What is the sum of the angles of a convex polygon. The sum of the angles of a triangle

Proof

Let be ABC " - an arbitrary triangle. Let's take you through the top B straight line parallel to straight line AC (such a line is called the Euclidean line). Let's mark a point on it D so that the points A and D lay on opposite sides of a straight line BC.Angles DBC and ACB equal as internal criss-crossing formed by the secant BC with parallel straight lines AC and BD... Therefore, the sum of the angles of the triangle at the vertices B and WITH equal to the angle ABD The sum of all three angles of a triangle is equal to the sum of the angles ABD and BAC... Since these corners are internal one-sided for parallel AC and BD at secant AB, then their sum is 180 °. The theorem is proved.

Consequences

The theorem implies that any triangle has two acute angles. Indeed, applying the proof by contradiction, suppose that a triangle has only one acute angle or no acute angles at all. Then this triangle has at least two angles, each of which is at least 90 °. The sum of these angles is not less than 180 °. And this is impossible, since the sum of all the angles of the triangle is 180 °. Q.E.D.

Generalization to a simplex theory

Where is the angle between i and j faces of the simplex.

Notes (edit)

• On a sphere, the sum of the angles of a triangle always exceeds 180 °, the difference is called a spherical excess and is proportional to the area of ​​the triangle.
• In the Lobachevsky plane, the sum of the angles of a triangle is always less than 180 °. The difference is also proportional to the area of ​​the triangle.

see also

Wikimedia Foundation. 2010.

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Targets and goals:

Educational:

• repeat and generalize knowledge about the triangle;
• prove the theorem on the sum of the angles of a triangle;
• practically verify the correctness of the statement of the theorem;
• learn to apply the knowledge gained in solving problems.

Developing:

• develop geometric thinking, interest in the subject, cognitive and creative activities of students, mathematical speech, the ability to independently acquire knowledge.

Educational:

• develop the personal qualities of students, such as dedication, perseverance, accuracy, the ability to work in a team.

Equipment: multimedia projector, colored paper triangles, "Living Mathematics" educational complex, computer, screen.

Preparatory stage: the teacher instructs the student to prepare a historical background on the theorem "The sum of the angles of a triangle".

Lesson type: learning new material.

During the classes

I. Organizational moment

Greetings. The psychological attitude of students to work.

II. Warm up

We got acquainted with the geometric shape "triangle" in the previous lessons. Let's reiterate what we know about the triangle?

Students work in groups. They are given the opportunity to communicate with each other, each to independently build the process of cognition.

What happened? Each group makes their suggestions, the teacher writes them down on the board. Results are discussed:

Picture 1

III. We formulate the task of the lesson

So, we already know a lot about the triangle. But not all. Each of you has triangles and protractors on your desk. What task do you think we can formulate?

Pupils formulate the task of the lesson - find the sum of the angles of a triangle.

IV. Explanation of the new material

Practical part(contributes to the updating of knowledge and self-knowledge skills) Take measurements of the angles using a protractor and find their sum. Write the results in a notebook (listen to the answers received). We find out that the sum of the angles is different for everyone (this can happen because the protractor was inaccurately attached, the calculation was careless, etc.).

Bend along the dotted lines and find out what else the sum of the angles of the triangle is equal to:

a)
Picture 2

b)
Figure 3

v)
Figure 4

G)
Figure 5

e)
Figure 6

After completing the practical work, the students formulate the answer: The sum of the angles of the triangle is equal to the degree measure of the unfolded angle, i.e. 180 °.

Teacher: In mathematics, practical work makes it possible only to make some statement, but it needs to be proved. The statement, the validity of which is established by proof, is called a theorem. What theorem can we formulate and prove?

Students: The angles of a triangle add up to 180 degrees.

Historical reference: The property of the sum of the angles of a triangle was established in ancient Egypt. The proof set forth in modern textbooks is contained in Proclus's commentaries on Euclid's Principles. Proclus claims that this proof (Fig. 8) was discovered by the Pythagoreans (5th century BC). In the first book of the Elements, Euclid sets out another proof of the theorem on the sum of the angles of a triangle, which is easy to understand with the help of the drawing (Fig. 7):

Figure 7

Figure 8

Drawings are displayed on the screen through a projector.

The teacher suggests using drawings to prove the theorem.

Then the proof is carried out using the CMK "Living Mathematics"... The teacher projects the proof of the theorem on the computer.

The theorem on the sum of the angles of a triangle: "The sum of the angles of a triangle is 180 °"

Figure 9

Proof:

a)

Figure 10

b)

Figure 11

v)

Figure 12

Students in a notebook makes a short note of the proof of the theorem:

Theorem: The angles of a triangle add up to 180 °.

Figure 13

Given:Δ ABC

Prove: A + B + C = 180 °.

Proof:

What was required to prove.

V. Phys. a minute.

Vi. Explanation of the new material (continued)

The consequence of the theorem on the sum of the angles of a triangle is deduced by students independently, this contributes to the development of the ability to formulate their own point of view, express and argue for it:

In any triangle, either all corners are acute, or two acute angles, and the third is obtuse or straight.

If all angles in a triangle are sharp, then it is called acute-angled.

If one of the corners of the triangle is obtuse, then it is called obtuse.

If one of the angles of a triangle is a straight line, then it is called rectangular.

The triangle sum theorem allows you to classify triangles not only by sides, but also by angles. (In the course of introducing the types of triangles, students fill in the table)

Table 1

Triangle view	Isosceles	Equilateral	Versatile
Rectangular
Obtuse
Acute-angled

Vii. Consolidation of the studied material.

1. Solve problems orally:

(Drawings are displayed on the screen through a projector)

In grade 8, in geometry lessons at school, students first become familiar with the concept of a convex polygon. They will soon find out that this figure has a very interesting property. No matter how complex it is, the sum of all the inner and outer angles of a convex polygon takes on a strictly defined value. In this article, a math and physics tutor talks about what the sum of the angles of a convex polygon is equal to.

The sum of the interior angles of a convex polygon

How can you prove this formula?

Before proceeding to the proof of this statement, let us recall which polygon is called convex. A convex polygon is a polygon that is entirely on one side of a straight line containing any of its sides. For example, the one shown in this figure:

If the polygon does not satisfy the specified condition, then it is called non-convex. For example, something like this:

The sum of the interior angles of a convex polygon is, where is the number of sides of the polygon.

The proof of this fact is based on the theorem on the sum of angles in a triangle, well known to all schoolchildren. I am sure that this theorem is familiar to you too. The sum of the interior angles of a triangle is.

The idea is to split the convex polygon into multiple triangles. This can be done in different ways. The evidence will differ slightly depending on which method we choose.

1. Divide the convex polygon into triangles by all possible diagonals drawn from some vertex. It is easy to understand that then our n-gon will split into triangles:

Moreover, the sum of all the angles of all the resulting triangles is equal to the sum of the angles of our n-gon. After all, each angle in the resulting triangles is a partial angle of some angle in our convex polygon. That is, the required amount is equal to.

2. You can also select a point inside the convex polygon and connect it to all vertices. Then our n-gon will be split into triangles:

Moreover, the sum of the angles of our polygon in this case will be equal to the sum of all the angles of all these triangles minus the central angle, which is equal to. That is, the required amount is again equal.

The sum of the outer angles of a convex polygon

Let us now ask ourselves the question: "What is the sum of the outer angles of a convex polygon?" The answer to this question is as follows. Each outer corner is adjacent to a corresponding inner corner. Therefore, it is equal to:

Then the sum of all external angles is. That is, it is equal.

That is, a very funny result is obtained. If we put all the outer corners of any convex n-gon sequentially one after another, then exactly the entire plane will be filled as a result.

This interesting fact can be illustrated as follows. Let's proportionally reduce all sides of some convex polygon until it merges into a point. After this happens, all the outer corners will be postponed from one another and thus fill the entire plane.

An interesting fact, isn't it? And there are a lot of such facts in geometry. So learn geometry, dear students!

The material on what the sum of the angles of a convex polygon is equal to was prepared by Sergey Valerievich

Sections: Maths

Presentation . (Slide 1)

Lesson type: a lesson in learning new material.

Lesson objectives:

• Educational:
  • consider the theorem on the sum of the angles of a triangle,
  • show the application of the theorem in solving problems.
• Educational:
  • fostering a positive attitude of students towards knowledge,
  • to educate students by means of a lesson in self-confidence.
• Developing:
  • development of analytical thinking,
  • development of "learning skills": to use knowledge, skills and abilities in the educational process,
  • development of logical thinking, the ability to clearly formulate their thoughts.

Equipment: interactive whiteboard, presentation, cards.

DURING THE CLASSES

I. Organizational moment

- Today in the lesson we will recall the definitions of rectangular, isosceles, equilateral triangles. Let's repeat the properties of the angles of triangles. Applying the properties of internal one-sided and internal criss-crossing angles, we will prove the theorem on the sum of the angles of a triangle and learn how to apply it to solving problems.

II. Orally(Slide 2)

1) Find rectangular, isosceles, equilateral triangles in the figures.
2) Define these triangles.
3) Formulate the properties of the angles of an equilateral and isosceles triangle.

4) In the figure KE II NH. (slide 3)

- Specify secants for these lines
- Find internal one-sided corners, internal criss-crossing corners, name their properties

III. Explanation of the new material

Theorem. The sum of the angles of the triangle is 180 °

According to the formulation of the theorem, the guys build a drawing, write down the condition, conclusion. Answering questions, they independently prove the theorem.

Given: Prove:

Proof:

1. Draw line BD II AC through vertex B of the triangle.
2. Specify secants for parallel lines.
3. What about CBD and ACB angles? (make a record)
4. What do we know about the CAB and ABD angles? (make a record)
5. Replace the CBD angle with the ACB angle
6. Make a conclusion.

IV. Complete the sentence.(Slide 4)

1. The sum of the angles of a triangle is ...
2. In a triangle, one of the angles is equal, the other, the third angle of the triangle is equal to ...
3. The sum of the acute angles of a right-angled triangle is ...
4. The angles of an isosceles right-angled triangle are ...
5. The angles of an equilateral triangle are ...
6. If the angle between the lateral sides of an isosceles triangle is 1000, then the angles at the base are ...

V. A little history.(Slides 5-7)

The sum of the interior angles of a triangle is 180 0. This is one of the fundamental axioms of Euclid's geometry. It is this geometry that students study. Geometry is defined by the science that studies the spatial forms of the real world.

What prompted the ancient Greeks to develop geometry? The need to measure fields, meadows - areas of the earth's surface. At the same time, the ancient Greeks accepted that the surface of the Earth is horizontal and flat. Taking this assumption into account, Euclid's axioms were created, including the sum of the interior angles of a triangle in 180 0.

An axiom is understood as a statement that does not require proof. How should this be understood? A wish is expressed that suits the person, and then it is confirmed by illustrations. But everything that is not proven is fiction, that which is not in reality.

Taking the earth's surface to be horizontal, the ancient Greeks automatically took the form of the Earth flat, but it is different - spherical. There are no horizontal planes and straight lines in nature at all, because gravity bends space. Straight lines and horizontal planes are found only in the brain of the human head.

Therefore, the geometry of Euclid, which explains the spatial forms of the fictional world, is a simulacrum - a copy that does not have an original.

One of Euclid's axioms says that the sum of the interior angles of a triangle is 180 0. In fact, in real curved space, or on the spherical surface of the Earth, the sum of the interior angles of a triangle is always greater than 180 0.

We reason like this. Any meridian on the globe intersects with the equator at an angle of 90 0. To get a triangle, you need to move another meridian away from the meridian. The sum of the angles of the triangle between the meridians and the side of the equator will be 180 0. But there will still be an angle at the pole. As a result, the sum of all the angles will be more than 180 0.

If at the pole the sides intersect at an angle of 90 0, then the sum of the interior angles of such a triangle will be 270 0. Two meridians intersecting with the equator at a right angle in this triangle will be parallel to each other, and at the pole, intersecting with each other at an angle of 90 0 will become perpendiculars. It turns out that two parallel lines on the same plane not only intersect, but I can be perpendiculars at the pole.

Of course, the sides of such a triangle will not be straight lines, but convex, repeating the spherical shape of the globe. But, just such a real world of space.

The geometry of real space, taking into account its curvature in the middle of the 19th century. developed by the German mathematician B. Riemann (1820-1866). But schoolchildren are not told about this.

So, Euclidean geometry, which takes the form of the Earth flat with a horizontal surface, which in fact does not exist, is a simulacrum. Nootic is Riemann's geometry, taking into account the curvature of space. The sum of the interior angles of the triangle in it is greater than 180 0.