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CHAPTER ONE LINES AND PLANES
V. DIHEDRAL ANGLES, A RIGHT ANGLE WITH A PLANE,
ANGLE OF TWO CROSSING RIGHTS, POLYHEDRAL ANGLES
dihedral angles
38. Definitions. The part of a plane lying on one side of a line lying in that plane is called half-plane. The figure formed by two half-planes (P and Q, Fig. 26) emanating from one straight line (AB) is called dihedral angle. The straight line AB is called edge, and the half-planes P and Q - parties or faces dihedral angle.
Such an angle is usually denoted by two letters placed at its edge (dihedral angle AB). But if there are no dihedral angles at one edge, then each of them is denoted by four letters, of which two middle ones are at the edge, and two extreme ones are at the faces (for example, the dihedral angle SCDR) (Fig. 27).
If, from an arbitrary point D, the edges AB (Fig. 28) are drawn on each face along the perpendicular to the edge, then the angle CDE formed by them is called linear angle dihedral angle.
The value of a linear angle does not depend on the position of its vertex on the edge. Thus, the linear angles CDE and C 1 D 1 E 1 are equal because their sides are respectively parallel and equally directed.
The plane of a linear angle is perpendicular to the edge because it contains two lines perpendicular to it. Therefore, to obtain a linear angle, it is sufficient to intersect the faces of a given dihedral angle with a plane perpendicular to the edge, and consider the angle obtained in this plane.
39. Equality and inequality of dihedral angles. Two dihedral angles are considered equal if they can be combined when nested; otherwise, one of the dihedral angles is considered to be smaller, which will form part of the other angle.
Like angles in planimetry, dihedral angles can be adjacent, vertical etc.
If two adjacent dihedral angles are equal to each other, then each of them is called right dihedral angle.
Theorems. 1) Equal dihedral angles correspond to equal linear angles.
2) A larger dihedral angle corresponds to a larger linear angle.
Let PABQ, and P 1 A 1 B 1 Q 1 (Fig. 29) be two dihedral angles. Embed the angle A 1 B 1 into the angle AB so that the edge A 1 B 1 coincides with the edge AB and the face P 1 with the face P.
Then if these dihedral angles are equal, then face Q 1 will coincide with face Q; if the angle A 1 B 1 is less than the angle AB, then the face Q 1 will take some position inside the dihedral angle, for example Q 2 .
Noticing this, we take some point B on a common edge and draw a plane R through it, perpendicular to the edge. From the intersection of this plane with the faces of dihedral angles, linear angles are obtained. It is clear that if the dihedral angles coincide, then they will have the same linear angle CBD; if the dihedral angles do not coincide, if, for example, the face Q 1 takes position Q 2, then the larger dihedral angle will have a larger linear angle (namely: / CBD > / C2BD).
40. Inverse theorems. 1) Equal linear angles correspond to equal dihedral angles.
2) A larger linear angle corresponds to a larger dihedral angle .
These theorems are easily proven by contradiction.
41. Consequences. 1) A right dihedral angle corresponds to a right linear angle, and vice versa.
Let (Fig. 30) the dihedral angle PABQ be a right one. This means that it is equal to the adjacent angle QABP 1 . But in this case, the linear angles CDE and CDE 1 are also equal; and since they are adjacent, each of them must be straight. Conversely, if the adjacent linear angles CDE and CDE 1 are equal, then the adjacent dihedral angles are also equal, i.e., each of them must be right.
2) All right dihedral angles are equal, because they have equal linear angles .
Similarly, it is easy to prove that:
3) Vertical dihedral angles are equal.
4) Dihedral angles with correspondingly parallel and equally (or oppositely) directed faces are equal.
5) If we take as a unit of dihedral angles such a dihedral angle that corresponds to a unit of linear angles, then we can say that a dihedral angle is measured by its linear angle.
In geometry, two important characteristics are used to study figures: the lengths of the sides and the angles between them. In the case of spatial figures, dihedral angles are added to these characteristics. Let's consider what it is, and also describe the method for determining these angles using the example of a pyramid.
The concept of dihedral angle
Everyone knows that two intersecting lines form an angle with the vertex at the point of their intersection. This angle can be measured with a protractor, or you can use trigonometric functions to calculate it. An angle formed by two right angles is called a linear angle.
Now imagine that in three-dimensional space there are two planes that intersect in a straight line. They are shown in the picture.
A dihedral angle is the angle between two intersecting planes. Just like linear, it is measured in degrees or radians. If to any point of the straight line along which the planes intersect, restore two perpendiculars lying in these planes, then the angle between them will be the desired dihedral. The easiest way to determine this angle is to use the general equations of planes.
The equation of planes and the formula for the angle between them
The equation of any plane in space in general terms is written as follows:
A × x + B × y + C × z + D = 0.
Here x, y, z are the coordinates of points belonging to the plane, the coefficients A, B, C, D are some known numbers. The convenience of this equality for calculating dihedral angles is that it explicitly contains the coordinates of the direction vector of the plane. We will denote it by n¯. Then:
The vector n¯ is perpendicular to the plane. The angle between two planes is equal to the angle between their n 1 ¯ and n 2 ¯. It is known from mathematics that the angle formed by two vectors is uniquely determined from their scalar product. This allows you to write a formula for calculating the dihedral angle between two planes:
φ = arccos (|(n 1 ¯ × n 2 ¯)| / (|n 1 ¯| × |n 2 ¯|)).
If we substitute the coordinates of the vectors, then the formula will be written explicitly:
φ = arccos (|A 1 × A 2 + B 1 × B 2 + C 1 × C 2 | / (√(A 1 2 + B 1 2 + C 1 2) × √(A 2 2 + B 2 2 + C 2 2))).
The modulo sign in the numerator is used to define only an acute angle, since a dihedral angle is always less than or equal to 90 o .
Pyramid and its corners
A pyramid is a figure that is formed by one n-gon and n triangles. Here n is an integer equal to the number of sides of the polygon that is the base of the pyramid. This spatial figure is a polyhedron or polyhedron, since it consists of flat faces (sides).
Pyramid polyhedra can be of two types:
- between the base and the side (triangle);
- between the two sides.
If the pyramid is considered correct, then it is not difficult to determine the named angles for it. To do this, according to the coordinates of three known points, an equation of the planes should be drawn up, and then use the formula given in the paragraph above for the angle φ.
Below we give an example in which we show how to find dihedral angles at the base of a quadrangular regular pyramid.
Quadrangular and the angle at its base
Suppose we are given a regular pyramid with a square base. The length of the side of the square is a, the height of the figure is h. Find the angle between the base of the pyramid and its side.
We place the origin of the coordinate system at the center of the square. Then the coordinates of points A, B, C, D, shown in the figure, will be equal to:
A = (a/2; -a/2; 0);
B = (a/2; a/2; 0);
C = (-a/2; a/2; 0);
Consider the planes ACB and ADB. Obviously, the direction vector n 1 ¯ for the plane ACB will be equal to:
To determine the direction vector n 2 ¯ of the ADB plane, we proceed as follows: we find arbitrary two vectors that belong to it, for example, AD¯ and AB¯, then we calculate their cross product. Its result will give the coordinates n 2 ¯. We have:
AD¯ = D - A = (0; 0; h) - (a/2; -a/2; 0) = (-a/2; a/2; h);
AB¯ = B - A = (a/2; a/2; 0) - (a/2; -a/2; 0) = (0; a; 0);
n 2 ¯ = = [(-a/2; a/2; h) × (0; a; 0)] = (-a × h; 0; -a 2 /2).
Since multiplication and division of a vector by a number does not change its direction, we transform the resulting n 2 ¯, dividing its coordinates by -a, we get:
We have defined direction vectors n 1 ¯ and n 2 ¯ for the base planes ACB and the lateral side ADB. It remains to use the formula for the angle φ:
φ = arccos (|(n 1 ¯ × n 2 ¯)| / (|n 1 ¯| × |n 2 ¯|)) = arccos (a / (2 × √h 2 + a 2 /4)).
Let's transform the resulting expression and rewrite it like this:
φ \u003d arccos (a / √ (a 2 + 4 × h 2)).
We have obtained the formula for the dihedral angle at the base for a regular quadrangular pyramid. Knowing the height of the figure and the length of its side, you can calculate the angle φ. For example, for the pyramid of Cheops, the side of the base of which is 230.4 meters, and the initial height was 146.5 meters, the angle φ will be equal to 51.8 o.
You can also determine the dihedral angle for a quadrangular regular pyramid using the geometric method. To do this, it suffices to consider a right-angled triangle formed by height h, half the length of the base a / 2 and the apothem of an isosceles triangle.
Dihedral angle. Linear angle of a dihedral angle. A dihedral angle is a figure formed by two half-planes that do not belong to the same plane and have a common boundary - a straight line a. The half-planes that form a dihedral angle are called its faces, and the common boundary of these half-planes is called the edge of the dihedral angle. The linear angle of a dihedral angle is the angle whose sides are the rays along which the faces of the dihedral angle intersect with a plane perpendicular to the edge of the dihedral angle. Each dihedral angle has as many linear angles as desired: through each point of an edge one can draw a plane perpendicular to this edge; the rays along which this plane intersects the faces of the dihedral angle, and form linear angles.
All linear angles of a dihedral angle are equal to each other. Let us prove that if the dihedral angles formed by the plane of the base of the pyramid KABC and the planes of its side faces are equal, then the base of the perpendicular drawn from the vertex K is the center of the circle inscribed in the triangle ABC.
Proof. First of all, we construct linear angles of equal dihedral angles. By definition, the plane of a linear angle must be perpendicular to the edge of a dihedral angle. Therefore, the edge of the dihedral angle must be perpendicular to the sides of the linear angle. If KO is perpendicular to the plane of the base, then we can draw OP perpendicular to AC, OR perpendicular to CB, OQ to perpendicular AB, and then connect points P, Q, R With point K. Thus, we will construct a projection of oblique RK, QK, RK so that the edges AC, CB, AB are perpendicular to these projections. Consequently, these edges are also perpendicular to the inclined ones. And therefore the planes of the triangles ROK, QOK, ROK are perpendicular to the corresponding edges of the dihedral angle and form those equal linear angles, which are mentioned in the condition. Right-angled triangles ROK, QOK, ROK are equal (since they have a common leg OK and the angles opposite to this leg are equal). Therefore, OR = OR = OQ. If we draw a circle with center O and radius OP, then the sides of the triangle ABC are perpendicular to the radii OP, OR and OQ and therefore are tangent to this circle.
Plane perpendicularity. Planes alpha and beta are called perpendicular if the linear angle of one of the dihedral angles formed at their intersection is 90". Signs of perpendicularity of two planes If one of the two planes passes through a line perpendicular to the other plane, then these planes are perpendicular.
The figure shows a rectangular parallelepiped. Its bases are rectangles ABCD and A1B1C1D1. And the side edges AA1 BB1, CC1, DD1 are perpendicular to the bases. It follows that AA1 is perpendicular to AB, i.e., the side face is a rectangle. Thus, it is possible to substantiate the properties of a cuboid: In a cuboid, all six faces are rectangles. In a cuboid, all six faces are rectangles. All dihedral angles of a cuboid are right angles. All dihedral angles of a cuboid are right angles.
Theorem The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions. Let us turn again to the figure, And we will prove that AC12 \u003d AB2 + AD2 + AA12 Since the edge CC1 is perpendicular to the base ABCD, then the angle AC1 is right. From the right triangle ACC1, according to the Pythagorean theorem, we obtain AC12=AC2+CC12. But AC is the diagonal of the rectangle ABCD, so AC2 = AB2+AD2. Also, CC1 = AA1. Therefore, AC12=AB2+AD2+AA12 The theorem is proved.
TEXT EXPLANATION OF THE LESSON:
In planimetry, the main objects are lines, segments, rays and points. Rays emanating from one point form one of their geometric shapes - an angle.
We know that a linear angle is measured in degrees and radians.
In stereometry, a plane is added to objects. The figure formed by the 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.
A dihedral angle, like a linear angle, can be named, measured, built. This is what we are going to find out in this lesson.
Find the dihedral angle on the ABCD tetrahedron model.
A dihedral angle with an edge AB is called CABD, where C and D points belong to different faces of the angle and the edge AB is called in the middle
Around us there are a lot of objects with elements in the form of a dihedral angle.
In many cities, special benches for reconciliation have been installed in parks. The bench is made in the form of two inclined planes converging towards the center.
In the construction of houses, the so-called gable roof is often used. The roof of this house is made in the form of a dihedral angle of 90 degrees.
The dihedral angle is also measured in degrees or radians, but how to measure it.
It is interesting to note that the roofs of the houses lie on the rafters. And the crate of the rafters forms two roof slopes at a given angle.
Let's transfer the image to the drawing. In the drawing, to find a dihedral angle, point B is marked on its edge. From this point, two beams BA and BC are drawn perpendicular to the edge of the angle. The angle ABC formed by these rays is called the linear angle of the dihedral angle.
The degree measure of a dihedral angle is equal to the degree measure of its linear angle.
Let's measure the angle AOB.
The degree measure of a given dihedral angle is sixty degrees.
Linear angles for a dihedral angle can be drawn in an infinite number, it is important to know that they are all equal.
Consider two linear angles AOB and A1O1B1. The rays OA and O1A1 lie in the same face and are perpendicular to the straight line OO1, so they are co-directed. Rays OB and O1B1 are also co-directed. Therefore, the angle AOB is equal to the angle A1O1B1 as angles with codirectional sides.
So a dihedral angle is characterized by a linear angle, and linear angles are acute, obtuse and right. Consider models of dihedral angles.
An obtuse angle is one whose linear angle is between 90 and 180 degrees.
A right angle if its linear angle is 90 degrees.
An acute angle, if its linear angle is between 0 and 90 degrees.
Let us prove one of the important properties of a linear angle.
The plane of a linear angle is perpendicular to the edge of the dihedral angle.
Let the angle AOB be the linear angle of the given dihedral angle. By construction, the rays AO and OB are perpendicular to the straight line a.
The plane AOB passes through two intersecting lines AO and OB according to the theorem: A plane passes through two intersecting lines, and moreover, only one.
The line a is perpendicular to two intersecting lines lying in this plane, which means that, by the sign of the perpendicularity of the line and the plane, the line a is perpendicular to the plane AOB.
To solve problems, it is important to be able to build a linear angle of a given dihedral angle. Construct the linear angle of the dihedral angle with the edge AB for the tetrahedron ABCD.
We are talking about a dihedral angle, which is formed, firstly, by the edge AB, one facet ABD, the second facet ABC.
Here is one way to build.
Let's draw a perpendicular from point D to the plane ABC, mark the point M as the base of the perpendicular. Recall that in a tetrahedron the base of the perpendicular coincides with the center of the inscribed circle in the base of the tetrahedron.
Let's draw an inclined line from point D perpendicular to the edge AB, mark the point N as the base of the inclined line.
In the triangle DMN, the segment NM will be the projections of the oblique DN onto the plane ABC. According to the three perpendiculars theorem, the edge AB will be perpendicular to the projection NM.
This means that the sides of the angle DNM are perpendicular to the edge AB, which means that the constructed angle DNM is the required linear angle.
Consider an example of solving the problem of calculating the dihedral angle.
Isosceles triangle ABC and regular triangle ADB do not lie in the same plane. The segment CD is perpendicular to the plane ADB. Find the dihedral angle DABC if AC=CB=2cm, AB=4cm.
The dihedral angle DABC is equal to its linear angle. Let's build this corner.
Let's draw an oblique CM perpendicular to the edge AB, since the triangle ACB is isosceles, then the point M will coincide with the midpoint of the edge AB.
The line CD is perpendicular to the plane ADB, which means it is perpendicular to the line DM lying in this plane. And the segment MD is the projection of the oblique SM onto the plane ADB.
The line AB is perpendicular to the oblique CM by construction, which means that by the three perpendiculars theorem it is perpendicular to the projection MD.
So, two perpendiculars CM and DM are found to the edge AB. So they form a linear angle СMD of a dihedral angle DABC. And it remains for us to find it from the right triangle СDM.
Since the segment SM is the median and the height of the isosceles triangle ASV, then according to the Pythagorean theorem, the leg of the SM is 4 cm.
From a right triangle DMB, according to the Pythagorean theorem, the leg DM is equal to two roots of three.
The cosine of an angle from a right triangle is equal to the ratio of the adjacent leg MD to the hypotenuse CM and is equal to three roots of three by two. So the angle CMD is 30 degrees.
