§ 1 The concept of a triangle
In this lesson, you will become familiar with shapes such as triangle and polygon.
If three points that do not lie on one straight line are connected by segments, then you get a triangle. The triangle has three vertices and three sides.
Before you triangle ABC, it has three vertices (point A, point B and point C) and three sides (AB, AC and CB).
By the way, these same sides can be called in another way:
AB = BA, AC = CA, CB = BC.
The sides of the triangle form three angles at the vertices of the triangle. In the picture you can see angle A, angle B, angle C.
Thus, a triangle is a geometric figure formed by three segments that connect three points that do not lie on one straight line.
§ 2 The concept of a polygon and its types
Besides triangles, there are quadrangles, pentagons, hexagons, and so on. In one word, they can be called polygons.
In the figure you can see the DMKE quadrilateral.
Points D, M, K and E are the vertices of the quadrilateral.
The segments DM, MK, KE, ED are the sides of this quadrangle. Just as in the case of a triangle, the sides of a quadrilateral form four corners at the vertices, as you guessed it, hence the name - quadrilateral. For this quadrilateral, you can see in the picture angle D, angle M, angle K and angle E.
What quadrangles do you already know?
Square and rectangle! Each of them has four corners and four sides.
Another type of polygons is the pentagon.
Points O, P, X, Y, T are the vertices of the pentagon, and the segments TO, OP, PX, XY, YT are the sides of this pentagon. The pentagon has five corners and five sides, respectively.
How many angles and sides do you think a hexagon has? That's right, six! Reasoning in a similar way, you can tell how many sides, vertices or corners a particular polygon has. And we can conclude that a triangle is also a polygon, which has exactly three corners, three sides and three vertices.
Thus, in this lesson, you got acquainted with such concepts as a triangle and a polygon. We learned that a triangle has 3 vertices, 3 sides and 3 corners, a quadrangle - 4 vertices, 4 sides and 4 corners, a pentagon - respectively 5 sides, 5 vertices, 5 corners and so on.
List of used literature:
- Mathematics grade 5. Vilenkin N.Ya., Zhokhov V.I. et al. 31st ed., erased. - M: 2013.
- Didactic materials in mathematics grade 5. Author - Popov M.A. - year 2013
- We calculate without errors. Works with self-test in mathematics, grades 5-6. Author - Minaeva S.S. - year 2014
- Didactic materials in mathematics grade 5. Authors: Dorofeev G.V., Kuznetsova L.V. - 2010
- Control and independent work in mathematics grade 5. Authors - Popov M.A. - year 2012
- Mathematics. Grade 5: textbook. for general education students. institutions / I. I. Zubareva, A. G. Mordkovich. - 9th ed., Erased. - M .: Mnemosina, 2009
Polygon is a geometric figure bounded by a closed polyline that does not have self-intersections.
The links of the broken line are called sides of the polygon, and its tops are the vertices of the polygon.
Corners polygon are called interior corners formed by adjacent sides. The number of corners of a polygon is equal to the number of its vertices and sides.
Polygons are named according to the number of sides. The polygon with the least number of sides is called a triangle, it has only three sides. A polygon with four sides is called a quadrangle, a polygon with five is called a pentagon, and so on.
The designation of a polygon is made up of the letters at its vertices, naming them in order (clockwise or counterclockwise). For example, they say or write: pentagon ABCDE :
In the pentagon ABCDE points A, B, C, D and E are the vertices of the pentagon, and the segments AB, BC, CD, DE and EA- the sides of the pentagon.
Convex and concave
The polygon is called convex if none of its sides, continued to a straight line, intersects it. V otherwise polygon is called concave:
Perimeter
The sum of the lengths of all sides of the polygon is called it perimeter.
Polygon perimeter ABCDE is equal to:
AB + BC+ CD + DE + EA
If a polygon has equal all sides and all angles, then it is called correct... Only convex polygons can be regular polygons.
Diagonal
Polygon diagonal is a line segment connecting the vertices of two corners that do not have common side... For example, the segment AD is the diagonal:
The only polygon that does not have any diagonal is a triangle, since it has no corners that have no common sides.
If all possible diagonals are drawn from any vertex of the polygon, then they will divide the polygon into triangles:
There will be exactly two less triangles than sides:
t = n - 2
where t is the number of triangles, and n- the number of parties.
Dividing a polygon into triangles using diagonals is used to find the area of a polygon, since to find the area of some polygon, you need to split it into triangles, find the area of these triangles and add the results obtained.
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
Sections: Mathematics
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 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: 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 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, 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: obtaining 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, connecting with 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.
The outer angle of 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, the 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, there is general idea on the topic under study.
Research work of students - filling out 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 regular polygon, each of the inner corners of which is equal to 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.
Knowledge of terminology, as well as knowledge of the properties of various geometric shapes will help in solving many problems in geometry. Studying such a section as planimetry, the student often comes across the term "polygon". What figure does this concept characterize?
Polygon - define a geometric shape
Closed broken line, all sections of which lie in the same plane and do not have self-intersection sections, forms a geometric figure called a polygon. The number of broken line links must be at least 3. In other words, a polygon is defined as a part of a plane, the boundary of which is a closed polyline.
In the course of solving problems with the participation of a polygon, such concepts as often appear:
- The side of the polygon. This term characterizes a segment (link) of a broken chain of the required figure.
- Polygon corner (inner) - the angle that is formed by 2 adjacent polyline links.
- The vertex of the polygon is defined as the vertex of the polyline.
- Polygon diagonal is a line segment connecting any 2 vertices (except for neighboring ones) of a polygonal shape.
In this case, the number of links and the number of vertices of the polyline within one polygon coincide. Depending on the number of corners (or polyline segments, respectively), the type of the polygon is also determined:
- 3 corners - triangle.
- 4 corners - a quadrangle.
- 5 corners - pentagon, etc.
If the polygonal shape has equal angles and, accordingly, the sides, then they say that the given polygon is regular.
Types of polygons
All polygonal geometric shapes are divided into 2 types - convex and concave.
- If any of the sides of the polygon, after continuing to a straight line, does not form intersection points with the actual figure, you have a convex polygonal figure in front of you.
- If, after extending a side (any), the resulting straight line intersects the polygon, we are talking about a concave polygon.
Polygon properties
Regardless of whether the studied polygonal figure is correct or not, it has the following properties. So:
- Its interior angles form a total of (p - 2) * π, where
π - radian measure of the unfolded angle, corresponds to 180 °,
p is the number of corners (vertices) of a polygonal figure (p-gon).
- The number of diagonals of any polygonal figure is determined from the ratio p * (p - 3) / 2, where
p is the number of sides of the p-gon.
