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Sumy and vertical kuti. Perpendicular straight lines. Bringing theorems about kuti, tied with a stake Yak

I propose at the end of the day to the top of the "proof marathon", as a result of which nine-graders are taught in the versions of the DPA in mathematics. Tied the stench of proving awkward, even at the same hour even more corny geometric facts. At the statty navmisno reports are not directed to the decision of the factory, depriving the deyak of handwriting and instructions. Try to complete the marathon distance independently, without pardon, in one go.

Zavdannya 1. Make sure that the bisectors of the summits are perpendicular.

Coot α values with one arc, β - two

Delivered: you can see from the baby α + α + β + β = 2α + 2β = 180 0 (Rozgornuty kut), from the same, α + β = 90 0 . It’s necessary to bring it up.

Zavdannya 2. two ways ACі BD change in points O, Yaka є in the middle of the skin of them. Bring the parity of tricycles ACDі CAB.

ABCD, zychayno, if a parallelogram, but not in the mind

Delivered: bichnі tricycles of rіvnі behind two sides and cuts between them ( BO = OD- behind the wash, AO = OC- behind the wash, ∠ DOC = ∠AOB- vertical), tobto ∠ ACD = ∠CAB, And splinters of stench є lay right next to AB, CD and vividly AC, then AB parallel DC... The parallelism of straight lines is similarly brought BCі AD. Otzhe, ABCD- the parallelogram for the value. BC = AD, AB = CD(In the parallelogram of the opposing sides of the pins), AC- home for tricycles ACDі CAB, Tom stinks rіvnі for three sides. It’s necessary to bring it up.

Zavdannya 3. Bring, that the median is carried out to the base of the femoral tricot, є bisectrix kut, the prototype of the base, and is also perpendicular to the base.

Kuti, approved by the median and past, called the "lower", the median and the lower sides - "upper"

Delivered: bichni tricycles for a little bit of equal for three sides, for which the vaping parity, in Persha, is the "upper" kutiv (brought, bisectrix), in another way, the "lower" kutiv, in addition to the sum, by the way, give 180 0 90 0 skin (adjusted perpendicularity). It’s necessary to bring it up.

Zavdannya 4. Bring the media to the side of the femur tricycle, rivn.

Trikutniki, approved by the medians, by the lower halves of the vertebral sides of the vernacular trikutnik, called "lower"

Delivered: kuti at the base of a half-length tricycle is equal, to that the "lower" tricycle is equal to the two sides and kutom between them, because of which the parity of carrying out median. It’s necessary to bring it up.

Zavdannya 5. Conduct, bisectrix, carried out from the peaks of the tricycle tricycle, rivn.

Try to be meaningful for a little kuti, obviously, even, if you want to mean by small arcs

Delivered:"Nizhniy" trikutnik of bisectors, which are vaping from the ryvnostі kutіv at the yo pіdstavі, "bichnі" tricytniks are rіvnі for the sides (rіvnі from the brought-up parts of the bisectors) and two kutаs (the first ones are dying) Rivnim one to one, and mean Rivni and Sami bisectrix whole. It’s necessary to bring it up.

Zavdannya 6. To inform, that the end of the two sides of the tricycle, the second half of the third side.

Clean side is called "pіdstavami", cross-seats - "bichny sides"

Delivered: bichni sides of the little and great tricycle on the little one are carried yak 1: 2, besides they have one zagalny kut, and it means the stench is similar for another familiarity with a factor of similarity 1: 2; bring.

Zavdannya 7. Make it clear that the diagonal of the parallelogram breaks it into two pryvnykh tricutnik.

Parallelogram z diagonally, more, mabut, dodati nichogo

Delivered: the opposite side of the parallelogram is equal, the diagonal is the sleeping side for tricycle tricycles, so the smell is equal for the three sides. It’s necessary to bring it up.

Zavdannya 8. Bring, that the median, rectangular tricycle, was carried out to hypotenuse, half of the hypotenuse.

In other words, the median is drawn from the top of the straight kut

Delivered: if near a given rectangular tricycle describe a number, then the inscriptions in the circle of the straight kut of a tricycle will be described on the basis of the hypothesis, that the hypothesis will be the diameter of the central circle, and the halves of the hypotenuse are given to us by the head of the media, It’s necessary to bring it up.

Zavdannya 9. Bring down the details of those who have been carried out to the cola from one point, equal.

Dodatkove pobudova: one point C from point O (thoughts)

Delivered: kuti Bі A straight (the radii of the circle, drawn to the point of the goydannya, perpendicular to the dot), means straight trikutniki AOCі BOC rіvnі by hypotenuse (the side is clear for them OC) That leg (radius circumference OB = OA), and hence AC = CB... It’s necessary to bring it up.

Zavdannya 10. Bring the diameter to pass through the middle of the chord circle, perpendicular to it.

Lіnіya, scho z'єdnu two points on the little one, є the median of the trikutnik, which is

Delivered: v trikotnik The median of the head is shown, which means the diameter, which is perpendicular to the chord. It’s necessary to bring it up.

Zavdannya 11. To make it clear that if the two circumferences are going along the chord, then it is straight, but goes through the center of the chord, perpendicular to the chord dan.

Thoughts all at once are all directed to a small point, a point to cross the horizontal and vertical is called H

Delivered: tricytes O 1 AO 2 i O 1 BO 2 rubles for three sides, from the same, ∠ HO 2 A = ∠HO 2 B, Todi trikutniki Hao 2 i HBO 2 pins on both sides and kutu between them, meaning ∠ AHO 2 = ∠BHO 2, and in the sum two rivny kuta can give 180 0 only in that case, as well as leather ones from them to 90 0. It is necessary to bring it.

Zavdannya 12. Inform that, if you can write in the chotirikutnik, then the sum of the other sides of the coin.

Descriptions of chotirikutnik. Let's call it ABCD. Drive M, E, X and L - dots

Delivered: victorian theorem about the derivation of the dots (zavdannya 9). VC = BP, Wed = CH, DX = DLі AT = AK... Pidsumovuєmo side ABі CD: AB + CD= (AM+ MB) + (DX+ XC) = AL+ BE+ DL+ CE= (AL+ LD) + (BE+ EC) = AD+ BC. It’s necessary to bring it up.

Zavdannya 13. To convey that how close a chotirikutnik can be described, sometimes it is possible to describe some of the other kutivs of the village.

the circumference is described

Delivered: according to the theorem about kut, the inscriptions of the sum of the protolezhnykh kutiv tsih chotirikutnik dorivnyuє 180 0, the stench at once spirals around, the degree of the world like 360 0. It is necessary to bring it.

Zavdannya 14. To convey how close the trapeze can be described, then the trapezium of the ravine.

Delivered: the sum of the protolezhny kutiv of the chotirikutnik, inscribed in the colo, dorivnyu α + β = 180 0 (div. Problem 13), the sum of kutiv at the normal side of the trapezium is also expensive α + γ = 180 0 (tsi kuti є one-sided with parallel submissions and the same bichnіy side), β = γ , Tobto kuti at the presentation of such a trapezium іvnі, і won dіysno іvnobedrena. It’s necessary to bring it up.

Zavdannya 15. In the square ABCD points Beforeі E- middle sides ABі AD for sure. Bring, scho KD perpendicular CE.

Theorem 1 ... The size of the inscribed kuta is half the size of the central kuta, which spirals to the same arc.

Dovedennya ... The list of inscriptions kut is visible ABC, side BC yakogo є diameter of stake, і central cut AOC(Fig. 5).

So yak vidrizki AOі BOє radius cola, then trikutnik AOB- wedges, і kut ABO dorivnyu kutu OAB... oskilki kut AOCє zovnishnіm tricycle cut AOB That is fair

In such a rank, if one of the sides of the inscribed kut passes through the center of the stake, Theorem 1 is completed.

It is now clear if the center of the stake lies in the middle of the inscribed kuta (Fig. 6).

and Theorem 1 has been brought up in a wide variety of ways.

If the center of the stake lies in the posture of an inscribed kuta, it is overlooked (Fig. 7).

In general, fairness

and completing the result by Theorems 1.

Theorem 2 ... The size of the kut, set by the intersecting chords, is the half of the sum of the values of the arcs, laid between the sides.

Dovedennya ... Babies are visible 8.

The size of the kuta AED E chords ABі CD... oskilki kut AED- zovnishniy kut trikutnik BED, A kuti CDBі ABD

it’s necessary to bring it.

Theorem 3 ... The size of the kut, set by the usual, cross-posture with a stake, half the difference in the values of the arcs, laid between the sides of the kut.

Dovedennya ... Little ones are visible 9.

The size of the kuta BED, Confirmed by the point E traditional ABі CD... oskilki kut ADC- zovnishniy kut trikutnik ADE, A kuti ADC , DCBі DABє inscribed kutas, then it is fair

it’s necessary to bring it.

Theorem 4 ... The size of the kut, set by the dotted and chord, is able to pass through the dot, half the size of the arc, laid between the sides.

Dovedennya ... Figure 10 is visible.

The size of the kuta BAC Dotted AB i chord AC... scraps AD diameter, go through the point doty, and the cut ACD- inscriptions kut, scho spiral into diameter, then kuti DABі DCA- straight. This is fair

it is necessary to bring

Theorem 5 ... The size of the kut, approved by the dotted and blue, half the difference in the values of the arcs, laid between the sides of the kut.

Dovedennya ... Figure 11 is visible.

The size of the kuta BED Dotted AB and vividly CD... Splendid, scho kut BDC- zovnishniy kut trikutnik DBE, A kuti BDCі BCDє inscribed kutas. Besides, kuti DBEі DCB, By virtue of Theorems 4, рівні. This is fair

Geometry is an okremiy subject to be taught by schoolchildren in the 7th grade. Geometric tasks to finish with an easy form and in the main that can be seen on the butts: the area of the room, the land plot, the gin and the height in the small objects, the flat objects and the inshe stink for an hour. At the beginning of the development of a non-mediocre geometry, there are first folds, such, for example, as a straightforward understanding, so as to wiggle the hands of a direct dumbness. It’s the simplest type of bagatokutniki to take revenge on all three kuti and three sides.

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The theme of trikutniks is one of the main importantі great topics school programs in geometry 7-9 classes. Having earned the goodness, it is possible to see it even more foldable. At the same time, it is possible to look at an absolutely іnshu geometric figure, and then to distribute її for handiness on the different triangular parts.

For robots over proof of equity Δ ABCі ΔA1B1C1 it is necessary to kindly acquire the signs of ryvnost і figur and in them to pass. Before the vivchennyam, the sign must be visible to start the parity sides and cabins of the simplest bagatokutniks.

To bring, how to cut tricycles, to help the offensive options:

∠ α = ∠ β go out from prompts and figures.
Given in mind.
With two parallel straight lines and obviousness, it can be asserted that the inner side is lying down, so it appears ∠ α = ∠ β.
Dodayuchi (vіdnіmayuchi) to (h) ∠ α = ∠ β rіvnі kuti.
Define more vertical lines ∠ α і ∠ β
Zagalny ∠ α, instantly lay Δ MNKі Δ MNH .
The bisector divides ∠ α into two equal values.
sum_zhny z ∠ 90 °- kut, pivny to vykhidnom.
Sumy rivnim kutah rivnі.
Visota will approve two sums ∠ 90 ° .
Isosceles Δ MNK when given ∠ α = ∠ β.
Rivnykh Δ MNKі Δ SDH type ∠ α = ∠ β.
Earlier the parity Δ MNKі Δ SDH .

Tse tsikavo: Yak know the perimeter of the trikutnik.

3 signs of equanimity of tricycles

prove the merits Δ ABCі ΔA1B1C1 even more manual robiti, spiraling on the main signs of the most simple bagatokutniks. There are three such signs. Smells are even more important when looking at solving geometric problems. Varto look skinny.

Pererakhovani vische signs є theorems and brought by the method of superimposing one figurine on the insha, the summit of the peaks of the previous coots and the cob of promenades. Prove the equanimity of tricycles in the 7th class described in even more accessible forms, albeit foldable by the schoolchildren for practical purposes, and to reveal a great number of elements, signified by the capital Latin letters. The price is not inviting for the bagatokh scholars at the time of the cobbing of the subject. Pillows wander in the names of sides, promenades, kutiv.

Three and three, more important is the theme "Similar to tricycles". The very meaning of "similarity" in geometry means similarity of form when dissolving rosemiriv. For the butt, you can take two squares, which are 4 cm wide on the third side, and the other 10 cm. If you see the chotirikutniks will be similar, and suddenly, mother will see, some other squares will be larger, and the skin side will be enlarged in the same way.

When looking at those details, 3 signs can also be induced:

The first one is about two different kinds of kutas, two different three-piece figurines.
The other is about vugillya and making the sides Δ MNK, Yakі dorіvnyuyut inіvіdnіe elements Δ SDH .
The third is based on the proportion of all the sides of the two required figures.

How to bring yak, who trikutniki podbnі? To reach the fastest one with a pererahovanih sign and correctly describe the whole process of proving the problem. Subject details Δ MNKі Δ SDH It’s easier to take schoolchildren out of the way because, until the moment of education, it’s already possible to repent of the significant elements in geometric motives, not to stray from the majestic number of names and read the chair.

Finishing the passage of the great those of trikutnykh geometric figures, Scientists have already thoroughly guilty of the nobility, how to bring the parity Δ MNK = Δ SDH on both sides, stand up two tricycles chi ni. I’ll look at the bagatokutnik, but there are at least three kuti - all of the geometrical figures that have been found, before the material is mastered, it’s going to go seriously, I will especially respect the other facts of the theory.

In the distant hours and donin poshuk signified the equanimity of figures to get involved in basic enterprises, as є the basis of the foundations of geometry; Hundreds of theorems are brought to the test as a sign of equanimity. Cleverness to bring the balance and flexibility of figures is more important in all spheres of life.

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Freezing of tips on practice

Admittedly, we have a figura, leaning on the archway of the paper. At the same time, we have a line and protractor, for the help of which we can freeze dozens of forms and cutters among them. Yak to transfer to another arkush the paper of the figur of the same size or the scale of two times.

We know that a trikotnik is a tse figure, it is possible to build up from three vidrizki, which are called parties, make kuti. In such a rank, there is a set of parameters - three sides and three kuti, which start a figuru.

However, having frozen the size of all three sides and kutiv, transferring the figure to the top of the surface will appear to the difficult employees. Besides, sense to supply power: but there will be little knowledge of the parameters of two sides and one cut, for all there are three sides.

Having frozen two sides and between them, then put the cut on a new sheet of paper, so I can add a three-piece. Let's get rid of, as the goal of the development, which can bring the signs, for which it is possible to respect the same, and for that, as the minimum number of parameters to reach the nobility, to remove the memory from that, however, the tricytesters.

Important! Figuri are called the same, as they are from one side to another, from one side to another. Let's call them figuri, on both sides and kuti proportions. In such a rank, parity is the price of the ratio with the ratio of proportionality 1.

There are signs of equanimity in tricycles, ladies and gentlemen:

Persha is a sign of equanimity: two trikutniks can be put in the same way, as well as on two sides, as well as kut between them.
another sign of equanimity of tricycles: two tricycles will be the same, as well as two tricycles, as well as the side between them.
the third sign of equanimity of tricycles : Trikutniki can be respected as the same, if all the parties can make the same meal.

Yak to bring, scho trikutniki pivnі. Hopefully proved the merits of tricycles.

Proof 1 signs

The most recent hour of the middle mathematicians was given a sign of engaging in axiom, prote, as it appeared, it can be geometrically brought, spiraling on more basic axioms.

Two tricycles are visible - KMN і K 1 M 1 N 1. The side of the KM can also be used for the same і K 1 M 1, and KN = K 1 N 1. But the MKN cut-out is for the KMN і M 1 K 1 N 1.

If you look at KM і K 1 M 1, KN і K 1 N 1 as there are two interchanges, which go from one point, then it is possible to say that there are two exchanges of the same kuti (the price is given by the theorem). A very parallel transfer of the exchange K 1 M 1 and K 1 N 1 from point K 1 to point K. During the previous transfer of the exchange K 1 M 1 and K 1 N 1 to the rear. It is shown on the K 1 M 1 walkway until the KM, so that I keep my ear at the point of K. Similarly, with KN and K 1 N 1. In this way, carrying K 1 M 1 N 1 so that the points K 1 and K are added, and the two sides are applied, it is possible to remove the additional zbig and the figures themselves.

Important! In the Internet, it is possible to prove the equanimity of tricycles behind two sides and a cube for additional algebraic and trigonometric convergences with the numerical values of sides and cuts. However, historically and mathematically, the Boolean theorem was formulated back to algebra and earlier, below trigonometry. For proving the signs of the theorems, vikoristovuvati is not correct, except for the basic axioms, it is incorrect.

Proof 2 signs

A friend of a sign of rynosti is brought along two kutas and sides, primed on the first.

Proof 2 signs

Clearly KMN and PRS. To the door P, N to the door S. The KN side can also be done, yak and PS. It is necessary to inform that KMN and PRS are the same.

The point M is imaginable when changing KN. Otrimanu point called L. When tsomu dozhina side KM = KL. NKL door PRS. KNL door RSP.

Oskilki sum kutiv to 180 degrees, then KLN to PRS, and to mean PRS and KLN - the same (similar) on the side and rose, it is well known for the first time.

Ale, so as KNL is road KMN, then KMN and PRS are two of the same figuri.

Proof 3 signs

Yak stand up, scho tricycle pivots. Tse is straightforward to prove other signs.

Dovzhina KN = PS. Oskilki K = P, N = S, KL = KM, if KN = KS, MN = ML, then:

Tse means, how offended figuri є take one to one. Ale is so yak on the side of the same, then the stench is equal.

It is a sign of eagerness and vividness of vaping without lingering inheritance. One of them is in the fact that, in order to be important, it is necessary for the nobility of their power, however:

all three sides;
offended parties and kut between them;
offense kuta and the side between them.

Vikoristannya signs of equanimity of trikutniks for the revision of the plant

Traces of the first sign

In the course of the proof, you can come to a number of tsikavi and korishnye heritage.

... The fact that the point of overturning the diagonals of the parallelogram is to spread them into two parts of the same is the inheritance signified by parity and should be given to the proof.
Yaksho є two rectangular tricot, For some of the same gostri kuti That stench podbnі. If you have the first leg of the first leg of the other, then the stench rіvnі. It is easy to reach the lightness - for any straight-haired tricycles є straight kut. That is a sign of equanimity for them more simplicity.
Two tricot shoes with straight kutas, for which two legs may have the same kind of dress, can be used as the same. It is tied together with two legs, which are tied up to 90 degrees. To the one behind the first acquaintance (on two sides and kutu between them), all tricycles are with straight kutas and the same legs - rivni.
Yaksho є two rectangular tricycles, and they have one leg and hypotenuse equal, which means they are the same.

Let us prove a simple theorem.

Є two rectangular knitwear. One side a, b, c, de c - hypotenuse; a, b - legs. The other side has n, m, l, de l - hypotenuse; m, n - legs.

According to the theorem of Pyphagoras, one of the catheters of the door:

;

In such a rank, if n = a, l = s (the parity of the legs and the hypotenses), apparently and other legs will be equal. Figuri, apparently, will be behind the third acquaintance (behind the three sides).

Significantly one more important heritage. Yaksho є two tricycles, and the stench is similar with a special function, so that there are two pairs of all sides of the same, then the space is placed on the road.

The first sign of the merits of the trikutniks. Geometry video tutorial 7th grade

Geometry 7 First sign of equanimity of trikutniks

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We have looked at the topic to help anyone learn more beautifully in the basic geometric concepts and advise their own tips in to find mathematics.

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