The purpose of the lesson: introduction of the concept of a dihedral angle and its linear angle.
Tasks:
Educational: to consider tasks for the application of these concepts, to form a constructive skill of finding the angle between planes;
Developing: development of creative thinking of students, personal self-development of students, development of students' speech;
Educational: education of the culture of mental work, communicative culture, reflective culture.
Lesson type: a lesson in learning new knowledge
Teaching methods: explanatory and illustrative
Equipment: computer, interactive whiteboard.
Literature:
Geometry. Grades 10-11: textbook. for 10-11 cells. general education institutions: basic and profile. levels / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev and others] - 18th ed. - M. : Education, 2009. - 255 p.
Lesson plan:
Organizational moment (2 min)
Updating knowledge (5 min)
Learning new material (12 min)
Consolidation of the studied material (21 min)
Homework (2 min)
Summing up (3 min)
During the classes:
1. Organizational moment.
Includes a greeting by the teacher of the class, preparation of the room for the lesson, checking absentees.
2. Actualization of basic knowledge.
Teacher: In the last lesson, you wrote an independent work. In general, the work was well written. Now let's repeat a little. What is called an angle on a plane?
Student: An angle in a plane is a figure formed by two rays emanating from one point.
Teacher: What is the angle between lines in space called?
Student: The angle between two intersecting lines in space is the smallest of the angles formed by the rays of these lines with the vertex at the point of their intersection.
Student: The angle between intersecting lines is the angle between intersecting lines, respectively, parallel to the data.
Teacher: What is the angle between a line and a plane called?
Student: Angle between line and planeAny angle between a straight line and its projection onto this plane is called.
3. Study of new material.
Teacher: In stereometry, along with such angles, another type of angles is considered - dihedral angles. You probably already guessed what the topic of today's lesson is, so open your notebooks, write down today's date and the topic of the lesson.
Writing on the board and in notebooks:
10.12.14.
Dihedral angle.
Teacher : To introduce the concept of a dihedral angle, it should be recalled that any straight line drawn in a given plane divides this plane into two half-planes(Fig. 1a)
Teacher : Imagine that we have bent the plane along a straight line so that two half-planes with the boundary turned out to be no longer lying in the same plane (Fig. 1, b). The resulting figure is the dihedral angle. A dihedral angle is a figure formed by a straight line and two half-planes with a common boundary that do not belong to the same plane. The half-planes forming a dihedral angle are called its faces. A dihedral angle has two faces, hence the name - dihedral angle. The straight line - the common boundary of the half-planes - is called the edge of the dihedral angle. Write the definition in your notebook.
A dihedral angle is a figure formed by a straight line and two half-planes with a common boundary that do not belong to the same plane.
Teacher : In everyday life, we often encounter objects that have the shape of a dihedral angle. Give examples.
Student : Half open folder.
Student : The wall of the room together with the floor.
Student : Gable roofs of buildings.
Teacher : Correctly. And there are many such examples.
Teacher : As you know, angles on a plane are measured in degrees. You probably have a question, but how are dihedral angles measured? This is done in the following way.We mark some point on the edge of the dihedral angle, and in each face from this point we draw a ray perpendicular to the edge. The angle formed by these rays is called the linear angle of the dihedral angle. Make a drawing in your notebooks.
Writing on the board and in notebooks.
O ∈ a, AO ⊥ a, VO ⊥ a, SABD- dihedral angle,∠ AOBis the linear angle of the dihedral angle.
Teacher : All linear angles of a dihedral angle are equal. Make yourself something like this.
Teacher : Let's prove it. Consider two linear angles AOB andPQR. Rays OA andQPlie on the same face and are perpendicularOQ, which means they are aligned. Similarly, the rays OB andQRco-directed. Means,∠ AOB= ∠ PQR(like angles with codirectional sides).
Teacher : Well, now the answer to our question is how the dihedral angle is measured.The degree measure of a dihedral angle is the degree measure of its linear angle. Redraw the drawings of an acute, right, and obtuse dihedral angle from the textbook on page 48.
4. Consolidation of the studied material.
Teacher : Make drawings for tasks.
№ 1 . Given: ΔABC, AC = BC, AB lies in the planeα, CD ⊥ α, C∉ a. Construct Linear Angle of Dihedral AngleCABD.
Student : Decision:CM ⊥ AB, DC ⊥ AB.∠ cmd - desired.
№ 2. Given: ΔABC, ∠ C= 90°, BC lies planeα, AO⊥ α, A∈ α.
Construct Linear Angle of Dihedral AngleAVSO.
Student : Decision:AB ⊥ BC, JSC⊥ Sun means OS⊥ Sun.∠ ACO - desired.
№ 3 . Given: ΔABC, ∠ C \u003d 90 °, AB lies in the planeα, CD⊥ α, C∉ a. Buildlinear dihedral angleDABC.
Student : Decision: CK ⊥ AB, DC ⊥ AB,DK ⊥ AB means∠ DKC - desired.
№ 4
. Given:DABC- tetrahedron,DO⊥
ABC.Construct the linear angle of the dihedral angleABCD.
Student : Decision:DM ⊥ sun,DO ⊥ BC means OM⊥ sun;∠ OMD - desired.
5. Summing up.
Teacher: What new did you learn at the lesson today?
Students : What is called dihedral angle, linear angle, how dihedral angle is measured.
Teacher : What did you repeat?
Students : What is called an angle on a plane; angle between lines.
6. Homework.
Writing on the board and in the diaries: item 22, no. 167, no. 170.
The concept of a dihedral angle
To introduce the concept of a dihedral angle, first we recall one of the axioms of stereometry.
Any plane can be divided into two half-planes of the line $a$ lying in this plane. In this case, the points lying in the same half-plane are on the same side of the straight line $a$, and the points lying in different half-planes are on opposite sides of the straight line $a$ (Fig. 1).
Picture 1.
The principle of constructing a dihedral angle is based on this axiom.
Definition 1
The figure is called dihedral angle if it consists of a line and two half-planes of this line that do not belong to the same plane.
In this case, the half-planes of the dihedral angle are called faces, and the straight line separating the half-planes - dihedral edge(Fig. 1).
Figure 2. Dihedral angle
Degree measure of a dihedral angle
Definition 2
We choose an arbitrary point $A$ on the edge. The angle between two lines lying in different half-planes, perpendicular to the edge and intersecting at the point $A$ is called linear angle dihedral angle(Fig. 3).
Figure 3
Obviously, every dihedral angle has an infinite number of linear angles.
Theorem 1
All linear angles of one dihedral angle are equal to each other.
Proof.
Consider two linear angles $AOB$ and $A_1(OB)_1$ (Fig. 4).
Figure 4
Since the rays $OA$ and $(OA)_1$ lie in the same half-plane $\alpha $ and are perpendicular to one straight line, they are codirectional. Since the rays $OB$ and $(OB)_1$ lie in the same half-plane $\beta $ and are perpendicular to one straight line, they are codirectional. Hence
\[\angle AOB=\angle A_1(OB)_1\]
Due to the arbitrariness of the choice of linear angles. All linear angles of one dihedral angle are equal to each other.
The theorem has been proven.
Definition 3
The degree measure of a dihedral angle is the degree measure of a linear angle of a dihedral angle.
Task examples
Example 1
Let us be given two non-perpendicular planes $\alpha $ and $\beta $ which intersect along the line $m$. The point $A$ belongs to the plane $\beta $. $AB$ is the perpendicular to the line $m$. $AC$ is perpendicular to the plane $\alpha $ (point $C$ belongs to $\alpha $). Prove that the angle $ABC$ is a linear angle of the dihedral angle.
Proof.
Let's draw a picture according to the condition of the problem (Fig. 5).
Figure 5
To prove this, we recall the following theorem
Theorem 2: A straight line passing through the base of an inclined one, perpendicular to it, is perpendicular to its projection.
Since $AC$ is a perpendicular to the $\alpha $ plane, then the point $C$ is the projection of the point $A$ onto the $\alpha $ plane. Hence $BC$ is the projection of the oblique $AB$. By Theorem 2, $BC$ is perpendicular to an edge of a dihedral angle.
Then, the angle $ABC$ satisfies all the requirements for defining the linear angle of a dihedral angle.
Example 2
The dihedral angle is $30^\circ$. On one of the faces lies the point $A$, which is at a distance of $4$ cm from the other face. Find the distance from the point $A$ to the edge of the dihedral angle.
Decision.
Let's look at Figure 5.
By assumption, we have $AC=4\ cm$.
By definition of the degree measure of a dihedral angle, we have that the angle $ABC$ is equal to $30^\circ$.
Triangle $ABC$ is a right triangle. By definition of the sine of an acute angle
\[\frac(AC)(AB)=sin(30)^0\] \[\frac(5)(AB)=\frac(1)(2)\] \
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Slides captions:
DOUBLE ANGLE Mathematics teacher GOU secondary school №10 Eremenko M.A.
The main objectives of the lesson: Introduce the concept of a dihedral angle and its linear angle Consider tasks for the application of these concepts
Definition: A dihedral angle is a figure formed by two half-planes with a common boundary line.
The value of a dihedral angle is the value of its linear angle. AF ⊥ CD BF ⊥ CD AFB is the linear angle of the dihedral angle ACD B
Let us prove that all linear angles of a dihedral angle are equal to each other. Consider two linear angles AOB and A 1 OB 1 . Rays OA and OA 1 lie on the same face and are perpendicular to OO 1, so they are co-directed. Rays OB and OB 1 are also co-directed. Therefore, ∠ AOB = ∠ A 1 OB 1 (as angles with codirectional sides).
Examples of dihedral angles:
Definition: The angle between two intersecting planes is the smallest of the dihedral angles formed by these planes.
Task 1: In the cube A ... D 1 find the angle between the planes ABC and CDD 1 . Answer: 90o.
Task 2: In the cube A ... D 1 find the angle between the planes ABC and CDA 1 . Answer: 45o.
Task 3: In the cube A ... D 1 find the angle between the planes ABC and BDD 1 . Answer: 90o.
Task 4: In the cube A ... D 1 find the angle between the planes ACC 1 and BDD 1 . Answer: 90o.
Task 5: In the cube A ... D 1 find the angle between the planes BC 1 D and BA 1 D . Solution: Let O be the midpoint of B D. A 1 OC 1 is the linear angle of the dihedral angle A 1 B D C 1 .
Problem 6: In the tetrahedron DABC all edges are equal, point M is the midpoint of edge AC. Prove that ∠ DMB is a linear angle of dihedral angle BACD .
Solution: Triangles ABC and ADC are regular, so BM ⊥ AC and DM ⊥ AC and hence ∠ DMB is a linear angle of dihedral angle DACB .
Task 7: From the vertex B of the triangle ABC, the side AC of which lies in the plane α, a perpendicular BB 1 is drawn to this plane. Find the distance from point B to the line AC and to the plane αif AB=2, ∠BAC=150 0 and the dihedral angle BACB 1 is 45 0 .
Solution: ABC is an obtuse triangle with an obtuse angle A, so the base of height BK lies on the extension of side AC. VC is the distance from point B to AC. BB 1 - distance from point B to plane α
2) Since AS ⊥VK, then AS⊥KV 1 (by the theorem converse to the three perpendiculars theorem). Therefore, ∠VKV 1 is the linear angle of the dihedral angle BACB 1 and ∠VKV 1 =45 0 . 3) ∆VAK: ∠A=30 0 , VK=VA sin 30 0 , VK =1. ∆VKV 1: VV 1 \u003d VK sin 45 0, VV 1 \u003d
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 required 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 ACB plane will be equal to:
To determine the direction vector n 2 ¯ of the ADB plane, we proceed as follows: find two arbitrary vectors that belong to it, for example, AD¯ and AB¯, then 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.
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.
Draw a slope from point D perpendicular to edge AB, mark point N as the base of the slope.
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.
