On the little one 18 it is shown, as for the additional line on the line, and with the cob point A, it is possible to add the vіrіzоk dоvzhini (3 cm).
Look at figure 19. a, continued beyond the cob point A, dividing the plane into two flats. It is shown on the little one, as for the help of the protractor, it is laid out in a straight line, and at the upper flat it is cut with a degree world (60 °).
The main power in the vіdkladannya vіdіrіzkіv i kutіv i nazvatimemo advancing authority:
VI. On be-yakіy pіvpryаіy vіd її pochatkovї point it is possible to add vіdrіzok given dolzhina, and only one.
VII. Vіd be-yakoy napіvprіvmoї in a givennapіvploschinait is possible to set a cut from a given degree approach, less than 180 °, and only one.
Manager (30). On the exchange AB, there is a vіdіzka АС, a smaller vіdrіzka AB. How can three points A, B, C lie between two others? Explain the point.
Solution (Fig. 20). The points of the point U and Z lie one half-line with the cob point A, all the points do not share the point A, that is, the point A lies between the points U and Z.
Can a point lie between points A and C? Yakby won lay between points A and C, then bulo b AB + BC = AC.
Ale is impossible, shards behind the mind Vіdrіzok AC is less than AB. This means that the point does not lie between points A and C.
Three points A, B, C, one lie between two others. Tom dot, dot Z lie between points A and B.
A. V. Pogorelov, Geometry for grades 7-11, Handyman for lighting installations
Geometry
The main powers of the simplest geometric figures
Appointment. Axioms
Geometry- tse science about the power of geometric positions. To give respect: a geometric figure - not only a tricot, a colo, a pyramid, etc., but rather an impersonal point.
Planimetry- tse razdіl geometry, yakom vychayutsya posts on the flat.
Dot, mottledі straightє basic concepts of planіmetry. Tse means that it is not possible to give a precise definition. It is possible to show them less, spying on the evidence that has resurrected their power.
The firmness, justice of which are accepted without proof, are called axioms. The stench is to revenge the formulation of the main powers of the simplest figures.
Firmness, how to bring, are called theorems.
Appointment- tse explanation of whether there is some kind of understanding, as if the main understanding is spiraling, what is the understanding, appointed earlier.
Designations: points are designated by great Latin letters; straight lines - in small Latin letters or two great Latin letters (as two points are marked on the straight line).
On a tiny dot A, B, C, N,M that straight aі b. Straight a can you tell me straight MN(otherwise NM).
The entry means that the dot M lie straight a. The entry means that the dot W do not lie straight a.
It is necessary to understand what is straight aі b they tinker with a little one, if we don’t want to, at the point.
The main power (axioms) belonging to points and straight lines on the plane
Axiom I. 1. What would not be a straight line, find points that lie straight, and points that do not lie.
2. Through two points, you can draw a straight line, and more than one. (You need to understand that there are two firms here: in the first place - the basis of such a straight line, and in a different way - unity.)
Axiom II. Three three points on a straight line, one and more than one lie between two others.
Vіdrіzkom part of the straight line is called, as it is formed from the points of the straight line, which lie between two given points. qi points are called kintsy vіdrіzka. A windmill is depicted on the little one AB(Vіdrіzok means, writing yoga kintsi).
The main powers (axioms) of vimіryuvannya vіdrіzkіv
Axiom III. 1. Leather vіdrіzok mає nevnu dovzhinu, greater than zero.
2. Dovzhina vіdrіzka dоrіvnyuє sumі dovzhin parts, on yakі vіn razbivaetsya be some kind of yoga point.
The main power of the location of the point should be straight on the plane
Axiom IV. Directly breaking the flat into two flats. Tse rozbittya may be such a power: if the borders of such a winding lie down to one flat, then the wind does not cross straight; if the corners of the windpipe lie on different flats, then the windshield will turn straight.
Pivdirect, or change, call the part on a straight line, as it is formed from the points of the straight line, which lie on one side of the given point on it. This point is called exchange point. Different lines of straight lines are called supplementary.
Changes are presented for the little one AB(wine AC), DA(otherwise D.B., DC), BC, CB(otherwise CA, CD), BA(otherwise BD), AD.
Promeni ABі AD, BCі BD- supplements. Promeni BDі AC not є dodatkovimi, to that there may be different legal points.
Kut- the whole figure, as it is formed from points - kuta peaks- and two different piv straight lines that come out of the center of the point, - sides of the kut.
Kut, representations of a little one, can be known as follows:,,.
Yakshcho the sides of the kuta є additional pіvstrami, kut call roar up:
Say what please pass between the sides of the kuta yakscho vin come out of the yogo tops and peretinaє yakiys vіdrіzok z kіntsami on yogo sides. For a roared kut, it’s important that it be a kind of promise, that you go out from the yogo peak and wind up on the yogo sides, pass between the sides of the kuta.
The main powers of the mimiryuvannya kutiv
Axiom V. 1. Kozhen kut may be the same degree of the world, greater than zero. Rozgornuty kut dorіvnyuє.
2. Degrees of the world kuta dovnyuє sum of degree entrances of the kutivs, on yakі vins break up be like a barter to pass between the sides.
The main authorities in the vіdkladannya vіdrіzkіv i kutіv
Axiom VI. On whether it is a direct view of the її pochatkovoї point, you can add a vіdrіzok dої dovzhina, and only one. Axiom VII. Vіd be-yakої pіvstramoї in qiu pіvploshchina can be placed kut zієyu degree world, less, and only one.
Trikutnik a figure is called, which is composed of three points, which does not lie on one straight line, and three winds, which combine the points in pairs. Krapki are called tops of trikutnik, and vіdrіzki - yogo parties.
Trikutnik the little one can be known as follows: abo, toshcho.
The main elements of nadannya vsheche trikutnik: sides AB, AC, BC(otherwise a, b, c); kuti (abo), , . ta - lie down to the side AC. - protilezhny botsі AC.
Trikutniks are called equal yakshcho they have the same sides of the river and the other side of the river. In case of any vіdpovіdnі kuti may lie against vіdpovіdnih storіn.
The entry means (div. babies) that:
; ;
; ;
; .
The main power of the foundation of equal knitwear
Axiom VIII. Yakim bi is not a tricutnik, is a tricutnik, which is a good yoma in a given placement at a straight line. Directly called parallel yakscho stinks do not peretinayutsya.
Parallel straight lines, depicting a little one, can be seen as follows: abo.
Axiom of parallel lines
Axiom IX. Through a point that does not lie on a given line, it is possible to draw no more than one straight line parallel to the given line on the plane. To restore respect: the axiom affirms the unity of such a straight line, but does not affirm the same reason.
Mutual expansion of straight lines on the flat
Two straight lines on the flat can: run away;
be parallel (so you don’t overtighten);
Mother one zagalnu point.
(Indeed, if two straight lines could have mated two double dots, then two different straight lines passed through these two points, so that axioms I, p. 2 supersede).
>>Mathematics Grade 7. More lessons >> Geometry: Vіdkladannya vіdrіzkіv i kutіv. More lessons
Vіdkladannya vіdrіzkіv i kutіv
It is depicted on the little one as if for help lines on the straight line a with the cob point A it is possible to add a 3 cm double ditch.
To whom the little one is depicted as if for help protractor vіdklasti vіd vіvіvіvraї a vіdklasti vіd navіvіvraї a u upper flat kut s degree world 60° 
Let's formulate the main authorities in the case of the deposit in the case of such kutiv:
- on be-yakіy pіvpryаіy vіd її pochatkovї point it is possible to add vіdrіzok given dolzhina and only one;
- in a given napіvploshchina it is possible to add a cut from a given degree approach, less than 180 °.
An example of solving problems.
On the exchange AB there is an AC windrow, a smaller AB windage. How can one of the three points A, B, C lie between the other two?
Solution.
The shards of the point B and C lie on the same half-line with the cob point A, and the point A is not the same, so the point A lies between the points B and C.
If point B lies between points A and C, then the equality would be correct: AB+BC=AC. It’s impossible, the shards for the mind of AC are smaller than those of AB. Again, point C does not lie between points A and C.
Three three points A, B, C are more than one to lie between the other two. In our view: point C is staggered between points A and B.
Promin.
We draw a straight line and that is significant on the point O (Fig. 11).
This point goes straight into two parts, the skin of some of them is called a change that comes out of the point (on little 11 one with a change of visions with a bold line). Point O is called the cob of skin change. Call promin either with a small Latin letter (for example, promin h for a baby 12, a), or two great Latin letters, the first of which means an ear of a prome, and a friend - a dot on a prome (for example, promin OA for a baby 12, b) .
Kut.
Guess what- this is a geometric figure, as it is formed from the points and two changes that come out of the points. The exchanges are called the sides of the kut, like the head of the cob - the top of the kut. On the little 13, a kut with a vertex O and sides h і k is depicted. Points A and B are marked on the sides.
Kut is called roaring yakscho insulting his side to lie on one straight line. It can be said that the skin side of the burnt kuta is the extension of the other side. Figure 14 shows a flared kut with a vertex C and sides p i q.
Be-yakiy kut podіlyaє flat on two parts. If the kut is not broken up, then one of the parts is called internal, and insha - old-fashioned the region of the tsy kut (Fig. 15 a). On the little one 15 b images of unopened kut. Points A, B, C lie in the middle of the kut (that is, at the inner distance of the kut), points D and E are on the sides of the kut, and points P and Q are in the pose of the kut (then at the outer distance of the kut). As a kut of rotting, then be it from two parts, on yak veins you can make a flat, you can enter the inner region of the kut. The figure that is formed from the kuta of that inner region is also called the kut.
As soon as you go out from the top of the unburnt kut and pass through the middle of the kut, divide this kut into two kuts. On the little one (16, a), promin OS subdivide the cut of AOB into two cuts: AOC and OWL. If the cut of the AOB is broken up, then be it a promin of the OS, which does not change with the exchanges of OA and OB, divide this cut into two cuts: AOS and SOV (Fig. 16, b). 
Por_vnyannya vіdrіzkіv and kutіv.
There are 20 on the little one, and two holes are shown. In order to install, chi equal stinks chi ni, we put one winding on the other one, so that the end of one winding will fall with the end of the other (Fig. 20, b). As if with him, two other kints would also go together, then the winds would go together, and, therefore, they would stink. If the other two kintsya do not sum up, then the lesser one is respected by that one, which becomes a part of the other. On the little one 20, at the ventricles of the AU, put a part of the bridle AB, then the ventricle of the AC is smaller than the ventricle of the AB (write like this: AC<АВ).
The point of the vіrіzka, scho to divide yogo navpіl, that into two equal vіrіzki, is called the middle of the vіrіzka. There are 21 points on the little one Z, the middle of the cut AB.
On the baby 22, and the images unburned kuti 1 and 2. In order to install, even if they are equal, we put one kut on the other one so that the side of one kut fell behind the side of the other, and the two others stumbled one by one in the direction of the sides, which sank (Fig. 22,b). If the other two sides are likewise going together, then they are going to go together again, and, oh, the stench is equal. If both sides do not come together, then the smaller one is respected by that kut, which becomes a part of the other. On the little one (22, b) cut 1 put a piece of cut 2 to volume 1<2.
Unburned kut warehouse part of the burnt(Fig. 23), therefore, an open kut is larger than an unopened kut. Be it two roared kuti, obviously equal.
Promin to come out of the top of the kuta and divide yoga into two equal kuti, is called bisectrix kuta. For a baby 24 promin l- bisector of kuta hk.
Power supply:
- How many degrees of hot cut?
- What is a bisectrix?
- Need to serve as a protractor?
List of vikoristani dzherel:
- P.I. Altinov, Geometry 7-9 class. Moscow. Vidavnichiy dіm "Bustbust", 2005.
- Programs for lighting installations. Geometry 7-9 classes. Orderly: S.A. Burmistrov. Moscow. "Osvita", 2009 rec.
- Newspaper "Mathematics" No. 19, 2000 rіk.
- Atanasyan, Geometry 7-9 grade.
- Pavlov A. N. Geometry: Planimetry in abstracts and solutions.
- Edited and added by Potunak S.A.
They practiced over the lesson:
Poturnak S.O.
At the heart of the system of teaching, as I once hacked at my lessons, lies the principle: the position of the teacher - to the class, not from the knowledge (ready knowledge, remember that beginners), but from the nutrition, the position of the teacher is for the knowledge of the world. The creation of an hour of lessons in the minds of the formation of intellectual intelligence and cognitive skills, which are at the core of the mind, the development of creative abilities and independent activity of learners, the formation of key competencies are good to go with the problem-search approach of learning. By myself, on the basis of learning through guidance, I try to learn all my lessons. From the first lessons in geometry in the 7th grade, I teach the lads tolerant and learned, by the method of "trials and pardons" to acquire unknown knowledge. Problematic nutrition, super-clear facts, which mutually turn on the points of view, or instructive, practical tasks, like to direct a search for unknown knowledge on the region, become a special management of the problem. I want to propagate some of the presentations of lessons on geometry in the 7th grade, as if they were inspired by the principles above.
Zavantage:
Front view:
To look ahead of the presentation, create your own Google post and see before: https://accounts.google.com
Captions before slides:
The main authorities in the vіdkladannya vіdrіzkіv i kutіv
1. Draw a straight line (horizontally), point to the next point O and B. 2. On the exchange of the OB at the first point of the cob, lay a winder, which is 5 cm thick. 3. How to change the RH at the bottom of the flat, put the BOA cut, which is good 50 ° How many times of the specified time can be applied to this direct line of the point? How many cuts of a given value (degree world) can be placed in a straight line in a given pressure plane? How many corners of a given degree setting can be included in a given direction?
O B C OS = 5 cm B O A 50 ° ∠ B O A = 50 ° O B C C "OS = 5 cm O V A B" 50 ° 50 ° ∠ B O A = 50 ° ∠ B ' OA = 50°
VI. On be-yakіy pіvpryаіy vіd її pochatkovї point it is possible to add vіdrіzok given dolzhina and only one. VII. In the case of a straight line in a given plane, it is possible to add a cut from a given degree approach, less than 180 °, and only one.