What is the sum of the angles. What is the sum of the angles of the convex polygon

The triangle is a polygon having three sides (three angle). Most often, the parties are denoted by small letters corresponding to capital letterswhich denote opposite vertices. In this article, we will get acquainted with the types of these geometric figuresTheorem that determines what the sum of the corners of the triangle is equal.

Types of corners

The following types of polygon with three vertices are distinguished:

• acute-deed, in which all the corners are sharp;
• rectangular, having one straight angle, with its formulations, are called categories, and the side, which is placed opposite to direct corner, is called hypotenuse;
• stupid when one;
• an isosceles, in which two sides are equal, and they are called side, and the third - the base of the triangle;
• equally, having all three equal side.

Properties

Allocate the main properties that are characteristic of each type of triangle:

• on the contrary, most sides are always a larger angle, and vice versa;
• opposite the equal size of the parties are equal angles, and vice versa;
• any triangle has two sharp corners;
• external angle more compared to any inner angle, not related to it;
• the amount of any two angles is always less than 180 degrees;
• the outer angle is equal to the sum of the other two angles that are not intertwined with it.

Theorem on the sum of the corners of the triangle

The theorem argues that if you add all the angles of a given geometric shape, which is located on the Euclidean plane, then their amount will be 180 degrees. Let's try to prove this theorem.

Let us have an arbitrary triangle with the vertices of the CMN.

Through the vertex, the CN will carry (still called direct Euclidea direct). It will note the point and thus that the point K and A is located from different sides of the straight line. We obtain equal angles of AMN and KNM, which, like internal, lie in the nearest and are formed by the sequential MN, together with direct CN and MA, which are parallel. It follows from this that the sum of the corners of the triangle located at the vertices of M and H is equal to the size of the CMA angle. All three angle constitute the amount that is equal to the amount of CMA and MCN angles. Since these angles are internal one-sided relative to parallel direct CN and MA with a sequential CM, their amount is 180 degrees. Theorem is proved.

Corollary

Of the above, the theorem follows the following consequence: any triangle has two sharp corners. To prove it, assume that this geometric figure has only one sharp angle. It can also be assumed that none of the corners are acute. In this case, there must be at least two angle, the magnitude of which is equal to or more than 90 degrees. But then the sum of the angles will be greater than 180 degrees. And this can not be, because according to the theorem, the sum of the corners of the triangle is 180 ° - no more and no less. That's what it was necessary to prove.

Property of external corners

What is the sum of the corners of the triangle, which are external? The answer to this question can be obtained by applying one of two ways. The first is that it is necessary to find the amount of the corners that are taken one at each vertex, that is, three angles. The second implies that you need to find the sum of all six corners at the tops. To begin with, we will deal with the first option. So, the triangle contains six external corners - with each vertex two.

Each pair has an equal angles, as they are vertical:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

In addition, it is known that the external angle in the triangle is equal to the sum of the two internal, which are not intertwined with it. Hence,

∟1 \u003d ∟A + ∟С, ∟2 \u003d ∟A + ∟V, ∟3 \u003d ∟В + ∟С.

It turns out that the amount of external angles that are taken one by one vertices will be equal to:

∟1 + ∟2 + ∟3 \u003d ∟A + ∟С + ∟A + ∟V + ∟V + ∟С \u003d 2 x (∟A + ∟V + ∟С).

Taking into account the fact that the amount of the angles is equal to 180 degrees, it can be argued that ∟a + ∟v + ∟c \u003d 180 °. This means that ∟1 + ∟2 + ∟3 \u003d 2 x 180 ° \u003d 360 °. If the second option is used, the sum of six corners will be, respectively, more than twice. That is, the sum of the external corners of the triangle will be:

∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 \u003d 2 x (∟1 + ∟2 + ∟2) \u003d 720 °.

Right triangle

What is the sum of the angles of the rectangular triangle, which are sharp? The answer to this question, again, follows from the theorem, which claims that the corners in the triangle in the amount are 180 degrees. And our statement sounds (property) so: in rectangular triangle Sharp corners in the amount give 90 degrees. We prove his truthfulness.

Let us give us a triangle of KMN, whose ∟n \u003d 90 °. It is necessary to prove that ∟k + ∟m \u003d 90 °.

So, according to the theorem on the sum of the angles of ∟k + ∟m + ∟n \u003d 180 °. In our condition it is said that ∟n \u003d 90 °. So it turns out, ∟k + ∟m + 90 ° \u003d 180 °. That is, ∟k + ∟m \u003d 180 ° - 90 ° \u003d 90 °. That is what we should prove.

In addition to the above properties of the rectangular triangle, you can add to the following:

• the angles that lie against the cathets are sharp;
• triangular hypotenuse is more than any of the cathets;
• the amount of cathets is more hypotenuse;
• the catat of the triangle, which lies opposite the angle of 30 degrees, is twice as fewer hypotenuses, that is, it equals her half.

As another property of this geometric shape, you can select the Pythagora theorem. It claims that in a triangle with an angle of 90 degrees (rectangular) the sum of the squares of the cathets is equal to the square of the hypotenuse.

The sum of the angles of an elevated triangle

Earlier, we said that the polygon with three vertices containing two equal sides is equally called. This property of this geometric shape is known: angles at its base are equal. We prove it.

Take the triangle of KMN, which is an equally chagrined, the book is its foundation.

We need to prove that ∟k \u003d ∟ So, let's say that Ma is the bisector of our triangle of KMN. The triangle of the ICA, taking into account the first sign of equality, is equal to the triangle of the MNA. Namely, according to the condition, it is given that KM \u003d NM, MA is a common party, ∟1 \u003d ∟2, since Ma is bisector. Using the fact of equality of these two triangles, it can be argued that ∟k \u003d ∟. So, the theorem is proved.

But we are interested in what the sum of the corners of the triangle (is an equilibrated). Since in this regard, he does not have its own features, will be repelled from the theorem discussed earlier. That is, we can argue that ∟k + ∟m + ∟n \u003d 180 °, or 2 x ∟k + ∟m \u003d 180 ° (since ∟k \u003d ∟n). We will not prove this property because the theorem on the sum of the corners of the triangle has been proven earlier.

In addition to the properties of the triangle corners, there are also such important allegations:

• which was omitted for the base, is simultaneously median, bisector angle, which is between equal parties, as well as its base;
• medians (bisector, heights), which have been carried out on the sides of such a geometric shape, are equal.

Equilateral triangle

It is also called correct, this is the triangle that all parties are equal. And therefore the angles are also equal. Each of them is 60 degrees. We prove this property.

Suppose we have a KMN triangle. We know that km \u003d nm \u003d kN. And this means that, according to the property of the angles, located at the base in an equilibried triangle, ∟k \u003d ∟m \u003d ∟. Since according to the theorem, the sum of the corners of the triangle is ∟k + ∟m + ∟n \u003d 180 °, then 3 x ∟k \u003d 180 ° or ∟k \u003d 60 °, ∟m \u003d 60 °, ∟n \u003d 60 °. Thus, the approval is proved.

As can be seen from the above proof on the basis of the theorem, the sum of the angles as the sum of the angles of any other triangle is 180 degrees. To prove this theorem to be needed.

There are still such properties characteristic of an equilateral triangle:

• median, bisector, height in such a geometric figure coincide, and their length is calculated as (and x √3): 2;
• if you describe around this polygon circle, its radius will be equal to (and x √3): 3;
• to enter a circle into an equilateral triangle, its radius will be (a x √3): 6;
• the area of \u200b\u200bthis geometric shape is calculated by the formula: (A2 x √3): 4.

Stupid triangle

According to the definition, one of its corners is between 90 to 180 degrees. But considering the fact that the other angle of this geometric shape is sharp, it can be concluded that they do not exceed 90 degrees. Consequently, the theorem on the sum of the corners of the triangle works when calculating the amount of the corners in the stupid triangle. It turns out, we can safely assert, relying on the aforementioned theorem that the sum of the angles of the stupid triangle is 180 degrees. Again, this theorem does not need re-evidence.

Evidence

Let be ABC " - Arbitrary triangle. Let's spend through the top B. straight, parallel direct AC (This direct is called direct Euclidea). Note on her point D. so that the points A. and D. lying on different sides from straight BC..Ugly DBC. and ACB. equal as internal closets lying, formed by the Sale BC. with parallel straight AC and BD.. Therefore, the sum of the corners of the triangle at the vertices B. and FROM equal to the corner ABD.Such every three triangle angles are equal to the sum of the corners ABD. and Bac.. Since these angles are internal one-sided for parallel AC and BD. under sech AB, Their amount is 180 °. Theorem is proved.

Corollary

From the theorem it follows that any triangle has two angle sharp. Indeed, applying proof from no other, assume that the triangle has only one sharp angle or there are no sharp corners at all. Then this triangle has at least two angle, each of which is not less than 90 °. The sum of these angles is not less than 180 °. And this is not possible, since the sum of all the corners of the triangle is 180 °. Q.E.D.

Generalization in the Simplex theory

Where -gol between the i and j are the edges of the simplicity.

Notes

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

see also

Wikimedia Foundation. 2010.

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The sum of the internal angles of the triangle is 180 0. This is one of the fundamental axes of the geometry of Euclide. It is this geometry that studies schoolchildren. Geometry is determined 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 - sections of the earth's surface. At the same time, the ancient Greeks took that the surface of the earth is horizontal, flat. Taking into account this assumption, the axioms of Euclide, including the sum of the internal corners of the triangle in 180 0, were created.

Under the axiom means a provision that does not require evidence. How do you need to understand? It is expressed by a wish that suits man, and further it is confirmed by illustrations. But all that is not proven - fiction, what is not in reality.

Taking ground surface Horizontal, the ancient Greeks automatically accepted the shape of the Earth flat, but it is different - spherical. There are no horizontal planes and straight lines in nature, because gravity twists space. Straight lines and horizontal planes are available only in the human brain.

Therefore, the geometry of the Euclide, explaining the spatial forms of the fictional world, is a simulacrome - a copy that does not have the original.

One of the axiom of Euclide states that the sum of the internal corners of the triangle is 180 0. In fact, in the real twist space, or on the spherical surface of the Earth, the sum of the internal angles of the triangle is always greater than 180 0.

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

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

Of course, the sides of such a triangle will be not straight lines, but by convex, repeating 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 XIX century. Developed a German mathematician B. Riman (1820-1866). But they do not speak schoolchildren.

So, Euclidova geometry, taking the shape of the Earth flat with a horizontal surface, which is not really no, is a simular. Nootik - Riemann geometry, which takes into account the curvature of space. The sum of the internal corners of the triangle in it is greater than 180 0.

RESEARCH

ON THE TOPIC:

"Is always the sum of the triangle angles equal to 180˚?"

Performed:

Student 7b class

MBOU INZEN SS №2

g. Inza, Ulyanovsk region

Malyshev Jan.

scientific adviser:

Bolshakova Lyudmila Yuryevna

TABLE OF CONTENTS

Introduction ....................................................................... ..3

The main part ................................................... 4

search for information

experiments

output

Conclusion ...................................................... ..12.

Introduction

This year I began to learn a new item-geometry. This science studies the properties of geometric shapes. At one of the lessons, we studied the theorem about the sum of the corners of the triangle. And with the help of proof, we concluded: the sum of the corners of the triangle is 180˚.

I thought, is there such triangles who have the amount of corners will not be 180˚?

Then I set myselfTARGET :

To find out when the sum of the triangle angles is not equal to 180˚?

Put the followingTASKS :

Get acquainted with the history of geometry;

Get acquainted with Euclidean geometry, Roman, Lobachevsky;

Prove the experimental way that the sum of the triangle angles may not be equal to 180˚.

MAIN PART

Geometry arose and developed in connection with the needs practical activities man. During the construction of even the most primitive structures, it is necessary to calculate how much material will go to the construction, calculate the distances between points in space and the corners between the planes. Trade and navigation development required the skills to navigate time and space.

For the development of geometry made a lot of scientists Ancient Greece. The first evidence of geometric facts is associated with the nameFalez Miletsky.

One of the most famous schools was Pythagorean, named after its founder, the author of the evidence of many theorems,Pythagora.

Geometry, which is studied at school, called Euclidean, namedEuclida - Ancient Greek scientist.

Euclid lived in Alexandria. He wrote the famous book "Beginning". The sequence and severity made this product with a source of geometric knowledge in many countries around the world during more than two millennia. Until recently, almost all school textbooks were largely similar to the "beginning."

But in the 19th century it was shown that the Euclideas axioms are not universal and are correct in any circumstances. The main discoveries of the geometric system in which the Axioms of Euclide are not correct, were made by Georg Riemann and Nikolai Lobachevsky. They are talking about how about the creators of non-child geometry.

And here, relying on the teachings of Euclid, Riemann and Lobachevsky, let's try to answer the question: is the amount of the triangle angles always 180˚?

Experiments

Consider a triangle from the point of view of geometryEuclidea.

To do this, take a triangle.

Fill its corners with red, green and blue colors.

We will spend a straight line. This is a detailed angle, it is 180 ˚.

Withdraw the corners of our triangle and put them to the unfolded corner. We see that the sum of three angles is 180˚.

One of the stages of the development of geometry was elliptical geometryRiemann. A special case of this elliptical geometry is geometry on the sphere. In the geometry of Riemann, the sum of the corners of the triangle is greater than 180˚.

So, this is the sphere.

Inside this sphere, a triangle is formed by meridians and the equator. Take this triangle, paint its corners.

Cut them and apply to the line. We see that the sum of three angles is greater than 180˚.

In geometryLobachevsky The sum of the corners of the triangle is less than 180˚.

This geometry is considered on the surface of the hyperbolic paraboloid (this is a concave surface resembling a saddle).

Examples of paraboloids can be found in architecture.

And even the "Pringle" chips -ample paraboloid.

Check the sum of the corners on the model of the hyperbolic paraboloid.

A triangle is formed on the surface.

Take this triangle, crate its corners, cut them down and put them on a straight line. Now we see that the sum of three angles is less than 180˚.

OUTPUT

Thus, we proved that the sum of the corners of the triangle is not always equal to 180˚.

It can be more, and less.

Conclusion

In conclusion, I want to say that it was interesting to work on this topic. I learned a lot of new things for myself and, in the future, I would be happy to learn this interesting geometry.

INFORMATION SOURCES

ru.wikipedia.org.

e-osnova.ru.

vestishki.ru.

yun.moluch.ru.

Evidence:

• Dan triangle ABC.
• Through the vertex b we will spend direct DK parallel to the base AC.
• \\ ANGLE CBK \u003d \\ Angle C as internal closer under the parallel DK and AC, and the securing BC.
• \\ ANGLE DBA \u003d \\ ANGLE A Internal closer under the DK \\ Parallel AC and the securing AB. DBK angle deployed and equal
• \\ ANGLE DBK \u003d \\ ANGLE DBA + \\ ANGLE B + \\ ANLE CBK
• Since the detailed angle is 180 ^ \\ CIRC, A \\ ANGLE CBK \u003d \\ ANLE C AND \\ ANGLE DBA \u003d \\ ANGLE A, I get 180 ^ \\ CIRC \u003d \\ ANGLE A + \\ ANGLE B + \\ ANGLE C.

Theorem is proved

The consequences of the theorem on the sum of the corners of the triangle:

1. The sum of the sharp corners of the rectangular triangle is equal to 90 °.
2. In an equilibried rectangular triangle, each sharp angle is equal 45 °.
3. In the equilateral triangle, every angle is equal 60 °.
4. In any triangle, either all the corners are sharp, or two angles are sharp, and the third is stupid or straight.
5. The outer angle of the triangle is equal to the sum of two internal angles, not related to it.

Theorem on the external triangle

The external angle of the triangle is equal to the sum of the two remaining triangle angles, not adjacent to this outer angle.

Evidence:

• Dan triangle ABC, where the ALD is an external angle.
• \\ Angle BAC + \\ Angle ABC + \\ Angle BCA \u003d 180 ^ 0
• From equal corner \\ Angle BCD + \\ Angle BCA \u003d 180 ^ 0
• Receive \\ Angle BCD \u003d \\ Angle BAC + \\ Angle ABC.