Elementary Geometry - Sholaster M.M. Signs of parallelism are two straight lines. The power of the parallel straight lines of Kuti, inscribed in the colo

Kuti.

Basic understanding.

kut- tse figure, set up by two interchanges, scho go from one point.

top kuta- the point is that there are two promenades, which store the cot.

bisector- the price, which goes from the top of the kut and extends the kut navpil.

rozgornuty kut- tse kut, the sides of which lie on the same area; door 180? and straight.

Straight kut- tse kut, half half of the open-cut kut; door 90?.

Gostriy Kut- tse kut, which is less than direct.

Stupid kut- tse kut, which is more direct, ale less flared.

The kut breaks the area into two parts. The skin in parts is called flat cut.

Flat kuti from the back sides are called dodatkovym.

Since the flat kut is a part of the square, then the degree world is called the degree world of the zvychayny kut with these sides.

If a flat kut is to revenge on a flat kut, then the first degree of the world is 360 º - α, de α is the degree of the world of a pre-flat kut.

Rivni kuti.

Tse kuti, which are lost when applied.

Sumіzhny kuti.

Two kutas are called sumy They have one side that is sleek, and the other sides of the tsikh kutiv are made up of additional interchanges.

For little kuti (ad)і (cd) sum_zhny. They have a side d zagalnaya, and sides aі c- additional changes.

theorem:

Sum of sumy kutiv road 180 º.

3 distillation theoremsє:

If there are two kuti rivni, then the sum of them is kuti rivni.

If the kut is not flared, then the degree of the world is less than 180 º.

Kut, sum_zhny with a straight kut, є straight kut.

Vertical kuti.

Two kutas are called vertical, Yakshho sides of one kut є with the addition of interchanges of the sides. The stench of the stems with a peretin two straight and not є snuggle down, the backward top and the same degree of the world.

On a little kuchi (A 1 B 1) and (A 2 B 2) vertical. Sides A 2 and B 2 of the other kut є with additional interchanges of sides A 1 and B 1 of the first kut.

theorem:

Vertical kuti pivn.

Central kut.

central cut in a circle is called a flat kut with a top in the center (Fig. 1).

A part of a circle, roztasovana in the middle of a flat cut, called arc cola, Vidpovidnoy central corner (in Fig. 1 arc AB є arc stake).

degree world the arc of the stake is called the degree world of the central kuta.

Kuti, inscribed in the colo.

Kut, the top of which lies on the number, and the sides cross the line, be called we will fit into the colo(Fig. 2).

power:

Kuti with a crossover two straight to the third.

When overflowing straight aі b sichnoyu c Confirm to see the kutiv, as the little one is numbered in numbers. Deyaki pari tsikh kutiv mayut special name:
vіdpovіdnі kuti: 1 і 5, 4 і 8, 2 і 6, 3 і 7;

navkhrest lying kuti: 3 і 5, 4 і 6;
one-sided kuti: 4 і 5, 3 і 6.

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Food 1. Yaki kuti are called sumy?
View. Two kutas are called summous, as they have one side of the bedroom, and the other sides of the tsikh kutis are preadatkovy interchanges.
There are 31 kuti per baby (a 1 b) and (a 2 b) sum. Their side b is backward, and the sides a 1 and a 2 є are pre-finished.

Food 2. Bring, that the sum of summery kutiv roads 180 °.
View. Theorem 2.1. Sum of summery kutiv roads 180 °.
Delivered. Nekhai kut (a 1 b) і kut (a 2 b) - danі sumіzhny kuti (div. Fig. 31). Beam b pass between sides a 1 and a 2 of the unfolded kut. To the sum of kutiv (a 1 b) i (a 2 b) to the unbent kuti, i.e. 180 °. It’s necessary to bring it up.

Food 3. To tell that if there are two kuti ryvni, then the sum of them kuti is also equal to it.
View.

the theorems 2.1 viplya, if there are two kuti rivni, then the sum of them kuti rivni.
Supposedly, kuti (a 1 b) і (c 1 d) рівні. We will need to inform you if kuti (a 2 b) and (c 2 d) are also equal.
Sum of summery kutiv roads 180 °. The third grade, a 1 b + a 2 b = 180 ° і c 1 d + c 2 d = 180 °. Zvidsi, a 2 b = 180 ° - a 1 b і c 2 d = 180 ° - c 1 d. So yak kuti (a 1 b) і (c 1 d) рівні, then we will not accept it, but a 2 b = 180 ° - a 1 b = c 2 d. According to the power of transitivity of the sign of equality of traces, scho a 2 b = c 2 d. It’s necessary to bring it up.

Food 4. What kut is called direct (gostry, stupid)?
View. Kut, scho dorіvnyuє 90 °, is called a straight kut.
Kut, less than 90 °, is called Gostim Kut.
Coot, greater than 90 ° and less than 180 °, is called stupid.

Food 5. Bring, that kut, sumy with direct, є straight kut.
View. From the theorems about the sum of summaries with a straight cut, є straight cut: x + 90 ° = 180 °, x = 180 ° - 90 °, x = 90 °.

Food 6. Yaki kuti are called vertical?
View. Two kutas are called vertical, like the sides of one kut є additional interchanges of the sides.

Food 7. Make sure that you are vertical kuti rivn.
View. Theorem 2.2. Vertical kuti pivn.
Delivered. Nekhai (a 1 b 1) і (a 2 b 2) - dany vertical kuti (fig. 34). Cut (a 1 b 2) є summed with cut (a 1 b 1) і with cut (a 2 b 2). Sounds according to the theorem about the sum of summery kutiv robimo visnovok, which kozen z kutiv (a 1 b 1) i (a 2 b 2) add kut (a 1 b 2) to 180 °, so that kuti (a 1 b 1) i (a 2 b 2) рівні. It’s necessary to bring it up.

Food 8. To bring it up that when there are two straight lines, one of the straight paths, then the three straight paths are also straight.
View. It is permissible that straight AB and CD interlace one one at the point O. Suppose that the cut AOD is 90 °. So, as the sum of sumy kutiv roadways is 180 °, then it is conceivable that AOC = 180 ° -AOD = 180 ° - 90 ° = 90 °. Kut COB vertical kutku AOD, to that stinks. Tobto kut COB = 90 °. Kut COA is a vertical Kutku BOD, so the stinks of the rіvnі. Tobto kut BOD = 90 °. In such a rank, all kuti are equal to 90 °, so that the stench of all is straight. It’s necessary to bring it up.

Food 9. Are straight lines called perpendicular? What is a sign of vikoristovu to mark the perpendicularity straight?
View. Two straight lines are called perpendicular, as the stench is overflowing with a straight cut.
The perpendicularity of the straight lines is denoted by \ (\ perp \). The entry \ (a \ perp b \) reads: "Line a is perpendicular to straight line b".

Food 10. To make it clear that it is possible to draw a straight point through a straight line perpendicular to a straight line, or just one.
View. Theorem 2.3. Through the dermal line, it is possible to draw a perpendicular line, and only one.
Delivered. Let a - given a straight line і A - given a point on them. Significantly through a 1 one of the straight lines a with the cob point A (Fig. 38). Approximately from the exchange a 1 cut (a 1 b 1), pivny 90 °. Todi is straight, like avenging b 1, if it is perpendicular to straight a.

It is acceptable that the line is straight, it can pass through the point A and is perpendicular to the straight line a. In terms of c 1, the half-line of the straight line, which lies in the same half-line with the exchange b 1.
Cut (a 1 b 1) і (a 1 c 1), equal to skin 90 °, in one section of the area from the exchange a 1. From the exchange a 1 into the given half-plane, it is possible to extend only one edge, equal to 90 °. This is not a straight line to pass through point A і perpendicular to straight line a. The theorem has been completed.

Food 11. So also the perpendicular to the straight line?
View. A perpendicular to a straight line is called a straight line, perpendicular to a line, which can be one of these points to cross. Tsey kinets vidrizka to be called before perpendicular.

Food 12. Explain why the pole is proof of a protylezhny.
View. A way of proving, as we have stated in Theorem 2.3, is called a proof of the opposite. Tsey is a way of proving the polarity in that which is robust at first, as opposed to the fact that it is solidified by the theorem. By the way of the world, spiraling on axioms and bringing theorems, we come to the point of supervising either mind theorems, or one with axioms, or the theorem brought earlier. At the time of presentation, we are robust, but our presentation is not true, which means that it’s true theorems.

Food 13. Who is called a bisectrix kuta?
View. The bisector of the kut is called the promin, which goes from the top of the kut, passes between the sides and extends the kut navpil.

Signs of parallelism of two straight lines

Theorem 1. Even if there are two straight lines,

navkhrest recumbent kuti pivni, abo

іdpovіdnі kuti іvnі, abo

sum of one-sided cuts on the doorstep 180 °, then

straight parallel(Fig. 1).

Delivered. Interchangeable with proof of the fall 1.

Do not go when you cross the straight a and b with a blue AB on the opposite side of the recumbent kuti rivni. For example, ∠ 4 = ∠ 6. Proved, scho a || b.

It is permissible that straight a and b are not parallel. That stench changes into a deyak_y point M і, from the same, one of the cuts 4 or 6 will be called the cuff of the AVM tricycle. Do not go for the value ∠ 4 is the name of the AVM tricycle, and ∠ 6 is the internal one. With the theorems about the calling out of the tricycle of the vipliv, 4 is more than ∠ 6, but not to overreact, it means that it’s straight and і 6 can’t change, because the stench is parallel.

Slidstvo 1. Two straight lines on a square, perpendicular to one and the same straight, parallel(Fig. 2).

Respect. The way, as they have brought Vipadok 1 by Theorem 1, is called the method of proving the type of antiquity, or reduced to non-dexterity. The first name is the way of otrimav to that, which on the cob of the world is to be afraid of popping, it is disgusting (opposed) to that which needs to be brought. Those who have been brought to a blindness are called as a result of the fact that they are growing at the end of a crushed stew, we come to a blindness (to the point of absurdity). Otrimannya such visnovka zmushuє us vіdkinuti otroblenaya from the selection of admission and accept those that would need to be brought.

Zavdannya 1. While staying straight, go through the given point M and parallel to the point of the straight line a, but not go through the point M.

Decision. Drawn through point M to straight line p perpendicular to straight line a (Fig. 3).

Then it is drawn through point M to straight line b perpendicular to straight line p. The straight line b is parallel to the straight line as follows from Theorems 1.

The most important visitor tasks are shown:
through a point, but not to lie on a straight line, you can always draw a straight line parallel to the line.

The main power of parallel straight lines is in the offensive.

Axiom of parallel straight lines. Through a given point, not to lie on a given straight line, to pass only one straight line, parallel to the given.

The actions of the power of parallel straight lines are visible, as they are drawn from the axiomi.

1) If there is a straight line overflowing one of two parallel straight lines, then it overflows and іnshu (Fig. 4).

2) If there are two straight lines parallel to the third straight, then the stench is parallel (Fig. 5).

The theorem is valid and fast.

Theorem 2. If two parallel lines are crossed with the other, then:

navkhrest recumbent kuti rivni;

vіdpovіdnі kuti rіvnі;

sum of one-sided doorways 180 °.

Slidstvo 2. If it is straight perpendicular to one of two parallel straight lines, then it is perpendicular to one(Div. Fig. 2).

Respect. Theorem 2 is called vortex theorem 1. Viznovok theorems 1 є viznovsky theorems 2. But viznov theorems 1 and viznovsky theorems 2. Not every theorem will vizorotnu, that is.

Explainable in the application of the theorems about vertical kutas. The tsiu theorem can be formulated as follows: if two kuti are vertical, then the stench is rivn. The zvorotniy їi theorem was used as follows: if there are two kuti rіvnі, then the stench is vertical. And tse, zychayno, not so. Two pivnih kuta it is not the crops that are crocheted, but they are vertical.

Butt 1. Two parallel straight lines cross the third. Seemingly, there is a difference between two internal one-sided roads 30 °. Know tsi kuti.

Decision. Don't mind seeing babies 6.

Edited by V.P. Ivanitskaya - M .: State educational institution of the Ministry of Education of the Russian Federation, 1959. - 272 p.
blackmail(Directly) : egnnsholaster1959.djvu Front 1 .. 11> .. >> Step

If sum_zhnі kuti is rіvnі, then kozen from them is called a direct kut. The back side is called the perpendicular to the straight line, approved by the two sides. You can also say that the bisector of the open-cut kuta is perpendicular to the straight line, approved by the sides.

Theorem. Yakshho kuti are rivni, then rivni and sumy with them kuti.

Come on (h, k) = ^. (I, t) і nay ^ (h!, K) і ^ (/ ", t) - similar to sumy kuti (Fig. 20). in (I, tri) .With a complete rusi, the flares ^ (h, K) appear as flared (I, / "). It is obvious that ^ (h ", k) appears in ^ (V, m), that is, ^ (H!, K) = ^ (V, m).

Theorem. Isnu bisectrix be-like kuta і before the same one.

Leave ^ (A, k) visible from the unfolded і inner region of the opucleus. Apparently on the th sides of the summit About the edges of the OA and OB (Fig. 21, a) and from the same point A and B. B trikotnik AOB A = ^ B (§ 8). Change the middle of C from AB to the point O, we can recognize the tricytes L OS і BOC Otzhe, AOC = VOC, and also promin OS - bisector (h, k).

If (h, k) is not є opucl (on the armchairs, the inner area is NOT shaded), then behind the front

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t ^

theorem yogo bisectrix є promin t, before the exchange /.

With respect to tricycles ACO and BCO, there is also the same ^ scho ^ ACO = BCO1 T. E.

Come on now given one-turn ^ (p,<7) (черт.21,6). Совершим движение, при котор ом р азвер нутый

ACB display in

(P, q). The CO ray is visualized when tsom in promin t. So yk ^ (p, t) = ^ lBCO, ^ BCO = ^ ACO і ^ ACO = = (q, t), then (p, t) = ^ (q, t), i.e. (p, q).

Nekhai / - bisectrix

(A, A), and G is a good place to go from the top of the kut and lie at the inner area of ​​the kut. If G lie near the inner region ^ (A, /), then ^ (A, / ")<^ (А, /) и ^ (А, Г) >^ (A, /). Otzhe, ^ (A, D)<^ (А, /"). Отсюда следует, что угол имеет единственную биссектрису. Теорема доказана.

Slide 1. There is one or only one perpendicular to the set of straight lines that go from the given point and lie in the set of the square, which is surrounded by a straight line.

Slidstvo 2. Half of the old kutiv are equal to themselves.

Surely, if ^ (A, A) = ^ (A ", A"), then іsnyє rukh /, for which one of them appears in іnshiy. For the brought theorem їх bisectrix / і Г, with a whole rusі, one is also guilty of in іnshu. To that ^ (A, /) = ^ (A ", Γ).

So, as all the razgornuty kuti are rivni, then we will surround it with the following two propositions: all straight kuti are equal to ourselves.

Straight a and a, which are set when the straight cut is overturned, are called perpendicular (a ± b).

Imaging straight ahead. Let’s lie straight but lie in the area. Comparison with a whole pivploshini is significant through X і р. (Fig. 22). Vizmemo on a straight line A

go from point O. According to the power of 6 rukhiv (§ 7), there is a single ruh, which appears to be h in itself and napivloshina X in napivloshina jx. All points of the whole exchange in power of 5 rukhiv appear in themselves. All points of exchange k, which add to the straight line h, may appear in themselves.

Otzhe, given Rus, all specks are straight and appear in themselves. Easy, dal, bachiti, scho po

Vіzmemo now point the pose straight a.

Theorem. Through a point, not to lie on a straight line, to pass one straight line, perpendicular to the given straight line.

Delivered. Let M be the point where you lie in a straight posture a (fig. 23). Straight and stretch the area, as it starts to go straight and

point M, on two areas: on the area X, to ave the point M, and on the area on jx. When imaged from a straight line a point M is mapped to a point M "of the half-plane jx. So the points M and M" lie in the other half-

stah, then straight MM "and a Chort 23

change into action

point M0, as when imaged, it appears in itself. Looks like a straight line MM "appears in itself, and that kuti / i 2, set by her with a straight line (div. Fig. 23), appears alone in the one.

the loop plane jx is displayed at the same location in the area X.

To look at the collapse to be called views from a straight line.

From the opening of the bisectrix of the unfolded kuta, go, through a point, lying on a straight line a, you can always draw a straight line of sight, perpendicular to a straight line.

This means that there is a stench, as well as a stench, in addition to the sum of money, then MM "± a. Now, through M, a straight line is drawn, as it crosses over to the straight line and to the deyak point Af0. appear in M ​​"N0M0. Otz, ^ 3 = ^ i4. But, by virtue of axiomi 1 (§ 2), the points M1 N0 and M" do not lie on the same straight line, because the sum of the cutouts is 3 and 4, that is, ^ MN0M " , not є unbuttoned kut. View from straight line MN0 will NOT be perpendicular to straight line a. straight line MM "є, in such a rank, in one straight line, perpendicular to a and go through point M.