Coot mіzh two straight lines theorem. Kut mіzh peresіchnym straight: visnachennya, put on znakhozhennya. Yak know the point to cross two straight

Two straight lines AB and CD are called parallel If the stench lies in the same area and does not change, but it doesn’t last (AB || CD). Kut mіzh parallel straight roads to zero.

Dovzhina is a perpendicular, laid between two parallel straight lines, - see mіzh them.

axiom: through a point, but not to lie on a straight line, you can draw only one straight line, parallel to a straight line.

The power of parallel straight lines:

1. If two are straight parallel to the third straight, then the stench is parallel to oneself.

2. If there are two straight perpendicular to the third straight, then the stench is parallel to one to one.

in case of overretin two parallel straight third straight, Settled up in kutiv (Fig. 13), which are called in pairs:

1) vіdpovіdnі kuti (1 і 5; 2 і 6; 3 і 7; 4 і 8 );

kuti in pairs: (Https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 src = "> 5; https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 "> 6; https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 "> 7; https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 "> 8 );

2) internal navkhrest lying kuti (4 і 5; 3 і 6 ); stink in pairs;

3) call navkrest lying kuti(1 і 8; 2 і 7 ); stench in pairs;

4) internal one-sided kuti (3 і 5; 4 і 6 ); sum of unilateral kutiv dorivnyu 180°

(Https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 "> 5 = 180 °; 4 + 6 = 180 °);

5) one-sided kuti (1 і 7; 2 і 8 ); їх sum dorіvnyuє 180 ° (https://pandia.ru/text/78/187/images/image003_66.gif "width =" 11 "height =" 10 "> 7 = 180 °; 2 + 8 = 180 °).

Thales' theorem. When the sides of the kut are overturned by parallel straight lines(Fig. 16) on the side of the cut to be divided into proportions:

More tricycles.

Two trikutniks are called pick up, Yakshko їkh kuti is equally equal to the sides of one tricycle in proportion to the same to the sides of the latter. more on the side of the little tricycles - the sides, where to lie on the opposite side of the pits.

https://pandia.ru/text/78/187/images/image006_51.gif "alt =" (! LANG: (! LANG: podbnі trikutniki" width="13" height="14">A = !} !} https://pandia.ru/text/78/187/images/image006_51.gif "alt =" (! LANG: (! LANG: podbnі trikutniki" width="13" height="14">B = B1, С = С1 !}!}і number k, Yake dorіvnyuє used to name function of requirements.

Signs of details:

1. Yaksho two kutis of one trikutnik take two kutamіншого, then the trip-ki are different.

2. Yaksho two sides one trikutnik proportional to the two sides of the other trikutnik i kuti, stacking between the sides, рівні, That trikutniki podіbnі.

3. yaksho three sides of one trikutnik proportional to the three sides of the These are tricytes podbnі.

Layers: 1. The area of ​​small tricycles is rendered as a square of performance for:

2. appointee perimeter junior tricycles and bisector, Median, height and middle perpendiculars of the road to the required function.

At the end of the day, we will give a correct value of co-directional changes and we bring a theorem about the equality of co-directional sides. Dal damo viznachennya kuta between intersecting straight lines and intersecting straight lines. It is easy to see, which can be two straight lines. In the end of the lesson, there is a vyrishimo kilka zavdan on the znakhozhennya kutiv mіzh with overwhelming straight lines.

Topic: Parallelism of straight lines and areas

Lesson: Cuti with co-directed sides. Kut mіzh two straight

Be-yaka straight, on the butt GO 1(Fig. 1.), the area is divided into two squares. how many exchanges OAі О 1 А 1 parallel and lie in one square, then the stench is called co-directional.

exchange О 2 А 2і OA not є co-directional (Fig. 1.). The stench is parallel, but not lie in one square.

If both sides are co-directed, then so are the coaches.

Dovedennya

Let us be given a parallel exchange OAі О 1 А 1 and parallel exchange OVі About 1 In 1(Fig. 2.). Tobto, mi maєmo two kuta AOBі A 1 O 1 B 1, Chiy side to lie on the co-directional walkways. Brought to you, scho tsі kuti rіvnі.

On the side of the exchange OAі О 1 А 1 vibrate point Aі A 1 so, schob vidrizki OAі О 1 А 1 bouly rіvnі. Similarly, points Vі IN 1 so vibe OVі About 1 In 1 bouli rіvnі.

Viewable chotirikutnik A 1 O 1 OA(Fig. 3.) OAі О 1 А 1 A 1 O 1 OA A 1 O 1 OA GO 1і AA 1 parallel and equal.

Viewable chotirikutnik В 1 О 1 ОВ... Tsyomu chotirikutniki side OVі About 1 In 1 parallel and equal. For acquaintance with a parallelogram, a chotirikutnik В 1 О 1 ОВє parallelogram. So yak В 1 О 1 ОВ- parallelogram, then sides GO 1і BB 1 parallel and equal.

I straight AA 1 parallel to straight GO 1, I straight BB 1 parallel to straight GO 1, Means straight AA 1і BB 1 parallel.

Viewable chotirikutnik B 1 A 1 AB... Tsyomu chotirikutniki side AA 1і BB 1 parallel and equal. For acquaintance with a parallelogram, a chotirikutnik B 1 A 1 ABє parallelogram. So yak B 1 A 1 AB- parallelogram, then sides ABі A 1 B 1 parallel and equal.

Tricytes are visible AOBі A 1 O 1 B 1. parties OAі О 1 А 1 rіvnі z inspire. parties OVі About 1 In 1є Induce. And they brought me, and the sides ABі A 1 B 1 also equal. Means, tricytes AOBі A 1 O 1 B 1 rіvnі for three sides. Rivny kuti lie on the opposite side of the tricycle. Means kuti AOBі A 1 O 1 B 1 rіvnі, which needs to be brought.

1) Overwhelming straight lines.

As it is straight, it is changing, then it’s my maєmo chotiri of the young kuta. Kutom mіzh two straight, It is called the best of the two straight lines. Kut mіzh pereichnymi straight aі b meaning α (Fig. 4.). Coot α is such, scho.

Small. 4. Kut mіzh two houses crossing straight lines

2) Perekhresny straight

don't be straight aі b peresnі. Viberemo to the point About... through the point About carried out directly a 1, Parallel to the straight a, I straight b 1, Parallel to the straight b(Fig. 5.). straight a 1і b 1 change in points About... Kut mіzh dvoma crossing straight a 1і b 1, Kut φ, i are called kut mіzh intersecting straight lines.

Small. 5. Kut mіzh dvoma perekhresnymi straight

Qi to lay the value of the kuta at the opposite point O? vibrate point About 1... through the point About 1 carried out directly a 2, Parallel to the straight a, I straight b 2, Parallel to the straight b(Fig. 6.). Kut mіzh pereіichnymi straight a 2і b 2 meaningfully φ 1... Todi kuti φ і φ 1 - kuti with co-directed sides. Yak they brought me, so kuti rіvnі mіzh myself. This means that the value of the kuta between the intersecting straight lines does not lie in the vibor of the point About.

straight OVі CD parallel, OAі CD clash. Know kut mіzh straight OAі CD, Yaksho:

1) ∠AOB= 40 °.

vibrate point Z... Pass straight through it CD... carried out CA 1 in parallel OA(Fig. 7.). Todi Kut A 1 CD- kut mіzh perekhresnymi straight OAі CD... According to the theorem about kuti with co-directed sides, kut A 1 CD dorivnyu kutu AOB, Tobto 40 °.

Small. 7. Know kut mіzh two straight

2) ∠AOB= 135 °.

Zrobimo the same pobudova (Fig. 8.). Todi kut mіzh perekhresnymi straight OAі CD the road is 45 °, so it’s the one who’s the one who found the door CDі CA 1.

3) ∠AOB= 90 °.

Zrobimo the same pobudova (Fig. 9.). Todi all kuti, as they go when they cross them straight CDі CA 1рівні 90 °. Shukaniy kut dorivnyuє 90 °.

1) To bring, that the middle of the sides of the spacious chotirikutnik є the tops of the parallelogram.

Dovedennya

Let us be given a spacious chotirikutnik ABCD. M,N,K,L- the middle of the ribs BD,AD,AC,BC as a matter of fact (Fig. 10.). It is necessary to bring, scho MNKL- parallelogram.

Tricycle tricycle ABD. МN МN parallel AB and the expensive halves.

Tricycle tricycle ABC. LK- middle line. According to the power of the middle line, LK parallel AB and the expensive halves.

І МN, і LK parallel AB... to mean, МN parallel LK by theorem about three parallel straight lines.

Otrimuєmo, scho in the chotirikutnik MNKL- sides МNі LK parallel і рівні, so yak МNі LK half AB... So, for the acquaintance of the parallelogram, the chotirikutnik MNKL- a parallelogram to bring the demand.

2) Know kut mіzh straight ABі CD, Yaksho kut MNK= 135 °.

Yak mi have already been brought, МN parallel to straight AB. NK- middle line of tricycle ACD, By power, NK parallel DC... So, through the point N pass two straight lines МNі NK, Yaki parallel perehresnyh straight ABі DC for sure. So, kut mіzh straight МNі NKє kutom mіzh intersecting straight ABі DC... We are given a stupid kut MNK= 135 °. Kut mіzh straight МNі NK- the one who has the least of the kut_v, otrimanih at the cross-flowing straight, tobto 45 °.

Again, we looked at the kuti from the co-directed sides and brought their parity. They looked at the kuti between the cross and the perekhresny straight lines, and they looked at the zakhozhennya kuta between the two straight lines. On the offensive level, we continue to solve problems and repeat the theory.

1. Geometry. 10-11 grades: handler for scholars of educational establishments (basic and professional) / І. M. Smirnova, V. A. Smirnov. - 5th view, corrected and updated - M .: Mnemozina, 2008 .-- 288 p. : Ill.

2. Geometry. 10-11 grades: Pidruchnik for outbound naval pawns / Sharigin I. F. - M .: Bustard, 1999 .-- 208 p.: Il.

3. Geometry. 10th grade: Pidruchnik for educational establishments for the funeral and professional education of mathematics / E.V. Potoskuev, L.I. Zvalich. - 6th vision, stereotype. - M .: Bustard, 008 .-- 233 p. : Ill.

V) BCі D 1 IN 1.

Small. 11. Know kut mіzh straight

4. Geometry. 10-11 grades: handler for scholars of educational establishments (basic and professional) / І. M. Smirnova, V. A. Smirnov. - 5th view, revised and updated - M .: Mnemosyna, 2008. - 288 p.: Il.

Zavdannya 13, 14, 15 bldg. 54

Oh-oh-oh-oh-oh ... well, hard, it’s dumb, you’ve read it yourself =) Well, then relaxation is more helpful, more than that, buying a variety of accessories. That can be done until the first time, I am encouraged, until the end of the statute, I will save the badges of the mood.

Again, roztashuvannya two straight

That vipadok, if the hall goes in chorus. Two straight can:

1) scammed;

2) but are parallel :;

3.)

Dovіdka for teapots : Be affectionate, forget the mathematical sign of the retinue, you will be taught even more often. The record means that it is a straight line to be interlaced with a straight line at a point.

Yak visnachiti in me roztashuvannya two straight?

For some reason from the first vypad:

There are two straight lines, todi and only todi, as long as there are certain factors in proportion, Tobto, also the number "lambda"

It is straightforward and easy to understand three times from the following factors :. Z dermal ryvnyannya slіd, scho, otzhe, danі straight zbіgajutsya.

Dyysno, as well as all the features multiply by -1 (change signs), and all features speed by 2, then you will see one and the same level :.

Another vipadok, if straight parallel:

Two straight parallel todes and only todies, if their performance is at a different proportion: , ale.

Yak butt is visible two straight. The overwhelming proportion of the current performance during the winter:

However, it is quite obvious that.

The І third type of admission, if it changes directly:

The two are straight, sometimes and only one, if the performance is not proportional, Tobto NOT have such a meaning "lambda"

So, for a straight line, the system can be stored:

From the first rivnyannya slid, scho, and from the other rіvnyannya:, it means, the system is insane(Rishen is dumb). In such a rank, the performance in the winter is not proportional.

Visnovok: straight overflow

At practical workers, it is possible to look at the visualization scheme only. Vaughn, before the speech, Velmi nagaduє algorithm for converting vectors to collinearity, which we looked at at urotsi Understanding the line (not) fallowness of vectors. basis vectors... Ale іsnu bolsh civilized packaging:

butt 1

Z'yasuvati instead of roztashuvannya straight:

Decision Snapshot on recent direct vectors in straightforward:

a) We know directly from the vectors directly: .

That means that the vectors are NOT collinear and straight.

I will put all kinds of vipadocs on the list with showcases:

Reshta re-brace the stone and follow the distance, right to Kashchey the Immortal =)

b) We know the direct vectors straight:

Straight may be one and the same direct vector, which means that it smells either parallel or stinks. There is no need for a vvazhnik vvazati.

Obviously, it’s efficiency when it’s unattended proportions.

Z'yasuєmo, whose parity is fair:

In such a rank,

c) We know the direct vectors straight:

It is numbered in the form of the addition of the coordinates of the given vectors:
, Otzhe, direct vectors are collinear. Straight or parallel or start.

The ratio of the proportionality of the "lambda" does not matter directly from the direction of the linear direct vectors. However, you can know it through the performance of the peasants themselves: .

Now z'yasuєmo, chi justly parity. Offended by the strong members of the null, to that:

Otriman's meaning is satisfying to this ryvnyannyu (you are happy with the number).

In such a rank, they are straightforward.

vidpovid:

Even sooner you will see (or else you will see) you will see the task asleep literally in a few seconds. I don’t need to think about it for a sense, but I don’t need it for an independent solution, more importantly, one more important thing is put into the geometric foundation:

How can I stay direct, parallel to the tribute?

For the ignorance of the most simple, the establishment of the suvoro karak Nightingale-Rozbiynik.

butt 2

The straight line is given to the equal. The flanks are parallel straight, yak pass through the point.

Decision: Meaningful to an unidentified direct letter. What is said about her in the mind? Pass straight through the point. And if it is straight parallel, then it is obvious that I will direct the vector of the straight "tse" along the path and to induce the direct "de".

Vityaguєmo will direct the vector from the line:

vidpovid:

Geometry to the butt of a viglyadє unmindfully:

Analytical changeover of the field in the coming crocs:

1) Reversion, if the straight lines have one and the same direct vector (if the straight line is not forgiven by the proper rank, then the vector will be collinear).

2) Perevryaєmo, chi is the point of the renounced rivnyannya.

Analytical change in the large number of types is easy to find sleepy. To marvel at two rivnyannya, and it’s quite a lot for you to quickly start parallelism straight without the need for a chair.

Apply for an independent solution to be creative this year. To that, you will have a chance to snatch up with Baba-Yagoi, and you know, she's a lover of all kinds of riddles.

butt 3

The flatness is straight, go through a point parallel to the straight, like

The most rational and no less rational way of broadcasting. Nykorotshy shlyakh - at the end of the lesson.

In parallel straight lines, the troches were stroked and we turn to them. There is a drop in the number of people who fall short, so the task is clear, as you know well with the school programs:

Yak know the point to cross two straight lines?

very straight change to points, then coordinates to solutions systems of line

Yak know the point to cross straight? Virishiti system.

Axis to you i geometric sense of a system of two lines- several two (most often) straight on the area.

butt 4

Know the point of overflow straight

Decision: There are two ways of viewing - graphic and analytical.

The graphical way of the field is that it is easy to cross the lines and know the point to cross without the front from the chair:

The axis is our point :. To reconcile the slid of presenting the coordinates in the skin straight, the stench is guilty go there, and there. In other words, the coordinates of the point є to the solution of the system. By the day, we looked at the graphical way of viewing systems of line with two family members, two homeless.

Graphical way, vividly, non-falsification, ale ісnu pіtnі minusi. Ні, on the right is not in the way that seventh-graders are so viral; Besides, it’s not so easy to do things straight, the very point of overflowing can be changed in the thirtieth kingdom beyond the borders of the zoshite leaf.

To that point, the point is repetitively shukati using the analytical method. Virishimo system:

For the revision of the system and the vicoristans, the method of term-by-term folding of rivnyans. Schob napratsyuvati show you the tips, see the lesson Yak virishiti system of rivnyans?

vidpovid:

The reversal is trivial - the coordinates of the point of overturning the blame for the satisfaction of the skin rivnyannya system.

butt 5

To know the point of overflowing straight at that vipadku, as the stench overflows.

Tse butt for an independent solution. Zavdannya hand-crafted into a chain of steps. Analysis must be given if necessary:
1) Right side of the flat.
2) Straight lines.
3) Z'yasuvati instead of roztashuvannya straight.
4) If you straighten straight, then you know the point of straightening.

The development of the algorithm is typical for geometrical problems, and I will repeatedly grind it down.

Outside the decision and in the end of the lesson:

Not yet a stop and a couple of chereviks, as they went to another section of the lesson:

Perpendicular straight lines. Go from the point to the straight line.
Kut mіzh straight

It is almost a typical and even more important zavdannya. At the first part, we knew that it would be direct, parallel to the tribute, and at the same time the hatinka on the chicken nizhni flared up by 90 degrees:

Yak staying straight, perpendicular to the given?

butt 6

The straight line is given to the equal. The flanks are perpendicular to the straight line, passing through the point.

Decision: For a wash I see, scho. It’s impossible to know a straight vector. Oscillations are straight perpendicular, focus of simplicity:

Rivnyannya "know" is a normal vector: which is a straight vector.

Equivalent straightforward according to the point and direction of the vector:

vidpovid:

Rosgornemo Geometric Study:

Hmmm ... Orange sky, orange sea, hot hot camel.

Analytical reconversion of solution:

1) Z rivnyan vityaguєmo direct vectors and for help scalar create vectors we come to visnovka, which is straight and perpendicular to :.

Before the speech, you can vikoristovuvati vectors of normal, the price is simpler.

2) Perevryaєmo, chi is the point of the renounced rivnyannya .

Perevirku, I know, it’s easy to go to sleep.

butt 7

Know the point of cross-flow of perpendicular straight lines, when vidomo rivnyannya and speck.

Tse butt for an independent solution. The tasks have a little more, so the solution is to be done by hand, point by point.

Ours will grumble on the triviaє:

Go from point to straight

Before us is the swarth of the little river and our zavdannya polyagaє in that, to go to her with the shortest way. There is no crossing, and the most optimal route will be perpendicular. Tobto, go from a point to a straight line - the chain of a perpendicular direction.

See in geometriya zzvychayut poznachayut walnut letter "ro", for example: - see from the point "eat" to the straight "de".

Go from point to straight bend the formula

butt 8

Know from a point to a straight line

Decision: Everything is necessary, it is necessary to accurately add numbers to the formula and carry out the calculation:

vidpovid:

Viconaєmo armchair:

You know it from a point to a straight line - just in the exactness of a heart's rate. How to arrange an armchair on a paper card in a scale of 1 od. = 1 cm (2 cells), then you can see it in a very line.

It is easy to see one of the following for the armchairs:

Zavdannya polyagaє in order to know the coordinates of a point, which is symmetrical to the point, always straight ... I will propose the viconati independently, I will defend the solution algorithm with the intermediate results:

1) It is known to the straight line, the yak is perpendicular to the straight line.

2) We know the point of overflow straight: .

Insult dії rosіbranі in detail within the framework of this lesson.

3) Point є in the middle of the line. We see the coordinates of the middle and one of the kints. per formulas for the coordinates of the middle of the known.

We will not be confused about the fact that there may be one 2.2 units.

It is difficult here to be determined in the calculations, but in the best way to use a micro calculator, which allows you to use special fractions. Repeatedly gladdening, I am glad and familiar.

How can you know whether there are two parallel straight lines?

butt 9

Know how to be between two parallel straight lines

Tse chergovy butt for an independent solution. Crumbs for me: there are a lot of ways to show up here. Pick up the favors at the end of the lesson, or try to help yourself more beautifully, I think your keenness has gone into the distance.

Kut mіzh two straight

It's not a kut, then a jamb:


In geometriya, behind the cut, there are two straight lines, the LESSER cut, which automatically boils, but we can't be stupid. On a little kut, meanings with a red arc, do not flatter yourself with a kut mіzh crossing straight lines. And to get involved in such a yogo "green" susid abo protesting arrangements"Malinoviy" kut.

If it is straight perpendicular, then for the kut between them you can take be-like from 4 kutiv.

Why do you get kuti? Orіntаtsієyu. In a way, we are important in principle є right through the "scrolling" of the kuta. In other words, negatively ordering kut to be registered with the sign "minus", for example, yaksho.

What am I going to get started? Nachebto you can get along with the zvychaynyy understand kuta. On the right, in the formulas, behind what we will know kuti, we can easily find a negative result, and it’s not guilty to catch you knowing. Coot with the sign "minus" is not better, but only a concrete geometric sense. On the armchair for a negative cut, it is necessary to make a bold oriєntation (for a year).

Yak know kut mіzh two straight? Check two working formulas:

butt 10

Know kut mіzh straight

Decisionі sposib first

Two straight lines, set by the girls in the backward view:

very straight NOT perpendicular, then organizing kut between them can be calculated according to the additional formula:

Naybilsh I will saw the respect for the beast on the banner - the price is accurate scalar tvir Straight vectors in straight:

Yaksho, then the denominator of the formula will approach zero, and the vector will be orthogonal and straight perpendicular. The very fact is shattered by the caution about non-perpendicularity straight into the formula.

Vyhodyachi from the foregoing, the solution is manually issued in two stages:

1) Numerically scalar tvir of direct vectors in straight lines:
, So, straight is not perpendicular.

2) Kut mіzh straight lines are known for the formula:

It is easy to know the kut itself for the additional funcional function. With a lot of vikorists, the arctangent is not paired (div. Graphs and power of elementary functions):

vidpovid:

In general, the value is probably more accurate, and also close to the value (bazhano in degrees, and in radians), calculated for an additional calculator.

Well, minus, so minus, nothing terrible. The axis is geometric illustration:

It is not surprising that kut viyshov has a negative attitude, and even in the minds of the first number, the first number is straight and the kuta respects itself from her.

I still want to reject the positive kut, it is necessary to remember it with direct mice, so that the efficiency of taking from another rivnyannya , And the efficiency of taking from the first rivnyannya. Shorter, it seems, you need to go straight .

ABі ZD re-enumeration of the third straight MN Then they pretended to be dressed in pairs with the following names:

vіdpovіdnі kuti: 1 і 5, 4 і 8, 2 і 6, 3 і 7;

internal navkrug reclining kuti: 3 і 5, 4 і 6;

call navkrest lying kuti: 1 і 7, 2 і 8;

internal one-sided kuti: 3 і 6, 4 і 5;

one-sided kuti: 1 і 8, 2 і 7.

So, ∠ 2 = ∠ 4 і ∠ 8 = ∠ 6, albeit according to the brought ∠ 4 = ∠ 6.

Otzhe, ∠ 2 = ∠ 8.

3. vіdpovіdnі kuti 2 and 6, however, oskilki ∠ 2 = ∠ 4, and ∠ 4 = ∠ 6. It is also a reconciliation in the parity of the existing houses.

4. soum internal one-way kutiv 3 and 6 bude 2d sumy huts 3 і 4 roads 2d = 180 0, and ∠ 4 can be replaced by an identical one ∠ 6. Also, the change is suma kutiv 4 and 5 doors 2d.

5. soum one-sided kutiv if it is 2d, then it’s obvious internal one-sided kutah, Yak kuti vertical.

I will otrimmo with the finished dressing gigantic theorems.

If, with a crossover, two straight lines, the third one is straightforward, scho:

1. Internal navkhrest recumbent kuti are the same;

or 2. The names of the new recumbent kuti are the same;

or 3. Similar cutouts;

or 4. Sum of internal one-sided kutiv dorivnyuє 2d = 180 0;

or 5. Sum of one-sided doorways 2d = 180 0 ,

then the first two straight parallel.

value. kutom mіzh transverse straight to be called kut mіzh peresіchnyi straight lines, parallel data of perekhresnyi straight lines.

butt... Given a cube ABCDA 1 B 1 C 1 D 1 ... Know kut mіzh perekhresnymi straight A 1 Bі C 1 D. In between CDD 1 C 1 conducted diagonal CD 1 ; CD 1 || BA 1  (A 1 B; C 1 D) = (CD 1 ; C 1 D) = 90 0 (cut between square diagonals).

D 1 Z 1 V 1 A 1

. Kut mіzh straight and square.

If it is straight parallel to the area, or if it lies in it, then it is between these straight lines and the area is embraced equal to 0 0.

value. Straight to be called perpendicular to the area If you are perpendicular to be straight, if you lie near the area. In a wide range of cuts between a straight line and an area, it is equal to 90 0.

value. Directly call yourself a kidnapping up to the deyakoi area, as it overtakes the area, it is not perpendicular to it. MK  MN- kidnapped before  KN – projection MN on 

value. I will cut it down to the area and the whole area to be called kut mіzh thief and projection on a given area.

(MN;) = (MN;KN) = MNK= 

Theorem 7 (about three perpendiculars ) ... The abduction to the area is perpendicular to the straight lines, but near the area, only todia and only todia, if the projection of the whole abduction to the whole area, is perpendicular to the given straight lines.

MK  MN- kidnapped before  KN – projection MN on  m  MNm KNm

... Vіdstanі in the open space.

value. I will go from the point to the straight line, If you do not take revenge on a point, you should be called a perpendicular, drawn from a point to a given area.

value. From the point to the area , Yaka does not take revenge on a point, to be called a perpendicular line drawn from a point to a given area.

Become parallel straight lines road from the same point from the same point straight to the right point.

Become a parallel area road from one point of one area to the other area.

Stand in a straight and parallel area road from the point of the center straight up to the area.

value. Vіdstannyu mіzh two houses with overwhelming straight to be called a perpendicular spil.

Become alternate straight lines road from the same point from the same point straight to the area, but to pass through another straight parallel to the first straight (in other words: there are two parallel areas, where there are straight lines).

V. Kut between areas. Two-sided kut.

If the area is parallel, then the kut between them is vvazhaєtsya equal to 0 0.

value. two-sided kut to be called a geometric figure, it is established that two squares with a cordon do not lie in the same area. named after facets , A їх spilny cordon edge of a dihedral kut .

value. Linear cut of a dihedral cut to be called a kut, otrimaniy when the crossover of a given dihedral kut is an area perpendicular to the th rib. All the line cut of this dihedral cut is equal to oneself. The size of the dihedral cut is the most important of the size of the first line cut.

butt... given pirama MABCD , The base of yakoi is a square ABCD from side 2, MAABC, MA = 2. Know kut nahila edge MBC area of ​​delivery.  (behind the familiar perpendicularity of a straight line and area). In such rank, flatness MAB peretinaє dihedral kut with edge BCі perpendicular to youmu. Otzhe, for the designation of the line kut:  MBA- line cut of this dihedral cut.