Preparing students for the exam in mathematics, as a rule, begins with a repetition of the basic formulas, including those that allow you to determine the angle between the planes. Despite the fact that this section of geometry is covered in sufficient detail within the framework of the school curriculum, many graduates need to repeat the basic material. Understanding how to find the angle between the planes, high school students will be able to quickly calculate the correct answer in the course of solving the problem and count on getting decent scores on the basis of the unified state exam.
Main nuances
So that the question of how to find the dihedral angle does not cause difficulties, we recommend that you follow the solution algorithm that will help you cope with the tasks of the exam.
First you need to determine the line along which the planes intersect.
Then on this line you need to choose a point and draw two perpendiculars to it.
The next step is to find the trigonometric function of the dihedral angle, which is formed by the perpendiculars. It is most convenient to do this with the help of the resulting triangle, of which the corner is a part.
The answer will be the value of the angle or its trigonometric function.
Preparation for the exam test together with Shkolkovo is the key to your success
In the process of studying on the eve of passing the exam, many students are faced with the problem of finding definitions and formulas that allow you to calculate the angle between 2 planes. A school textbook is not always at hand exactly when it is needed. And in order to find the necessary formulas and examples of their correct application, including for finding the angle between planes on the Internet online, sometimes you need to spend a lot of time.
Mathematical portal "Shkolkovo" offers a new approach to preparing for the state exam. Classes on our website will help students identify the most difficult sections for themselves and fill gaps in knowledge.
We have prepared and clearly presented all the necessary material. Basic definitions and formulas are presented in the "Theoretical Reference" section.
In order to better assimilate the material, we also suggest practicing the corresponding exercises. A large selection of tasks of varying degrees of complexity, for example, on, is presented in the Catalog section. All tasks contain a detailed algorithm for finding the correct answer. The list of exercises on the site is constantly supplemented and updated.
Practicing in solving problems in which it is required to find the angle between two planes, students have the opportunity to save any task online to "Favorites". Thanks to this, they will be able to return to him the necessary number of times and discuss the progress of his solution with a school teacher or tutor.
Back forward
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Lesson Objectives: introduce the concept of a dihedral angle and its linear angle;
During the classes
I. Organizational moment.
Inform the topic of the lesson, form the objectives of the lesson.
II. Actualization of students' knowledge (slide 2, 3).
1. Preparation for the study of new material.
What is called an angle on a plane?
What is the angle between lines in space called?
What is the angle between a line and a plane called?
Formulate the three perpendiculars theorem
III. Learning new material.
- The concept of a dihedral angle.
The figure formed by two half-planes passing through the line MN is called a dihedral angle (slide 4).
Half-planes are faces, straight line MN is an edge of a dihedral angle.
What objects in everyday life have the shape of a dihedral angle? (Slide 5)
- The angle between the planes ACH and CHD is the dihedral angle ACND, where CH is an edge. Points A and D lie on the faces of this angle. Angle AFD is the linear angle of the dihedral angle ACHD (slide 6).
- Algorithm for constructing a linear angle (slide 7).
1 way. Take any point O on the edge and draw perpendiculars to this point (PO DE, KO DE) and get the angle ROCK - linear.
2 way. Take a point K in one half-plane and drop two perpendiculars from it to the other half-plane and an edge (KO and КР), then by the inverse TTP theorem PODE
- All linear angles of a dihedral angle are equal (slide 8). Proof: the rays OA and O 1 A 1 are co-directed, the rays OB and O 1 B 1 are also co-directed, the angles BOA and B 1 O 1 A 1 are equal as angles with co-directed sides.
- The degree measure of a dihedral angle is the degree measure of its linear angle (slide 9).
IV. Consolidation of the studied material.
- Problem solving (orally according to ready-made drawings). (Slides 10-12)
1. RAVS - pyramid; the angle ACB is 90°, the straight line PB is perpendicular to the plane ABC. Prove that angle PCB is a linear angle of a dihedral angle with
2. RAVS - pyramid; AB \u003d BC, D is the midpoint of the segment AC, the straight line PB is perpendicular to the plane ABC. Prove that angle PDB is a linear angle of a dihedral angle with edge AC.
3. PABCD - pyramid; line PB is perpendicular to plane ABC, BC is perpendicular to DC. Prove that angle PKB is a linear angle of a dihedral angle with edge CD.
- Tasks for constructing a linear angle (slides 13-14).
1. Construct a linear angle of a dihedral angle with an edge AC, if in the pyramid RABC the face ABC is a regular triangle, O is the intersection point of the medians, the line RO is perpendicular to the plane ABC
2. Rhombus ABCD is given. The straight line PC is perpendicular to the plane ABCD.
Construct a linear angle of a dihedral angle with edge BD and a linear angle of a dihedral angle with edge AD.
- Computational task. (Slide 15)
In the parallelogram ABCD, the angle ADC is 120 0, AD = 8 cm,
DC = 6 cm, straight line PC is perpendicular to the plane ABC, PC = 9 cm.
Find the value of the dihedral angle with the edge AD and the area of the parallelogram.
V. Homework (slide 16).
P. 22, No. 168, 171.
Used Books:
- Geometry 10-11 L.S. Atanasyan.
- The system of tasks on the topic “Dihedral angles” by M.V. Sevostyanova (Murmansk), journal Mathematics at school 198 ...
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 SM 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.
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.
