Geometric shapes that are not polygons. Types of polygons" within the framework of the technology "Development of critical thinking through reading and writing

Topic: "Polygons. Types of polygons"

Grade 9

SL №20

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

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

Development 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 ability to work in pairs and groups; develop research and cognitive activity;

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

Equipment: interactive whiteboard (presentation)

During the classes

Show presentation: "Polygons"

“Nature speaks the language of mathematics, the letters of this language ... mathematical figures.” G. Gallilei

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

1. Call stage-

a) updating students' knowledge on the topic;

b) the awakening of interest in the topic under study, the motivation of each student for learning activities.

Reception: The game "Do you believe that ...", organization of work with text.

Forms of work: frontal, group.

“Do you believe that….”

1. ... the word "polygon" indicates that all the figures of this family have "many corners"?

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

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

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

2. Stage of comprehension

Purpose: obtaining new information, its comprehension, selection.

Reception: zigzag.

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

Each group is given a text on the topic of the lesson, and the text is designed 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 hasn't heard of the mysterious Bermuda Triangle, where ships and planes disappear without a trace? But the triangle familiar to us from childhood is fraught with a lot of interesting and mysterious things.

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

The word "polygon" indicates that all the figures of this family have "many corners". 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 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)

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

Substitute in the word “polygon” instead of the “many” part a specific number, for example 3. You will get a triangle. Or 5. Then - a pentagon. Note that there are as many angles as there are sides, 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 regions: internal and external (Fig. 6).

A plane polygon or polygonal region is a finite part of a plane bounded by a polygon.

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

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

Although smallest number sides of a polygon - 3. But triangles, connecting with each other, can form other figures, which in turn are also polygons.

Segments connecting non-neighboring 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 that vertex.

Let's prove the theorem (on the sum of angles of a convex n-gon): The sum of the angles of a convex n-gon is equal to 1800*(n - 2).

Proof. In the case n=3 the theorem is valid. Let А1А2…А n be a given convex polygon and n>3. Let's draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals divide it into n - 2 triangles. The sum of the angles of the polygon is the same as the sum of the angles of all these triangles. The sum of the angles of each triangle is 1800, and the number of these triangles is n - 2. Therefore, the sum of the angles of a convex n - angle A1A2 ... A n is 1800 * (n - 2). The theorem has been proven.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at that vertex.

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

So the square can be called differently - a regular quadrilateral. Equilateral triangles are also regular. Such figures have long been of interest to the masters who decorated the buildings. They made beautiful patterns, for example, on the parquet. But not all regular polygons could be used to form parquet. Parquet cannot be formed from regular octagons. The fact is that they have each angle equal to 1350. And if any point is the vertex of two such octagons, then they will have 2700, and there is nowhere for the third octagon to fit: 3600 - 2700 \u003d 900. But this is enough for a square. Therefore, it is possible to fold the parquet from regular octagons and squares.

The stars are correct. Our five-pointed star is a regular pentagonal star. And if you rotate the square around the center by 450, you get a regular octagonal star.

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

Which broken line is called simple?

Which broken line is called closed?

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

What is a flat polygon? 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?

What is a convex polygon?

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

What is a regular polygon? Give examples of regular polygons.

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

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

3. Stage of reflection -

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

b) understanding and appropriation of the received information.

Reception: research work.

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

The working groups are experts in the answers to 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 their questions. In the group there is an exchange of information of all members of the working group. Thus, in each working group, thanks to the work of experts, a general idea is formed on the topic under study.

Research work students- filling in the table.

Regular polygons Drawing Number of sides Number of vertices Sum of all internal angles Degree measure of internal. angle Degree measure of external angle Number of diagonals

A) a triangle

B) quadrilateral

B) five-hole

D) hexagon

E) n-gon

Solution interesting tasks on the subject of the lesson.

1) How many sides does a regular polygon have, each of internal corners which is equal to 1350?

2) In a certain polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be: 3600, 3800?

3) Is it possible to build a pentagon with angles of 100,103,110,110,116 degrees?

Summing up the lesson.

Recording homework: STR 66-72 №15,17 AND PROBLEM: in a QUADRANGLE, DRAW A DIRECT SO THAT SHE DIVIDES IT INTO THREE TRIANGLES.

Reflection in the form of tests (on an interactive whiteboard)

The part of the plane bounded by a closed broken line is called a polygon.

The segments of this broken line are called parties polygon. AB, BC, CD, DE, EA (Fig. 1) - sides of the polygon ABCDE. The sum of all the sides of a polygon is called its perimeter.

The polygon is called convex, if it is located on one side of any of its sides, extended indefinitely beyond both vertices.

The polygon MNPKO (Fig. 1) will not be convex, since it is located on more than one side of the straight line KP.

We will consider only convex polygons.

The angles formed by two adjacent sides of a polygon are called its internal corners, and their tops - polygon vertices.

A line segment connecting two non-adjacent vertices of a polygon is called a diagonal of the polygon.

AC, AD - diagonals of the polygon (Fig. 2).

The corners adjacent to the internal corners of the polygon are called the external corners of the polygon (Fig. 3).

Depending on the number of angles (sides), a polygon is called a triangle, quadrilateral, pentagon, etc.

Two polygons are said to be equal if they can be superimposed.

Inscribed and circumscribed polygons

If all the vertices of a polygon lie on a circle, then the polygon is called inscribed into a circle, and the circle described near the polygon (fig.).

If all sides of a polygon are tangent to a circle, then the polygon is called described around the circle, and the circle is called inscribed into a polygon (fig.).

Similarity of polygons

Two polygons of the same name are called similar if the angles of one of them are respectively equal to the angles of the other, and the similar sides of the polygons are proportional.

Polygons with the same name are called the same number sides (corners).

The sides of similar polygons are called similar if they connect the vertices of correspondingly equal angles (Fig.).

So, for example, for the polygon ABCDE to be similar to the polygon A'B'C'D'E', it is necessary that: E = ∠E' and, in addition, AB / A'B' = BC / B'C' = CD / C'D' = DE / D'E' = EA / E'A' .

Perimeter ratio of similar polygons

First, consider the property of a series of equal ratios. Let's have, for example, relations: 2 / 1 = 4 / 2 = 6 / 3 = 8 / 4 =2.

Let's find the sum of the previous members of these relations, then - the sum of their subsequent members and find the ratio of the received sums, we get:

$$ \frac(2 + 4 + 6 + 8)(1 + 2 + 3 + 4) = \frac(20)(10) = 2 $$

We get the same if we take a number of some other relations, for example: 2 / 3 = 4 / 6 = 6 / 9 = 8 / 12 = 10 / 15 = 2 / 3 and then we find the ratio of these sums, we get:

$$ \frac(2 + 4 + 5 + 8 + 10)(3 + 6 + 9 + 12 + 15) = \frac(30)(45) = \frac(2)(3) $$

In both cases, the sum of the previous members of a series of equal relations is related to the sum of subsequent members of the same series, as the previous member of any of these relations is related to its next one.

We deduced this property by considering a number of numerical examples. It can be deduced strictly and in general form.

Now consider the ratio of the perimeters of similar polygons.

Let the polygon ABCDE be similar to the polygon A'B'C'D'E' (fig.).

It follows from the similarity of these polygons that

AB / A'B' = BC / B'C' = CD / C'D' = DE / D'E' = EA / E'A'

Based on the property of a series of equal relations we have derived, we can write:

The sum of the previous terms of the relations we have taken is the perimeter of the first polygon (P), and the sum of the subsequent terms of these relations is the perimeter of the second polygon (P '), so P / P ' = AB / A'B '.

Consequently, the perimeters of similar polygons are related as their corresponding sides.

Ratio of areas of similar polygons

Let ABCDE and A'B'C'D'E' be similar polygons (fig.).

It is known that ΔABC ~ ΔA'B'C' ΔACD ~ ΔA'C'D' and ΔADE ~ ΔA'D'E'.

Besides,

;

Since the second ratios of these proportions are equal, which follows from the similarity of polygons, then

Using the property of a series of equal ratios, we get:

where S and S' are the areas of these similar polygons.

Consequently, the areas of similar polygons are related as the squares of similar sides.

The resulting formula can be converted to this form: S / S '= (AB / A'B ') 2

Area of an arbitrary polygon

Let it be required to calculate the area of an arbitrary quadrilateral ABDC (Fig.).

Let's draw a diagonal in it, for example AD. We get two triangles ABD and ACD, the areas of which we can calculate. Then we find the sum of the areas of these triangles. The resulting sum will express the area of \u200b\u200bthe given quadrangle.

If you need to calculate the area of a pentagon, then we proceed in the same way: we draw diagonals from one of the vertices. We get three triangles, the areas of which we can calculate. So we can find the area of this pentagon. We do the same when calculating the area of any polygon.

Polygon projection area

Recall that the angle between a line and a plane is the angle between a given line and its projection onto the plane (Fig.).

Theorem. The area of the orthogonal projection of the polygon onto the plane is equal to the area of the projected polygon multiplied by the cosine of the angle formed by the plane of the polygon and the projection plane.

Each polygon can be divided into triangles, the sum of the areas of which is equal to the area of the polygon. Therefore, it suffices to prove the theorem for a triangle.

Let ΔABC be projected onto the plane R. Consider two cases:

a) one of the sides ΔABS is parallel to the plane R;

b) none of the sides ΔABC is parallel R.

Consider first case: let [AB] || R.

Draw through the (AB) plane R 1 || R and project orthogonally ΔABC onto R 1 and on R(rice.); we get ΔABC 1 and ΔA’B’C’.

By the projection property, we have ΔABC 1 (cong) ΔA’B’C’, and therefore

S ∆ ABC1 = S ∆ A'B'C'

Let's draw ⊥ and the segment D 1 C 1 . Then ⊥ , a $\overbrace(CD_1C_1)$ = φ is the angle between the plane ΔABC and the plane R one . That's why

S ∆ ABC1 = 1 / 2 | AB | | C 1 D 1 | = 1 / 2 | AB | | CD 1 | cos φ = S ∆ ABC cos φ

and, therefore, S Δ A'B'C' = S Δ ABC cos φ.

Let's move on to consideration second case. Draw a plane R 1 || R through that vertex ΔАВС, the distance from which to the plane R the smallest (let it be vertex A).

Let's design ΔABC on the plane R 1 and R(rice.); let its projections be respectively ΔAB 1 C 1 and ΔA’B’C’.

Let (BC) ∩ p 1 = D. Then

S Δ A'B'C' = S ΔAB1 C1 = S ΔADC1 - S ΔADB1 = (S ΔADC - S ΔADB) cos φ = S Δ ABC cos φ

Other materials

Polygon Properties

A polygon is a geometric figure, usually defined as a closed polyline without self-intersections (a simple polygon (Fig. 1a)), but sometimes self-intersections are allowed (then the polygon is not simple).

The vertices of the polyline are called the vertices of the polygon, and the segments are called the sides of the polygon. The vertices of a polygon are called neighbors if they are the ends of one of its sides. Line segments connecting non-neighboring vertices of a polygon are called diagonals.

An angle (or internal angle) of a convex polygon at a given vertex is the angle formed by its sides converging at this vertex, and the angle is considered from the side of the polygon. In particular, the angle may exceed 180° if the polygon is not convex.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at that vertex. In general, the outside angle is the difference between 180° and the inside angle. From each vertex of the -gon for > 3 go out - 3 diagonals, therefore total number the diagonals of a -gon are equal.

A polygon with three vertices is called a triangle, with four - a quadrilateral, with five - a pentagon, and so on.

Polygon with n peaks is called n- square.

A flat polygon is a figure that consists of a polygon and the finite part of the area bounded by it.

A polygon is called convex if one of the following (equivalent) conditions is met:

1. it lies on one side of any straight line connecting its neighboring vertices. (i.e., the extensions of the sides of a polygon do not intersect its other sides);
2. it is the intersection (i.e. common part) of several half-planes;
3. any segment with ends at points belonging to the polygon belongs entirely to it.

A convex polygon is called regular if all sides are equal and all angles are equal, for example, an equilateral triangle, a square, and a pentagon.

A convex polygon is said to be inscribed about a circle if all its sides are tangent to some circle

A regular polygon is a polygon in which all angles and all sides are equal.

Polygon properties:

1 Each diagonal of a convex -gon, where >3, decomposes it into two convex polygons.

2 The sum of all angles of a convex -gon is equal to.

D-in: Let's prove the theorem by the method of mathematical induction. For = 3 it is obvious. Assume that the theorem is true for a -gon, where <, and prove it for -gon.

Let be a given polygon. Draw a diagonal of this polygon. By Theorem 3, the polygon is decomposed into a triangle and a convex -gon (Fig. 5). By the induction hypothesis. On the other hand, . Adding these equalities and taking into account that (- inner beam angle ) and (- inner beam angle ), we get. When we get: .

3 About any regular polygon it is possible to describe a circle, and moreover, only one.

D-in: Let a regular polygon, and and be the bisectors of the angles, and (Fig. 150). Since, therefore, * 180°< 180°. Отсюда следует, что биссектрисы и углов и пересекаются в некоторой точке O. Let's prove that O = OA 2 = O =… = OA P . Triangle O isosceles, therefore O= O. According to the second criterion for the equality of triangles, therefore, O = O. Similarly, it is proved that O = O etc. So the point O equidistant from all vertices of the polygon, so the circle with the center O radius O is circumscribed about a polygon.

Let us now prove that there is only one circumscribed circle. Consider some three vertices of a polygon, for example, BUT 2 , . Since only one circle passes through these points, then about the polygon … You cannot describe more than one circle.

4 In any regular polygon, you can inscribe a circle and, moreover, only one.
5 A circle inscribed in a regular polygon touches the sides of the polygon at their midpoints.
6 The center of a circle circumscribing a regular polygon coincides with the center of a circle inscribed in the same polygon.
7 Symmetry:

A figure is said to be symmetric (symmetric) if there is such a movement (not identical) that transforms this figure into itself.

7.1. A general triangle has no axes or centers of symmetry, it is not symmetrical. An isosceles (but not equilateral) triangle has one axis of symmetry: the perpendicular bisector to the base.
7.2. An equilateral triangle has three axes of symmetry (perpendicular bisectors to the sides) and rotational symmetry about the center with a rotation angle of 120°.

7.3 Any regular n-gon has n axes of symmetry, all of which pass through its center. It also has rotational symmetry about the center with a rotation angle.

Even n some axes of symmetry pass through opposite vertices, others through the midpoints of opposite sides.

For odd n each axis passes through the vertex and midpoint of the opposite side.

The center of a regular polygon with an even number of sides is its center of symmetry. A regular polygon with an odd number of sides has no center of symmetry.

8 Similarity:

With similarity, and -gon goes into a -gon, half-plane - into a half-plane, therefore convex n-gon becomes convex n-gon.

Theorem: If the sides and angles of convex polygons and satisfy the equalities:

where is the podium coefficient

then these polygons are similar.

8.1 The ratio of the perimeters of two similar polygons is equal to the coefficient of similarity.
8.2. The ratio of the areas of two convex similar polygons is equal to the square of the similarity coefficient.

polygon triangle perimeter theorem

Topic polygons - 8th grade:

A line of adjacent segments that do not lie on the same straight line is called broken line.

The ends of the segments are peaks.

Each cut- link.

And all the sums of the lengths of the segments make up the total length broken line. For example, AM + ME + EK + KO = polyline length

If the segments are closed, then polygon(see above) .

The links in a polygon are called parties.

The sum of the lengths of the sides - perimeter polygon.

Vertices on the same side are neighboring.

A line segment connecting non-adjacent vertices is called diagonal.

Polygons called by number of sides: pentagon, hexagon, etc.

Everything inside the polygon is inner part of the plane, and everything outside - outer part of the plane.

Note! The picture below- this is NOT a polygon, since there are additional common points on the same straight line for non-adjacent segments.

Convex polygon lies on one side of each line. To determine it mentally (or drawing) we continue each side.

In a polygon as many angles as there are sides.

In a convex polygon sum of all interior angles is equal to (n-2)*180°. n is the number of corners.

The polygon is called correct if all its sides and angles are equal. So the calculation of its internal angles is carried out according to the formula (where n is the number of angles): 180° * (n-2) / n

Below are the polygons, the sum of their angles, and what one angle is equal to.

The exterior angles of convex polygons are calculated as follows:

Subject, age of students: geometry, grade 9

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

Learning task: to update, expand and generalize students' knowledge of polygons; form an idea of the “components” of a polygon; conduct a study of the number of constituent elements of regular polygons (from a 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; develop research and educational activities;

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

During the classes: a quote is written on the blackboard

“Nature speaks the language of mathematics, the letters of this language ... mathematical figures.” G. Gallilei

At the beginning of the lesson, the class is divided into working groups (in our case, the division into groups of 4 people 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) the awakening of interest in the topic under study, the motivation of each student for learning activities.

Reception: The game "Do you believe that ...", organization of work with text.

Forms of work: frontal, group.

“Do you believe that….”

1. ... the word "polygon" indicates that all the figures of this family have "many corners"?

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

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

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 the figures of this family have "many corners". But this is not enough to characterize the figure.

A broken line A 1 A 2 ... A n is a figure that consists of points A 1, A 2, ... A n and segments A 1 A 2, A 2 A 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 the same straight line (Fig. 5).

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 regions: internal and external (Fig. 6).

A plane polygon or polygonal region is a finite part of a plane bounded by a polygon.

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

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

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

Segments connecting non-neighboring 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 straight 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 that vertex.

Let's prove the theorem (on the sum of angles of a convex n-gon): The sum of the angles of a convex n-gon is equal to 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. Let's draw diagonals in it (from one vertex). Since the polygon is convex, these diagonals divide it into n - 2 triangles. The sum of the angles of the 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 - angle A 1 A 2 ... A n is 180 0 * (n - 2). The theorem has been proven.

The exterior angle of a convex polygon at a given vertex is the angle adjacent to the interior angle of the polygon at that vertex.

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

So the square can be called differently - a regular quadrilateral. Equilateral triangles are also regular. Such figures have long been of interest to the masters who decorated the buildings. They made beautiful patterns, for example, on the parquet. But not all regular polygons could be used to form parquet. Parquet cannot be formed from regular octagons. The fact is that they have each angle equal to 135 0. And if any point is the vertex of two such octagons, then they will have 270 0, and there is nowhere for the third octagon to fit: 360 0 - 270 0 \u003d 90 0. But enough for a square. Therefore, it is possible to fold the parquet from regular octagons and squares.

The stars are 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.

1 group

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

Which broken line is called simple?

Which broken line is called closed?

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

2 group

What is a flat polygon? 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?

3 group

What is a convex polygon?

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

What is a regular polygon? Give examples of regular polygons.

4 group

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

3. Stage of reflection -

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

b) understanding and appropriation of the received information.

Reception: research work.

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

The working groups are experts in the answers to each of the sections of the proposed questions.

Research work of students - filling in the table.

Regular polygons	Drawing	Number of sides	Number of peaks	Sum of all internal angles	Degree measure int. angle	Degree measure of external angle	Number of diagonals
A) a triangle
B) quadrilateral
B) five-wall
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 whose interior angles is equal to 135 0 ?
In a certain polygon, all interior angles are equal to each other. Can the sum of the interior angles of this polygon be: 360 0 , 380 0 ?

Summing up the lesson. Recording homework.