get after addition:
2dn - 4d < 2dn - x .
Since in differences 2 dn - 4d and 2 dn - x the 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 the deductible X ; that 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) along an equal dihedral angle enclosed between two correspondingly equal and identically spaced plane angles, or
2) along an equal plane angle enclosed between two correspondingly equal and identically spaced dihedral angles.
1) Let S and S 1 be two trihedral angles (Fig. 42), for which ∠ASB = ∠A 1 S 1 B 1, ∠ASC = ∠A 1 S 1 C 1 (and these equal angles identically located) and the dihedral angle AS is equal to the dihedral angle A 1 S 1 .
Let us insert the angle S 1 into the angle S so that their points S 1 and S, straight lines S 1 A 1 and SA and 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 the angles A 1 S 1 B 1 and ASB), the plane A 1 S 1 C 1 will go along ASC (by virtue of the equality of dihedral angles) and the edge S 1 C 1 will go along the edge SC (due to the equality of angles A 1 S 1 C 1 and ASC). Thus, the trihedral angles will coincide with all their edges, i.e. they will be equal.
2) The second sign, like the first, is proved by embedding.
Symmetrical polyhedral angles
As is known, vertical angles are equal when we are talking about angles formed by straight lines or planes. Let's see if this statement is true in relation to polyhedral angles.
Let us 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 relative to the first angle. It is easy to see that both angles have equal flat and dihedral angles, respectively, but both are located in reverse order. Indeed, if we imagine an observer looking from outside a polyhedral angle at its vertex, then the edges SA, SB, SC, SD, SE will seem to him to be located in a counterclockwise direction, whereas, looking at the angle SA 1 B 1 C 1 D 1 E 1, he sees the edges SA 1, SB 1, ..., located in a clockwise direction.
Polyhedral angles with correspondingly equal plane and dihedral angles, but located in the opposite order, generally cannot be combined when nested; that means they are not equal. Such angles are called symmetrical(relative to vertex S). The symmetry of figures in space will be discussed in more detail below.
Other materialsLet 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 made up of three flat angles (ab), (bc) and (ac) (Fig. 2). These angles are called the faces of a trihedral angle, and their sides are called edges; the common vertex of flat angles is called 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 defined similarly (Fig. 3).
Polyhedron
In stereometry, figures in space called bodies are studied. A visual (geometric) body must be imagined as a part of space occupied physical body and limited by the 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 is located on one side of the plane of every plane 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 this using the example of a familiar cube (Fig. 5). A cube is a convex polyhedron. Its surface consists of six squares: ABCD, BEFC, .... These are its faces. The edges of the cube are the sides of these squares: AB, BC, BE,.... The vertices of a cube are the vertices of the squares: A, B, C, D, E, .... The cube has six faces, twelve edges and eight vertices.
For 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 body. They will be defined as geometric figures indicating all the points in space belonging to them. Concept geometric body and its surface in general case will be given later.