Lesson “Dihedral angle. Dihedral angles and formula for calculating them. Dihedral angle at the base of a quadrangular regular pyramid

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.

The magnitude of the angle between two different planes can be determined for any relative position of the planes.

A trivial case if the planes are parallel. Then the angle between them is considered equal to zero.

A non-trivial case if the planes intersect. This case is the subject of further discussion. First we need the concept of a dihedral angle.

9.1 Dihedral angle

A dihedral angle is two half-planes with a common straight line (which is called the edge of the dihedral angle). In Fig. 50 shows a dihedral angle formed by half-planes and; the edge of this dihedral angle is the straight line a, common to these half-planes.

Rice. 50. Dihedral angle

The dihedral angle can be measured in degrees or radians in a word, enter the angular value of the dihedral angle. This is done as follows.

On the edge of the dihedral angle formed by the half-planes and, we take an arbitrary point M. Let us draw rays MA and MB, respectively lying in these half-planes and perpendicular to the edge (Fig. 51).

Rice. 51. Linear dihedral angle

The resulting angle AMB is the linear angle of the dihedral angle. The angle " = \AMB is precisely the angular value of our dihedral angle.

Definition. The angular magnitude of a dihedral angle is the magnitude of the linear angle of a given dihedral angle.

All linear angles of a dihedral angle are equal to each other (after all, they are obtained from each other by a parallel shift). Therefore, this definition is correct: the value " does not depend on the specific choice of point M on the edge of the dihedral angle.

9.2 Determining the angle between planes

When two planes intersect, four dihedral angles are obtained. If they all have the same size (90 each), then the planes are called perpendicular; The angle between the planes is then 90.

If not all dihedral angles are the same (that is, there are two acute and two obtuse), then the angle between the planes is the value of the acute dihedral angle (Fig. 52).

Rice. 52. Angle between planes

9.3 Examples of problem solving

Let's look at three problems. The first is simple, the second and third are approximately at level C2 on the Unified State Examination in mathematics.

Problem 1. Find the angle between two faces of a regular tetrahedron.

Solution. Let ABCD be a regular tetrahedron. Let us draw the medians AM and DM of the corresponding faces, as well as the height of the tetrahedron DH (Fig. 53).

Rice. 53. To task 1

Being medians, AM and DM are also altitudes of equilateral triangles ABC and DBC. Therefore, the angle " = \AMD is the linear angle of the dihedral angle formed by the faces ABC and DBC. We find it from the triangle DHM:

Answer: arccos 1 3 .

Problem 2. In a regular quadrangular pyramid SABCD (with vertex S), the side edge is equal to the side of the base. Point K is the middle of edge SA. Find the angle between the planes

Solution. Line BC is parallel to AD and thus parallel to plane ADS. Therefore, plane KBC intersects plane ADS along straight line KL parallel to BC (Fig. 54).

Rice. 54. To task 2

In this case, KL will also be parallel to line AD; therefore, KL is the midline of triangle ADS, and point L is the midpoint of DS.

Let's find the height of the pyramid SO. Let N be the middle of DO. Then LN is the middle line of triangle DOS, and therefore LN k SO. This means LN is perpendicular to plane ABC.

From point N we lower the perpendicular NM to the straight line BC. The straight line NM will be the projection of the inclined LM onto the ABC plane. From the three perpendicular theorem it then follows that LM is also perpendicular to BC.

Thus, the angle " = \LMN is the linear angle of the dihedral angle formed by the half-planes KBC and ABC. We will look for this angle from the right triangle LMN.

Let the edge of the pyramid be equal to a. First we find the height of the pyramid:

Solution. Let L be the intersection point of lines A1 K and AB. Then plane A1 KC intersects plane ABC along straight line CL (Fig.55).

A C

Rice. 55. To problem 3

Triangles A1 B1 K and KBL are equal in leg and acute angle. Therefore, the other legs are equal: A1 B1 = BL.

Consider triangle ACL. In it BA = BC = BL. Angle CBL is 120; therefore, \BCL = 30 . Also, \BCA = 60 . Therefore \ACL = \BCA + \BCL = 90 .

So, LC? AC. But line AC serves as a projection of line A1 C onto plane ABC. By the theorem of three perpendiculars we then conclude that LC ? A1 C.

Thus, angle A1 CA is the linear angle of the dihedral angle formed by the half-planes A1 KC and ABC. This is the desired angle. From the isosceles right triangle A1 AC we see that it is equal to 45.

This lesson is intended for independent study of the topic “Dihedral Angle”. In this lesson, students will become familiar with one of the most important geometric shapes, the dihedral angle. Also in the lesson we will learn how to determine the linear angle of the geometric figure in question and what the dihedral angle is at the base of the figure.

Let us repeat what an angle on a plane is and how it is measured.

Rice. 1. Plane

Let's consider the plane α (Fig. 1). From point ABOUT two rays emanate - OB And OA.

Definition. A figure formed by two rays emanating from one point is called an angle.

Angle is measured in degrees and radians.

Let's remember what a radian is.

Rice. 2. Radian

If we have a central angle whose arc length is equal to the radius, then such a central angle is called an angle of 1 radian. ,∠ AOB= 1 rad (Fig. 2).

Relationship between radians and degrees.

glad.

We get it, I'm glad. (). Then,

Definition. Dihedral angle a figure formed by a straight line is called A and two half-planes with a common boundary A, not belonging to the same plane.

Rice. 3. Half-planes

Let's consider two half-planes α and β (Fig. 3). Their common border is A. This figure is called a dihedral angle.

Terminology

Half-planes α and β are the faces of a dihedral angle.

Straight A is an edge of a dihedral angle.

On a common edge A dihedral angle, choose an arbitrary point ABOUT(Fig. 4). In the half-plane α from the point ABOUT restore the perpendicular OA to a straight line A. From the same point ABOUT in the second half-plane β we construct a perpendicular OB to the edge A. Got an angle AOB, which is called the linear angle of the dihedral angle.

Rice. 4. Dihedral angle measurement

Let us prove the equality of all linear angles for a given dihedral angle.

Let us have a dihedral angle (Fig. 5). Let's choose a point ABOUT and period O 1 on a straight line A. Let's construct a linear angle corresponding to the point ABOUT, i.e. we draw two perpendiculars OA And OB in planes α and β respectively to the edge A. We get the angle AOB- linear angle of the dihedral angle.

Rice. 5. Illustration of proof

From point O 1 let's draw two perpendiculars OA 1 And OB 1 to the edge A in planes α and β respectively and we obtain the second linear angle A 1 O 1 B 1.

Rays O 1 A 1 And OA codirectional, since they lie in the same half-plane and are parallel to each other like two perpendiculars to the same line A.

Likewise, rays About 1 In 1 And OB are co-directed, which means ∠ AOB =∠ A 1 O 1 B 1 as angles with codirectional sides, which is what needed to be proven.

The plane of the linear angle is perpendicular to the edge of the dihedral angle.

Prove: A ⊥ AOB.

Rice. 6. Illustration of proof

Proof:

OA ⊥ A by construction, OB ⊥ A by construction (Fig. 6).

We find that the line A perpendicular to two intersecting lines OA And OB out of plane AOB, which means it's straight A perpendicular to the plane OAV, which was what needed to be proven.

A dihedral angle is measured by its linear angle. This means that as many degrees radians are contained in a linear angle, the same number of degrees radians are contained in its dihedral angle. In accordance with this, the following types of dihedral angles are distinguished.

Acute (Fig. 6)

A dihedral angle is acute if its linear angle is acute, i.e. .

Straight (Fig. 7)

A dihedral angle is right when its linear angle is 90° - Obtuse (Fig. 8)

A dihedral angle is obtuse when its linear angle is obtuse, i.e. .

Rice. 7. Right angle

Rice. 8. Obtuse angle

Examples of constructing linear angles in real figures

ABCD- tetrahedron.

1. Construct a linear angle of a dihedral angle with an edge AB.

Rice. 9. Illustration for the problem

Construction:

We are talking about a dihedral angle formed by an edge AB and edges ABD And ABC(Fig. 9).

Let's make a direct DN perpendicular to the plane ABC, N- the base of the perpendicular. Let's draw an inclined DM perpendicular to a straight line AB,M- inclined base. By the theorem of three perpendiculars we conclude that the projection of an oblique NM also perpendicular to the line AB.

That is, from the point M two perpendiculars to the edge were restored AB on two sides ABD And ABC. We got the linear angle DMN.

notice, that AB, an edge of a dihedral angle, perpendicular to the plane of the linear angle, i.e., the plane DMN. The problem is solved.

Comment. The dihedral angle can be denoted as follows: DABC, Where

AB- edge, and points D And WITH lie on different sides of the angle.

2. Construct a linear angle of a dihedral angle with an edge AC.

Let's draw a perpendicular DN to the plane ABC and inclined DN perpendicular to a straight line AC. Using the three perpendicular theorem, we find that НN- oblique projection DN to the plane ABC, also perpendicular to the line AC.DNH- linear angle of a dihedral angle with an edge AC.

In a tetrahedron DABC all edges are equal. Dot M- middle of the rib AC. Prove that the angle DMV- linear dihedral angle YOUD, i.e. a dihedral angle with an edge AC. One of its faces is ACD, second - DIA(Fig. 10).

Rice. 10. Illustration for the problem

Solution:

Triangle ADC- equilateral, DM- median, and therefore height. Means, DM ⊥ AC. Likewise, triangle AINC- equilateral, INM- median, and therefore height. Means, VM ⊥ AC.

Thus, from the point M ribs AC dihedral angle restored two perpendiculars DM And VM to this edge in the faces of the dihedral angle.

So, ∠ DMIN is the linear angle of the dihedral angle, which is what needed to be proven.

So we've defined the dihedral angle, the linear angle of the dihedral angle.

In the next lesson we will look at the perpendicularity of lines and planes, then we will learn what a dihedral angle is at the base of figures.

List of references on the topic "Dihedral angle", "Dihedral angle at the base of geometric figures"

1. Geometry. Grades 10-11: textbook for general education institutions / Sharygin I. F. - M.: Bustard, 1999. - 208 pp.: ill.
2. Geometry. 10th grade: textbook for general education institutions with in-depth and specialized study of mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M.: Bustard, 2008. - 233 p.: ill.

1. Yaklass.ru ().
2. E-science.ru ().
3. Webmath.exponenta.ru ().
4. Tutoronline.ru ().

Homework on the topic "Dihedral angle", determining the dihedral angle at the base of figures

Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and expanded - M.: Mnemosyne, 2008. - 288 pp.: ill.

Tasks 2, 3 p. 67.

What is linear dihedral angle? How to build it?

ABCD- tetrahedron. Construct a linear angle of a dihedral angle with an edge:

A) IND b) DWITH.

ABCD.A. 1 B 1 C 1 D 1 - cube Construct Linear Angle of Dihedral Angle A 1 ABC with rib AB. Determine its degree measure.

The magnitude of the angle between two different planes can be determined for any relative position of the planes.

A trivial case if the planes are parallel. Then the angle between them is considered equal to zero.

A non-trivial case if the planes intersect. This case is the subject of further discussion. First we need the concept of a dihedral angle.

9.1 Dihedral angle

A dihedral angle is two half-planes with a common straight line (which is called the edge of the dihedral angle). In Fig. 50 shows a dihedral angle formed by half-planes and; the edge of this dihedral angle is the straight line a, common to these half-planes.

Rice. 50. Dihedral angle

The dihedral angle can be measured in degrees or radians in a word, enter the angular value of the dihedral angle. This is done as follows.

On the edge of the dihedral angle formed by the half-planes and, we take an arbitrary point M. Let us draw rays MA and MB, respectively lying in these half-planes and perpendicular to the edge (Fig. 51).

Rice. 51. Linear dihedral angle

The resulting angle AMB is the linear angle of the dihedral angle. The angle " = \AMB is precisely the angular value of our dihedral angle.

Definition. The angular magnitude of a dihedral angle is the magnitude of the linear angle of a given dihedral angle.

All linear angles of a dihedral angle are equal to each other (after all, they are obtained from each other by a parallel shift). Therefore, this definition is correct: the value " does not depend on the specific choice of point M on the edge of the dihedral angle.

9.2 Determining the angle between planes

When two planes intersect, four dihedral angles are obtained. If they all have the same size (90 each), then the planes are called perpendicular; The angle between the planes is then 90.

If not all dihedral angles are the same (that is, there are two acute and two obtuse), then the angle between the planes is the value of the acute dihedral angle (Fig. 52).

Rice. 52. Angle between planes

9.3 Examples of problem solving

Let's look at three problems. The first is simple, the second and third are approximately at level C2 on the Unified State Examination in mathematics.

Problem 1. Find the angle between two faces of a regular tetrahedron.

Solution. Let ABCD be a regular tetrahedron. Let us draw the medians AM and DM of the corresponding faces, as well as the height of the tetrahedron DH (Fig. 53).

Rice. 53. To task 1

Being medians, AM and DM are also altitudes of equilateral triangles ABC and DBC. Therefore, the angle " = \AMD is the linear angle of the dihedral angle formed by the faces ABC and DBC. We find it from the triangle DHM:

Answer: arccos 1 3 .

Problem 2. In a regular quadrangular pyramid SABCD (with vertex S), the side edge is equal to the side of the base. Point K is the middle of edge SA. Find the angle between the planes

Solution. Line BC is parallel to AD and thus parallel to plane ADS. Therefore, plane KBC intersects plane ADS along straight line KL parallel to BC (Fig. 54).

Rice. 54. To task 2

In this case, KL will also be parallel to line AD; therefore, KL is the midline of triangle ADS, and point L is the midpoint of DS.

Let's find the height of the pyramid SO. Let N be the middle of DO. Then LN is the middle line of triangle DOS, and therefore LN k SO. This means LN is perpendicular to plane ABC.

From point N we lower the perpendicular NM to the straight line BC. The straight line NM will be the projection of the inclined LM onto the ABC plane. From the three perpendicular theorem it then follows that LM is also perpendicular to BC.

Thus, the angle " = \LMN is the linear angle of the dihedral angle formed by the half-planes KBC and ABC. We will look for this angle from the right triangle LMN.

Let the edge of the pyramid be equal to a. First we find the height of the pyramid:

Solution. Let L be the intersection point of lines A1 K and AB. Then plane A1 KC intersects plane ABC along straight line CL (Fig.55).

A C

Rice. 55. To problem 3

Triangles A1 B1 K and KBL are equal in leg and acute angle. Therefore, the other legs are equal: A1 B1 = BL.

Consider triangle ACL. In it BA = BC = BL. Angle CBL is 120; therefore, \BCL = 30 . Also, \BCA = 60 . Therefore \ACL = \BCA + \BCL = 90 .

So, LC? AC. But line AC serves as a projection of line A1 C onto plane ABC. By the theorem of three perpendiculars we then conclude that LC ? A1 C.

Thus, angle A1 CA is the linear angle of the dihedral angle formed by the half-planes A1 KC and ABC. This is the desired angle. From the isosceles right triangle A1 AC we see that it is equal to 45.