The sum of kutiv tricoutnik - why is it so expensive? Suma kutiv trikutnik. Learn more

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The sum of the inner kutіv of the tricoutnik is as old as 1800. It is one of the main axioms of the geometry of Euclid. The very same geometry is taught by schoolchildren. Geometry is signified by science, as weaving the expanses of the real world.

What prompted the ancient Greeks to develop geometry? The need to win the field, bows - the earth's surface. From this the ancient Greeks accepted that the Earth is horizontal, flat. With a glance at the whole, the axioms of Euclid, zokrem and the sum of the inner kutivs of the tricot were created in 180 0 .

Under the axiom, the camp is conceived, as if we need to prove it. How can you understand? There is an honor, like a powerful person, and further it is confirmed by illustrations. Ale, everything that is not brought is a guess, those that are not true.

Accepting the earth's surface horizontally, the ancient Greeks automatically formed the shape of the Earth flat, but otherwise - spherical. There are no horizontal planes and straight lines in nature, but gravity bends the space. Straight lines and horizontal planes are less than a human head near the brain.

Therefore, the geometry of Euclid, as it explains the expanses of the form of the divine world, is a simulacrum - a copy that does not have the original.

One of the axioms of Euclid is to say that the sum of the inner hoods of the tricot tree is older than 1800. In fact, in a real curved space, or on the spherical surface of the Earth, the sum of the inner hoods of the tricot tree is greater than 180 0 .

Mirkuemo so. Whether a meridian on the globe tumbles with the equator at a 90° angle. Sob take off the tricot, you need to take the other meridian out of the meridian. The sum of kutiv trikutnik between meridians and the side of the equator becomes 180 0 . And yet to lose the kut of the pole. Through war, the sum of all kutivs and become more than 180 0.

If you turn on the poles of the side under the kut 90 0, then the sum of the inner kutiv of such a tricot will be 270 0. Two meridian, which intertwine with the equator under a straight kutom at that trikutnik, will be parallel one to one, and on the poles, which interchange one with one under the kutom 90 0, become perpendicular. To come out, two parallel lines on one plane do not only overlap, but I can be perpendicular to the poles.

Obviously, the sides of such a knitwear will not be straight lines, but bulging, which repeat the spherical shape of the earth's kuli. Ale same so real world space.

The geometry of the real space with the improvement of its curvature in the middle of the 19th century. developed by the German mathematician B. Riman (1820–1866). Ale about tse schoolchildren do not seem.

Otzhe, Euclidean geometry, which forms the shape of the Earth flat horizontal surface, What is not true, it is a simulacrum. Nootik is the geometry of Rome, which vrakhovu the curvature of space. The sum of the inner kutіv of the tricot in the new one is more for 180 0 .

proof

Come on ABC" - Dovіlny trikutnik. Passing through the top B straight, parallel straight AC (such a straight line is called straight Euclidean). Significantly on the spot D so, sob points A і D lay on different sides in a straight line BC.Kuti DBCі ACB rivnі yak vnutrіshnі navhrest lying, utavnі sіkoi BC with parallel lines ACі BD. To that sum of kutiv trikutnik at the tops Bі W dorivnyuє kuta ABD.The sum of all three kutiv of the tricutnik is more than the sum of kutiv. ABDі BAC. Oskіlki tsі kuti vnutrіshnі unilateral for parallel ACі BD at sіchnіy AB, their sum is 180°. The theorem has been completed.

Lasts

From the theorem it is obvious that a tricoutnik has two kuti gostrі. Actual, stubborn proof of the unacceptable, it is acceptable that the tricoutnik has only one hospitable kut, or there are no hospitable kutivs. The same tricot can take two kuti, the skin of which is not less than 90 °. The sum of these cuts is not less than 180°. But it’s impossible, the shards of the sum of all the kutivs of the tricot tree reach 180 °. What did it take to bring.

An overview of the simplex theory

De-cut between i and j faces of the simplex.

Notes

On the sphere of the sum of the cutivs of the tricot, it always exceeds 180 °, the difference is called spherical and too proportional to the area of the trikut.
At the area of Lobachevsky sum, the tricutnik is less than 180 °. The price is also proportional to the area of the tricot.

Div. also

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Wonder what is the "Theorem about the sum of kutіv trikutnik" in other dictionaries:

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NON-EUCLIDEAN GEOMETRY- Geometry, similar to the geometry of Euclid in that, in it, the ruh figures are assigned, but it is also considered in the Euclidean geometry, because one of the five postulates (the other or the five) substitutions for the lists. A list of one of the Euclidean postulates. Collier Encyclopedia

Proof:

Danish tricoutnik ABC.
Draw a straight line DK through the vertex B parallel to the base AC.
\angle CBK= \angle C as an inner crossover to lie at parallel DK and AC, and parallel BC.
\angle DBA = \angle A inner crosswise lying at DK \parallel AC and AB. Kut DBK roaring and equal
\angle DBK = \angle DBA + \angle B + \angle CBK
Oskіlki rasgornuty kut 180 ^\circ , and \angle CBK = \angle C and \angle DBA = \angle A , then we take 180 ^\circ = \angle A + \angle B + \angle C.

Theorem finished

Lessons from the theorem about the sum of kutіv trikutnik:

The sum of gostrikh kutiv of a straight-cut tricutnik is dorіvnyuє 90°.
At the equal-femoral straight-cut tricoutnik of the skin, the gostry kut dorіvnyuє 45°.
At the equal-sided tricotnik, the skin is cool 60°.
For a knitter, either all kuti gostrі, or two kuti gostrі, and the third one is blunt or straight.
Zovnishhnіy kut tricutnika is the sum of two internal kutіv, not the sum of them.

theorem

The old kut of the tricot tree is worth the sum of two kutiv of the tricot tree, which are lost, not the sum of the winter kut

Proof:

Given trikutnik ABC, de BCD - ovnishnіy kut.
\angle BAC + \angle ABC +\angle BCA = 180^0
Z of equalities kut \angle BCD + \angle BCA = 180^0
Acceptable \angle BCD = \angle BAC+\angle ABC.

Back to the other side:

Gramo with a mosaic under a fairy tale with geometry:

There lived boules trikutniks. So similar, it's just a copy of one of the same.
They began to stink like an order on a straight line. And the fact that the stench was all the same growth -
then the tops of their boules were on the same level, under the line:

Trikutniks loved to flip and stand on their heads. Weakly at the top row and stood on a tuft, mov acrobats.
And we already know - if the stink to stand upright in line,
then they have the same feet along the line - for if it’s the same height, then the wine and upside down are the same height!

All the stench was the same - and the height is the same, and the feet are one to one,
and weights on the sides - one steep, іnsha larger canopy - on the same
and sickened them however. Well, just twins! (only in casual clothes, on the skin of your little piece of the puzzle).

- De trikutniks toss the same sides? And where are the kutochki the same?

The tricutniks stood on their heads, they stood, they vyrishili z_skovznuti and lie down in the bottom row.
They shackled and slammed like a girka; and the weights in them are the same!
Axis and moved around amidst the lower tricots, without gaps and didn’t disturb anyone.

Trikutniks looked around and remembered the specialty of the cicava.
Skrіz, de їhnі kuti zіyshlisya at once - all three kuti unanimously chirped:
the largest - "kut-head", the highest kut and the third, the middle one after the size of the kut.
The stench navitt lines of color were tied up, so that it was commemorative, de yaky.

І vyishlo, scho three kuti trikutnik, yakscho ї podnati -
to create one great kut, "kut navstizh" - like the lining of a book opened,

______________________about ___________________

vin is called like this: roaring kut.

Whether a tricoutnik has a passport: three kuti at once is equal to a torn kutka.
Someone comes to you: - knock-knock, I'm a trickster, let me spend the night!
And wi youmu - Present the sum of kutiv to the roaring one!
I suddenly realized - chi tse spravzhnіy trikutnik chi impostor.
I didn’t go through the revision - Turn around a hundred and fifty degrees and go get out!

If it seems "to turn 180 ° - tse means to turn back to front and
go straight to the gate.

Those same in larger virazes, without "lived boules":

Zrobimo parallel transfer of the tricot ABC vzdovzh osі OH
per vector AB Rіvniy dovzhіnі foundations AB.
A straight line, DF, which can pass through the vertices С and С 1 trikutnikov
parallel to the axis OX, to that perpendicular to the axis OX
vіdrіzki h а h 1 (vіsoty rіvnih trikutnikіv) rіvnі.
In this order, the base of the tricot A 2 B 2 C 2 is parallel to the base AB
and dorivnyuy yoma on dozhina (because the top Z 1 is shifted to Z by the value of AB).
Trikutniki A 2 B 2 Z 2 і ABC are built on three sides.
And also kuti ∠А 1 ∠В ∠С 2, which make the rotting kut, finish the kutam of the trikutnik ABC.
=> Suma kutіv trikutnika dorіvnyuє 180 °

With ruhi - "translations" so called, the proof is shorter and more accurate,
on the small pieces of the mosaic, you can wind the little ones, but you can understand.

Natomist traditional school:

which spiral on the alignment of internal cross-lying coils, which appear on parallel lines

valuable tim that we give a statement about those - why is it so,
why is the sum of kutiv of a tricoutnik dorivnyuy to a burnt kut?

To that, otherwise, parallel lines would not be small for our light of power.

Theorems work in the wrong direction. From the axioms about parallel lines follow
rivnіst navhrest lying and vertical kutіv, and їх - the sum of kutіv trikutnik.

Alé vіrno i zvorotne: until the cuti of the tricoutnik to become 180 ° - to establish parallel lines
(Thus, it is possible to draw a single straight line through a point not lying on a straight line | | given).
Every once in a while a tricoutnik appears in the world, in some sum of kutivs it’s not good for a torn kutka.
then the parallels will cease to be parallel, the whole world will twist and twist.

Like smuga with an ornament from trikutniks, roztashuvati one over one -
you can cover the entire field with a repeating visor, nibi under the tile:

you can circle on such a page different figures - six-pieces, rhombuses,
zіrkovі bagatokutniki i otrimuvati raznі parquet

The pavement of the area with parquets is not less than a cicava gra, but it is also an actual mathematical problem:

________________________________________ _______________________-------__________ ________________________________________ ______________
/\__||_/\__||_/\__||_/\__||_/\__|)0(|_/\__||_/\__||_/\__||_/\__||_/\=/\__||_/ \__||_/\__||_/\__||_/\__|)0(|_/\__||_/\__||_/\__||_/\__||_/\

Oskіlki kozhen chotirikutnik - a rectangle, a square, a rhombus and іn.
can be folded from two tricots,
apparently the sum of cutiv chotirikutnik: 180 ° + 180 ° = 360 °

However, even-femoral tricots are folded in squares in different ways.
Small square of 2 parts. Middle of 4. І largest of 8.
Skіlki on armchair figures, scho folded out of 6 trikutnikіv?

In the 8th grade, for an hour of lessons in geometry, the schoolchildren learn first to understand the understanding of the swollen bagatokutnik. Nezabar stinks know that this figure can even be a cicada of power. She wouldn’t have been like a folding one, the sum of all the inner and outer kutivs of the bulging bagatokutnik is gaining a strictly singing meaning. In this article, a tutor in mathematics and physics tells about those who are worth the sum of the kutiv of a swollen bagatokutnik.

The sum of the inner folds of the swollen bagatokutnik

How to bring this formula?

Let's move on to the proof of which firmness, guess what, some kind of bug-bearer is called opuklim. Vipuklim is called such a bagatokutnik, which is supposed to be known on one side of the straight line, in order to take revenge on that side. For example, such, such an image for this little one:

Well, if the bagatokutnik is not satisfied with the appointment of the mind, the wine is called inconsistent. For example, like this:

The sum of the inner folds of the bulging hornwort is more dense, the number of sides of the hornwort.

The proof of which fact is based on goodness is given to the schoolchildren of the theorem about the sum of kutivs for a trickster. Upevneniy, what is the theorem you know. The sum of the inner kutiv of the tricoutnik is good.

The idea is to split the puffy bagatokutnik into a sprat of trikutnikov. You can do it in a different way. It’s worth it, whatever way we choose, prove the trochs will be resurrected.

1. Rozіb'єmo puffy bagatokutnik on tricutniks with all possible diagonals, drawn from the vertices. It is easy to understand that our n-kutnik rose to trikutnik:

Moreover, the sum of all kutivs of all trikutniks, which came out, is more expensive than the sum of kutivs of our n-kutnik. Adzhe kozhen kut in trikutniks, sho wiyshli, є private kuta in our swollen bagatoknik. Tobto shukana is a sum of money.

2. You can also choose a point in the middle of the swollen bagatokutnik and at the bottom with mustache peaks. Todi our n-kutnik rose on trikutnikov:

Moreover, the sum of the kutivs of our bagatokutnik in this way is more expensive than the sum of all the kutivs of our trikutniks for the virahuvannyam of the central kut, which is more beautiful. Tobto shukana suma I’m still good.

The sum of the ovnishnіkh kutіv of the puffy bagatokutnik

Now let's put the food: "Why do you need the sum of the beautiful kutivs of the swollen bagatokutnik?" Vіdpovіsti tse pitanya can be so. Leather ovnіshnіy kut є sumіzhnim іz vіdpovіdny vіdnіshіnіm. That's why vin dorivnyuє:

To the sum of all the zovnishnіh kutіv dorіvnyuє. That's good.

Tobto vyhodzhe kumedny result. If one by one, one after another, one by one, lay all the oval kuti of a swollen n-kutnik, then as a result, the entire area will be completely covered.

Tsey cicavia fact can be illustrated in this way. Let's proportionately change all the sides of such a puffy bagatokutnik to quiet fir, until the veins do not turn into a speck. After that, as it happens, all the names of kuti will appear as one kind of one and fill the entire area with such a rank.

Ts_kavy fact, chi not so? Geometry has a lot of such facts. So learn geometry, dear students!

Material about those, to which the sum of kutiv of a swollen bagatokutnik is rich, prepared by Sergiy Valeriyovich