Dihedral angle. Linear 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 - straight line a. The half-planes forming 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 an angle whose sides are the rays along which the faces of the dihedral angle are intersected by a plane perpendicular to the edge of the dihedral angle. Each dihedral angle has any number of linear angles: 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 a dihedral angle 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 lateral faces are equal, then the base of the perpendicular drawn from vertex K is the center of the inscribed circle in triangle ABC.
Proof. First of all, let's construct linear angles of equal dihedral angles. By definition, the plane of a linear angle must be perpendicular to the edge of the dihedral angle. Therefore, the edge of a dihedral angle must be perpendicular to the sides of the linear angle. If KO is perpendicular to the base plane, then we can draw OR perpendicular AC, OR perpendicular SV, OQ perpendicular AB, and then connect points P, Q, R WITH point K. Thus, we will construct a projection of inclined RK, QK, RK so that the edges AC, NE, AB are perpendicular to these projections. Consequently, these edges are perpendicular to the inclined ones themselves. And therefore the planes of triangles ROK, QOK, ROK are perpendicular to the corresponding edges of the dihedral angle and form those equal linear angles that are mentioned in the condition. Right triangles ROK, QOK, ROK are congruent (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 triangle ABC are perpendicular to the radii OP, OR and OQ and therefore are tangent to this circle.
Perpendicularity of planes. The alpha and beta planes are called perpendicular if the linear angle of one of the dihedral angles formed at their intersection is equal to 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 ribs 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, we can justify the properties of a rectangular parallelepiped: In a rectangular parallelepiped, all six faces are rectangles. In a rectangular parallelepiped, all six faces are rectangles. All dihedral angles of a rectangular parallelepiped are right angles. All dihedral angles of a rectangular parallelepiped 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 prove that AC12 = AB2 + AD2 + AA12 Since edge CC1 is perpendicular to the base ABCD, angle ACC1 is right. From the right triangle ACC1, using the Pythagorean theorem, we obtain AC12 = AC2 + CC12. But AC is a diagonal of rectangle ABCD, so AC2 = AB2 + AD2. In addition, CC1 = AA1. Therefore AC12= AB2+AD2+AA12 The theorem is proven.
The purpose of the lesson: introduction of the concept of dihedral angle and its linear angle.
Tasks:
Educational: consider tasks on the application of these concepts, develop the constructive skill of finding the angle between planes;
Developmental: development of creative thinking of students, personal self-development of students, development of students’ speech;
Educational: nurturing a culture of mental work, communicative culture, reflective culture.
Lesson type: lesson in learning new knowledge
Teaching methods: explanatory and illustrative
Equipment: computer, interactive whiteboard.
Literature:
Geometry. Grades 10-11: textbook. for 10-11 grades. general education institutions: basic and profile. levels / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, etc.] - 18th ed. – M.: Education, 2009. – 255 p.
Lesson plan:
Organizational moment (2 min)
Updating knowledge (5 min)
Learning new material (12 min)
Reinforcement of learned material (21 min)
Homework (2 min)
Summing up (3 min)
During the classes:
1. Organizational moment.
Includes the teacher greeting the class, preparing the room for the lesson, and checking on absentees.
2. Updating basic knowledge.
Teacher: In the last lesson you wrote an independent work. In general, the work was written well. Now let's repeat it a little. What is an angle in a plane called?
Student: An angle on 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 straight line and a plane called?
Student: The angle between a straight line and a planeAny angle between a straight line and its projection onto this plane is called.
3. Studying new material.
Teacher: In stereometry, along with such angles, another type of angle 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.
Write 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. 1, a)
Teacher : Let’s imagine that we have bent the plane along a straight line so that two half-planes with a boundary no longer lie 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 sides, 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-opened folder.
Student : The wall of the room is together with the floor.
Student : Gable roofs of buildings.
Teacher : Right. And there are a huge number of such examples.
Teacher : As you know, angles in a plane are measured in degrees. You probably have a question, how are dihedral angles measured? This is done as follows.Let's mark some point on the edge of the dihedral angle and draw a ray perpendicular to the edge from this point on each face. The angle formed by these rays is called the linear angle of the dihedral angle. Make a drawing in your notebooks.
Write on the board and in notebooks.
ABOUT ∈ a, JSC ⊥ a, VO ⊥ a, SABD– dihedral angle,∠ AOB– linear angle of the dihedral angle.
Teacher : All linear angles of a dihedral angle are equal. Make yourself another drawing 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 co-directed. Similarly, the rays OB andQRco-directed. Means,∠ AOB= ∠ PQR(like angles with aligned 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 images of an acute, right and obtuse dihedral angle from the textbook on page 48.
4. Consolidation of the studied material.
Teacher : Make drawings for the tasks.
№ 1 . Given: ΔABC, AC = BC, AB lies in the planeα, CD ⊥ α, C∉ α. Construct linear angle of dihedral angleCABD.
Student : Solution:C.M. ⊥ AB, DC ⊥ AB.∠ CMD - sought after.
№ 2. Given: ΔABC, ∠ C= 90°, BC lies on the planeα, JSC⊥ α, A∈ α.
Construct linear angle of dihedral angleABCO.
Student : Solution:AB ⊥ B.C., JSC⊥ BC means OS⊥ Sun.∠ ACO - sought after.
№ 3 . Given: ΔABC, ∠ C = 90°, AB lies in the planeα, CD⊥ α, C∉ α. Buildlinear dihedral angleDABC.
Student : Solution: CK ⊥ AB, DC ⊥ AB,DK ⊥ AB means∠ DKC - sought after.
№ 4
. Given:DABC- tetrahedron,DO⊥
ABC.Construct the linear angle of the dihedral angleABCD.
Student : Solution:DM ⊥ sun,DO ⊥ VS means OM⊥ Sun;∠ OMD - sought after.
5. Summing up.
Teacher: What new did you learn in class today?
Students : What is called dihedral angle, linear angle, how is dihedral angle measured.
Teacher : What did they repeat?
Students : What is called an angle on a plane; angle between straight lines.
6.Homework.
Write on the board and in your diaries: paragraph 22, No. 167, No. 170.
TEXT TRANSCRIPT 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 linear angle is measured in degrees and radians.
In stereometry, a plane is added to objects. A figure formed by a 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. Straight line a is an edge of a dihedral angle.
A dihedral angle, like a linear angle, can be named, measured, and constructed. This is what we have to find out in this lesson.
Let's find the dihedral angle on the ABCD tetrahedron model.
A dihedral angle with edge AB is called CABD, where points C and D belong to different faces of the angle and edge AB is called in the middle
There are quite a lot of objects around us with elements in the form of a dihedral angle.
In many cities, special benches for reconciliation are installed in parks. The bench is made in the form of two inclined planes converging towards the center.
When building houses, the so-called gable roof is often used. On this house the roof is made in the form of a dihedral angle of 90 degrees.
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 rest on the rafters. And the rafter sheathing 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 rays BA and BC are drawn perpendicular to the edge of the angle. The angle ABC formed by these rays is called the linear 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.
An infinite number of linear angles can be drawn for a dihedral angle; it is important to know that they are all equal.
Let's consider two linear angles AOB and A1O1B1. Rays OA and O1A1 lie on the same face and are perpendicular to straight line OO1, so they are codirectional. Beams OB and O1B1 are also co-directed. Therefore, angle AOB is equal to angle A1O1B1 as angles with co-directional sides.
So a dihedral angle is characterized by a linear angle, and linear angles are acute, obtuse and right. Let's consider models of dihedral angles.
An obtuse angle is if its 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 from 0 to 90 degrees.
Let us prove one of the important properties of a linear angle.
The plane of the linear angle is perpendicular to the edge of the dihedral angle.
Let angle AOB be the linear angle of a given dihedral angle. By construction, the rays AO and OB are perpendicular to 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 only one.
Line a is perpendicular to two intersecting lines lying in this plane, which means, based on the perpendicularity of the line and the plane, straight line a is perpendicular to the plane AOB.
To solve problems, it is important to be able to construct a linear angle of a given dihedral angle. Construct a linear angle of a dihedral angle with edge AB for tetrahedron ABCD.
We are talking about a dihedral angle, which is formed, firstly, by edge AB, one face ABD, and the second face ABC.
Here's one way to build it.
Let's draw a perpendicular from point D to plane ABC. Mark 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 at the base of the tetrahedron.
Let's draw an inclined line from point D perpendicular to edge AB, mark point N as the base of the inclined line.
In the triangle DMN, the segment NM will be the projection of the inclined DN onto the plane ABC. According to the theorem of three perpendiculars, 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 desired linear angle.
Let's consider an example of solving a problem of calculating a 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=2 cm, AB= 4 cm.
The dihedral angle of DABC is equal to its linear angle. Let's build this angle.
Let us draw the inclined CM perpendicular to the edge AB, since the triangle ACB is isosceles, then point M will coincide with the middle of the edge AB.
The straight line CD is perpendicular to the plane ADB, which means it is perpendicular to the straight line DM lying in this plane. And the segment MD is a projection of the inclined CM onto the plane ADV.
The straight line AB is perpendicular to the inclined CM by construction, which means, by the theorem of three perpendiculars, it is perpendicular to the projection MD.
So, two perpendiculars CM and DM are found to the edge AB. This means they form a linear angle CMD of the dihedral angle DABC. And all we have to do is find it from the right triangle CDM.
So the segment SM is the median and the altitude of the isosceles triangle ACB, then according to the Pythagorean theorem, the leg SM is equal to 4 cm.
From the 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 times two. This means that the angle CMD is 30 degrees.
Concept of dihedral angle
To introduce the concept of a dihedral angle, let us first 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, points lying in the same half-plane are on one side of the straight line $a$, and 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 edges, and the straight line separating the half-planes is dihedral edge(Fig. 1).
Figure 2. Dihedral angle
Degree measure of dihedral angle
Definition 2
Let us choose an arbitrary point $A$ on the edge. The angle between two straight lines lying in different half-planes, perpendicular to an edge and intersecting at point $A$ is called linear 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.
Let's 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 the same straight line, then they are codirectional. Since the rays $OB$ and $(OB)_1$ lie in the same half-plane $\beta $ and are perpendicular to the same straight line, then 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 the linear angle of a dihedral angle.
Sample problems
Example 1
Let us be given two non-perpendicular planes $\alpha $ and $\beta $ which intersect along the straight line $m$. Point $A$ belongs to the plane $\beta$. $AB$ is perpendicular to line $m$. $AC$ is perpendicular to the plane $\alpha $ (point $C$ belongs to $\alpha $). Prove that angle $ABC$ is a linear angle of a dihedral angle.
Proof.
Let's draw a picture according to the conditions of the problem (Fig. 5).
Figure 5.
To prove it, recall the following theorem
Theorem 2: A straight line passing through the base of an inclined one is perpendicular to it, perpendicular to its projection.
Since $AC$ is perpendicular to the plane $\alpha $, then point $C$ is the projection of point $A$ onto the plane $\alpha $. Therefore, $BC$ is a projection of the oblique $AB$. By Theorem 2, $BC$ is perpendicular to the edge of the dihedral angle.
Then, angle $ABC$ satisfies all the requirements for defining a linear dihedral angle.
Example 2
The dihedral angle is $30^\circ$. On one of the faces lies a point $A$, which is located at a distance of $4$ cm from the other face. Find the distance from the point $A$ to the edge of the dihedral angle.
Solution.
Let's look at Figure 5.
By condition, 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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