Which refers to polygons. Lesson "Polygons

Subject, student age: geometry, grade 9

The purpose of the lesson: the study of the types of polygons.

Learning task: to update, expand and generalize students' knowledge about polygons; to form an idea of “ constituent parts"Polygon; conduct research on the number constituent elements regular polygons (from triangle to n - gon);

Developing task: to develop the ability to analyze, compare, draw conclusions, develop computational skills, oral and written mathematical speech, memory, as well as independence in thinking and learning activities, the ability to work in pairs and groups; to develop research and cognitive activities;

Educational task: to educate independence, activity, responsibility for the assigned work, perseverance in achieving the set goal.

During the classes: there is a quote on the blackboard

"Nature speaks in the language of mathematics, the letters of this language ... mathematical figures." G.Galliley

At the beginning of the lesson, the class is divided into working groups (in our case, division into groups of 4 people in each - the number of group members is equal to the number of question groups).

1.Call stage -

Goals:

a) updating students' knowledge on the topic;

b) awakening interest in the topic under study, motivating each student for educational activities.

Technique: The game “Do you believe that ...”, the organization of work with the text.

Forms of work: frontal, group.

"Do you believe that ...."

1.… the word “polygon” indicates that all shapes in this family have “many angles”?

2. ... a triangle belongs to a large family of polygons, distinguished among many different geometric shapes on surface?

3.… is a square a regular octagon (four sides + four corners)?

Today's lesson will focus on polygons. We learn that this figure is bounded by a closed polyline, which in turn is simple, closed. Let's talk about the fact that polygons are flat, regular, convex. One of the flat polygons is a triangle, with which you have been familiar for a long time (you can demonstrate to students posters with the image of polygons, a broken line, show them different kinds, you can also use TCO).

2. Stage of comprehension

Purpose: receiving new information, its comprehension, selection.

Reception: zigzag.

Forms of work: individual-> pair-> group.

Each of the group is given a text on the topic of the lesson, and the text is composed in such a way that it includes both information already known to students and completely new information. Together with the text, students receive questions, the answers to which must be found in this text.

Polygons. Types of polygons.

Who has not heard of the mysterious Bermuda Triangle, in which ships and planes disappear without a trace? But the triangle, familiar to us from childhood, is fraught with a lot of interesting and mysterious.

In addition to the types of triangles already known to us, divided by sides (versatile, isosceles, equilateral) and corners (acute-angled, obtuse, right-angled), a triangle belongs to a large family of polygons, distinguished among many different geometric shapes on the plane.

The word “polygon” indicates that all shapes in this family have “many angles”. But this is not enough to characterize the figure.

A broken line А 1 А 2 ... А n is a figure that consists of points А 1, А 2, ... А n and the segments А 1 А 2, А 2 А 3, ... connecting them. The points are called the vertices of the polyline, and the segments are called the links of the polyline. (fig. 1)

A broken line is called simple if it does not have self-intersections (Fig. 2, 3).

A broken line is called closed if its ends coincide. The length of a broken line is the sum of the lengths of its links (Fig. 4).

A simple closed broken line is called a polygon if its adjacent links do not lie on one straight line (Fig. 5).

Substitute a specific number in the word “polygon” instead of the part “many”, for example 3. You will get a triangle. Or 5. Then - a pentagon. Note that there are as many sides as there are angles, so these figures could well be called multilaterals.

The vertices of the polyline are called the vertices of the polygon, and the links of the polyline are called the sides of the polygon.

The polygon divides the plane into two areas: internal and external (Fig. 6).

A flat polygon or polygonal region is the end portion of a plane bounded by a polygon.

Two vertices of a polygon that are the ends of one side are called adjacent. Vertices that are not the ends of one side are not adjacent.

A polygon with n vertices, and hence with n sides, is called an n-gon.

Although the smallest number of sides of a polygon is 3. But triangles, joining each other, can form other shapes, which in turn are also polygons.

Line segments connecting non-adjacent vertices of a polygon are called diagonals.

A polygon is called convex if it lies in one half-plane with respect to any line containing its side. In this case, the line itself is considered to belong to the half-plane.

The angle of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex.

Let us prove the theorem (on the sum of the angles of a convex n - gon): The sum of the angles of a convex n - gon is 180 0 * (n - 2).

Proof. In the case n = 3, the theorem is valid. Let А 1 А 2 ... А n be a given convex polygon and n> 3. Draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals split it into n - 2 triangles. The sum of the angles of a polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 180 0, and the number of these triangles is n - 2. Therefore, the sum of the angles of a convex n - gon А 1 А 2 ... А n is equal to 180 0 * (n - 2). The theorem is proved.

Outside corner A convex polygon at a given vertex is the angle adjacent to the inner corner of the polygon at this vertex.

A convex polygon is called regular if all sides of it are equal and all angles are equal.

So the square can be called in another way - a regular quadrangle. Equilateral triangles are also regular. Such figures have long been of interest to masters who decorate buildings. They made beautiful patterns, for example on the parquet. But not all regular polygons could be folded into parquet. Parquet cannot be folded from regular octagons. The fact is that each angle of them is equal to 135 0. And if any point is the vertex of two such octagons, then their share will be 270 0, and there is nowhere for the third octagon to fit there: 360 0 - 270 0 = 90 0. But this is enough for a square. Therefore, it is possible to fold the parquet from regular octagons and squares.

The stars are also correct. Our five-pointed star is a regular pentagonal star. And if you rotate the square around the center by 45 0, you get a regular octagonal star.

1st group

What is called a broken line? Explain what the vertices and links of a polyline are.

Which polyline is called simple?

Which polyline is called closed?

What is called a polygon? What are the vertices of a polygon? What are the sides of a polygon?

2nd group

Which polygon is called flat? Give examples of polygons.

What is n - gon?

Explain which vertices of the polygon are adjacent and which are not.

What is the diagonal of a polygon?

Group 3

Which polygon is called convex?

Explain which corners of the polygon are external and which are internal?

Which polygon is called regular? Give examples of regular polygons.

4 group

What is the sum of the angles of a convex n-gon? Prove it.

Students work with the text, looking for answers to the questions posed, after which expert groups are formed, the work in which is on the same issues: students highlight the main thing, make up a supporting summary, present information in one of the graphic forms. At the end of the work, students return to their work groups.

3. Stage of reflection -

a) assessment of their knowledge, challenge to the next step of knowledge;

b) comprehension and appropriation of the information received.

Reception: research work.

Forms of work: individual-> pair-> group.

In the working groups, there are specialists in answering each of the sections of the proposed questions.

Returning to the working group, the expert introduces the other members of the group with the answers to his questions. In the group, information is exchanged between all members of the working group. Thus, in each working group, thanks to the work of experts, is general idea on the topic under study.

Research students - filling in the table.

Regular polygons	Drawing	Number of sides	Number of vertices	Sum of all inside corners	Degree measure int. corner	Outside angle measure	Number of diagonals
A) triangle
B) quadrangle
C) fivewolnik
D) hexagon
E) n-gon

Solving interesting problems on the topic of the lesson.

In the quadrilateral, draw a line so that it divides it into three triangles.
How many sides does a regular polygon have, each of inner corners which is 135 0?
In some polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be equal to: 360 0, 380 0?

Summing up the lesson. Homework recording.

In the course of geometry, we will study the properties of geo-metric figures and have already considered the simplest of them: triangles-ni-ki and surroundings. At the same time, we discussed specific cases of these figures, such as rectangular, equal tre-coal-ni-ki. Now the time has come to talk about more general and complex figs - a lot of coal-no-kah.

In a particular case many-coal-nikov we already know-to-we are a triangle (see Fig. 1).

Rice. 1. Tre-charcoal-nick

In the name itself, it is already under-cher-ki-va-et-sya, that this is fi-gu-ra, which has three corners. Left-to-wa-tel-but, in a lot of coal-no-ke there can be many of them, i.e. more than three. For example, the image of a five-coal-nickname (see Fig. 2), ie fi-gu-ru with five corners-la-mi.

Rice. 2. Five-sided nickname. You-bunch-a-yogo-coal-nick

Definition.Polygon- fi-gu-ra, consisting of several points (more than two) and co-with-the-vet-u-u-u-l-a-th-a-th kov, some of them after-va-tel-but co-unite. These points are na-zy-va-ut-Xia ver-shi-na-mi a lot of coal-no, but from-cut - sto-ro-na-mi... At the same time, no two adjacent sides lie on the same straight line and no two non-adjacent sides do not cross ...

Definition.Correct multi-coal-nick- this is you-bunch-a-lot-coal-nick, at which-ro-go all sides and angles are equal.

Any polygon div-de-la-et the plane-bone into two regions: inner-ren-nu and outer-nu. The inner-ren-nyu area is also related to a lot of coal-no-ku.

In other words, for example, when they talk about five-coal-no-ke, they mean its entire inner-nyu region, and tsu. And all the points that lie inside a lot of coal, i.e. the point also refers to the five-coal-ni-ku (see Fig. 2).

A lot of coal-ni-ki still sometimes na-zy-va-yut n-coal-ni-ka-mi, to underline that ras-smat-ri-va-is-Xia common the case of an unknown number of corners (n pieces).

Definition. Per-ri-meter many-coal-no-ka- the sum of the lengths of the sides of a lot of coal.

Now it is necessary to know-to-know-to-go with we-yes-many-coal-nikov. They are on you-fart-ly and not-fart-ly... For example, the polygonal nickname depicted in Fig. 2 is bulging, and in Fig. 3 non-bunched.

Rice. 3. Neva-bunch-ly-go-coal-nick

2. Convex and non-convex polygons

Definition-de-le-tion 1. Polygon na-zy-wa-et-Xia you-bunch-lym if, when pro-ve-de-ny, straight through any of its sides all polygon lies only one side of this straight line. Neva-bunch-ly-mi all the rest mogo-coal-ni-ki.

It is easy to imagine that when any side of the five-coal in Fig. 2 he will all be one hundred percent from this straight line, i.e. he is a bunch. But with the pro-ve-de-ny of a straight line through in four-you-rekh-coal-no-ke in Fig. 3 we already see that she divides it into two parts, i.e. he is not-fart-ly.

But there is also another definition of you-bunch-of-a lot of coal-no-ka.

Definition-de-le-tion 2. Polygon na-zy-wa-et-Xia you-bunch-lym if, when you select any two of its inner points and when they are connected, all points from the cut are also inner -n-mi-dots-mi-a-go-coal-ni-ka.

Demonstration of the use of this definition can be seen on the example of the construction of cut-off points in Fig. 2 and 3.

Definition. Dia-go-na-liu a lot of coal-no-ka-zy-va-is-Xia any ot-zok, co-one-nya-yu-shi-two not contiguity of its vertices.

3. The theorem on the sum of the interior angles of a convex n-gon

For a description of the properties of many-coal-nikov, there are two important theo-re-we about their angles: theo-re-ma about the sum of the inner-corners of the vy-bunch-lo-th-many-coal-no-ka and theo-re-ma about the sum of the outer angles of the bunch... Examine them.

Theorem. About the sum of the inner angles of the bunch of many-coal-no-ka (n-coal-no-ka).

Where is the number of its corners (sides).

Do-ka-tel-tstvo 1. Image-zim in Fig. 4 you-bunch n-gon-nick.

Rice. 4. You-bunch-n-gon-nick

From the top-shi-we pro-we-dem all possible dia-go-na-li. They divide the n-gon-nick into the tri-gon-nick, because each of the sides has a lot of coal-no-ka-ra-zu-et a triangle, except for the sides that come to the top. It is easy to see from the drawing that the sum of the angles of all these triangles will be exactly equal to the sum of the inner angles of n-coals. Since the sum of the angles of any triangle is, then the sum of the inner angles of n-coal is:

Do-ka-tel-tstvo 2. Possibly, and another do-ka-tel-tiality of this theo-re-we. Pictured-ra-winter analogous n-gon in Fig. 5 and co-unite any of its internal points with all vertices.

We have-got-chi-whether raz-bi-e-n-coals-n-coals into n-coals (how many sides, so many coals ). The sum of all their angles is equal to the sum of the inner angles of the polygon and the sum of the angles at the inner point, and this is the angle. We have:

Q.E.D.

Do-ka-za-but.

According to the adjusted theory, it can be seen that the sum of the angles of n-coal-ni-ka-sits on the number of its sides (from n). For example, in a triangle, and the sum of the angles. In th-you-rekh-coal-no-ke, and the sum of the angles - etc.

4. The theorem on the sum of the outer angles of a convex n-gon

Theorem. About the sum of the outer angles of a bunch of a lot of coal-no-ka (n-coal-no-ka).

Where is the number of its corners (sides), and, ..., are the outer corners.

Proof. Pictured-ra-winter vy-bump-n-gon-nick in Fig. 6 and denote its inner and outer corners.

Rice. 6. You-bump-ly n-gon-nick with denoted outer corners

Because the outer corner is connected with the inner corner as adjacent, then and similarly for the rest of the outer corners. Then:

In the course of pre-ob-ra-zo-va-ny, we used-to-use-zo-va-li already reached the theorem about the sum of the inner angles n-coal-ni- ka.

Do-ka-za-but.

From the top-of-the-line theo-re-we follow the interesting fact that the sum of the outer angles of the bunch of n-coal is equal to from the number of its corners (sides). By the way, in contrast to the sum of the inner angles.

Further, we will work in more detail with the particular case of a lot of coal-nikov - che-you-rekh-coal-ni-ka-mi. In the next lesson, we will get to know such a fi-gu-swarm as pa-ra-le-lo-gram, and discuss its properties.

A SOURCE

http://interneturok.ru/ru/school/geometry/8-klass/chyotyrehugolniki/mnogougolniki

http://interneturok.ru/ru/school/geometry/8-klass/povtorenie/pryamougolnye-treugolniki

http://interneturok.ru/ru/school/geometry/8-klass/povtorenie/treugolniki-2

http://nsportal.ru/shkola/geometriya/library/2013/10/10/mnogougolniki-urok-v-8-klasse

https://im0-tub-ru.yandex.net/i?id=daa2ea7bbc3c92be3a29b22d8106e486&n=33&h=190&w=144

In this lesson, we will start already to new topic and introduce a new concept for us "polygon". We will cover the basic concepts associated with polygons: sides, vertices, corners, convexity, and non-convexity. Then we prove the most important facts, such as the theorem on the sum of the inner angles of a polygon, the theorem on the sum of the outer angles of a polygon. As a result, we will come close to studying special cases of polygons, which will be considered in further lessons.

Theme: Quadrangles

Lesson: Polygons

In the geometry course, we study the properties of geometric shapes and have already considered the simplest of them: triangles and circles. At the same time, we also discussed specific special cases of these figures, such as rectangular, isosceles and regular triangles... Now it's time to talk about more general and complex shapes - polygons.

With a special case polygons we are already familiar - this is a triangle (see Fig. 1).

Rice. 1. Triangle

The name itself already emphasizes that this is a figure with three corners. Therefore, in polygon there can be many of them, i.e. more than three. For example, let's draw a pentagon (see Fig. 2), ie. a figure with five corners.

Rice. 2. Pentagon. Convex polygon

Definition.Polygon- a figure consisting of several points (more than two) and the corresponding number of segments that connect them in series. These points are called peaks polygon, and the line segments - parties... Moreover, no two adjacent sides lie on one straight line and no two non-adjacent sides intersect.

Definition.Regular polygon is a convex polygon with all sides and angles equal.

Any polygon divides the plane into two areas: internal and external. The inner area is also referred to as polygon.

In other words, for example, when they talk about a pentagon, they mean both its entire inner region and its border. And all points that lie inside the polygon also belong to the inner region, i.e. the point also belongs to the pentagon (see Fig. 2).

Polygons are also sometimes called n-gons to emphasize that the general case of the presence of some unknown number of corners (n pieces) is considered.

Definition. Polygon perimeter- the sum of the lengths of the sides of the polygon.

Now we need to get acquainted with the types of polygons. They are divided into convex and non-convex... For example, the polygon shown in Fig. 2 is convex, and in Fig. 3 non-convex.

Rice. 3. Nonconvex polygon

Definition 1. Polygon called convex if, when drawing a straight line through any of its sides, the entire polygon lies only on one side of this straight line. Non-convex are all the rest polygons.

It is easy to imagine that when extending either side of the pentagon in Fig. 2 it will all be on one side of this straight line, i.e. it is convex. But when drawing a straight line through in a quadrangle in Fig. 3 we already see that she divides it into two parts, i.e. it is non-convex.

But there is another definition of the convexity of a polygon.

Definition 2. Polygon called convex if, when choosing any two of its interior points and connecting them with a segment, all points of the segment are also interior points of the polygon.

A demonstration of the use of this definition can be seen on the example of constructing segments in Fig. 2 and 3.

Definition. Diagonal a polygon is any line segment that connects two non-adjacent vertices.

To describe the properties of polygons, there are two important theorems about their angles: the theorem on the sum of the interior angles of a convex polygon and the theorem on the sum of the outer angles of a convex polygon... Let's consider them.

Theorem. On the sum of the interior angles of a convex polygon (n-gon).

Where is the number of its corners (sides).

Proof 1. We depict in Fig. 4 convex n-gon.

Rice. 4. Convex n-gon

Draw all possible diagonals from the top. They divide the n-gon into triangles, because each side of the polygon forms a triangle, except for the sides adjacent to the apex. It is easy to see from the figure that the sum of the angles of all these triangles will just be equal to the sum of the interior angles of the n-gon. Since the sum of the angles of any triangle is, then the sum of the interior angles of an n-gon is:

Q.E.D.

Proof 2. Another proof of this theorem is also possible. Let's draw a similar n-gon in Fig. 5 and connect any of its internal points with all vertices.

Rice. 5.

We have obtained a partition of an n-gon into n triangles (as many sides as there are triangles). The sum of all their angles is equal to the sum of the inner angles of the polygon and the sum of the angles at the inner point, and this is the angle. We have:

Q.E.D.

Proven.

By the proved theorem, it is clear that the sum of the angles of an n-gon depends on the number of its sides (on n). For example, in a triangle, and the sum of the angles. In a quadrangle, and the sum of the angles is, etc.

Theorem. On the sum of the outer angles of a convex polygon (n-gon).

Where is the number of its corners (sides), and,…, are the outer corners.

Proof. Let's draw a convex n-gon in Fig. 6 and designate its inner and outer corners.

Rice. 6. Convex n-gon with marked external corners

Because the outer corner is related to the inner corner as adjacent, then and similarly for the rest of the outer corners. Then:

In the course of the transformations, we used the already proved theorem on the sum of the interior angles of an n-gon.

Proven.

The theorem proved above implies interesting fact that the sum of the outer angles of a convex n-gon is from the number of its corners (sides). By the way, in contrast to the sum of the interior angles.

Bibliography

Alexandrov A.D. and others. Geometry, grade 8. - M .: Education, 2006.
Butuzov V.F., Kadomtsev S.B., Prasolov V.V. Geometry, grade 8. - M .: Education, 2011.
Merzlyak A.G., Polonskiy V.B., Yakir S.M. Geometry, grade 8. - M .: VENTANA-GRAF, 2009.

Profmeter.com.ua ().
Narod.ru ().
Xvatit.com ().

Homework

Topic: "Polygons. Types of polygons"

Grade 9

ShL No. 20

Teacher: Kharitonovich T.I. The purpose of the lesson: the study of the types of polygons.

Learning task: update, expand and generalize students' knowledge of polygons; to form an idea of the "constituent parts" of the polygon; conduct a study of the number of constituent elements of regular polygons (from a triangle to n - a gon);

Developing task: develop the ability to analyze, compare, draw conclusions, develop computational skills, oral and written mathematical speech, memory, as well as independence in thinking and learning activities, the ability to work in pairs and groups; develop research and cognitive activity;

Educational task: to bring up independence, activity, responsibility for the assigned task, perseverance in achieving the set goal.

Equipment: interactive whiteboard (presentation)

During the classes

Presentation showing: "Polygons"

"Nature speaks in the language of mathematics, the letters of this language ... mathematical figures." G.Galliley

At the beginning of the lesson, the class is divided into working groups (in our case, the division into 3 groups)

1.Call stage -

a) updating students' knowledge on the topic;

b) awakening interest in the topic under study, motivating each student for educational activities.

Technique: The game “Do you believe that ...”, the organization of work with the text.

Forms of work: frontal, group.

"Do you believe that ...."

1.… the word “polygon” indicates that all shapes in this family have “many angles”?

2.… a triangle belongs to a large family of polygons, distinguished among a multitude of different geometric shapes on a plane?

3.… is a square a regular octagon (four sides + four corners)?

Today's lesson will focus on polygons. We learn that this figure is bounded by a closed polyline, which in turn is simple, closed. Let's talk about the fact that polygons are flat, regular, convex. One of the flat polygons is a triangle, which you are familiar with for a long time (you can demonstrate to students posters with the image of polygons, a broken line, show their various types, you can also use TCO).

2. Stage of comprehension

Purpose: obtaining new information, its comprehension, selection.

Reception: zigzag.

Forms of work: individual-> pair-> group.

Polygons. Types of polygons.

The word “polygon” indicates that all shapes in this family have “many angles”. But this is not enough to characterize the figure.

A broken line A1A2 ... An is a figure that consists of points A1, A2, ... An and the segments A1A2, A2A3, ... connecting them. The points are called the vertices of the polyline, and the segments are called the links of the polyline. (FIG. 1)

A broken line is called simple if it does not have self-intersections (Fig. 2, 3).

A broken line is called closed if its ends coincide. The length of a broken line is the sum of the lengths of its links (Fig. 4)