How to determine which sign of equality of triangles. The third sign of equality of triangles

For many who study Russian, the real stumbling block is spelling "not" with adjectives - fused or separate. You can't just remember how each word is spelled. But there are several simple rules, and they will help determine the order of writing in each case.

Separate use - as a negative particle

"Not" is placed separately, as an independent particle, in the following cases:

• When used in a sentence with other particles or words expressing negation. For example, not at all interesting book, an unremarkable face, not an exciting film at all.
• When used together with quality adjectives in comparative degree. For example - no harder than ice, no sweeter than sugar, no bolder than a lion.
• If there is some opposition in the phrase, most often transmitted using the particle "a". For example - a person is not evil, but kind, the weather is not sunny, but cloudy, the teacher is not old, but young.
• If there is a denial of any qualitative, possessive or relative name adjective. For example - "this is not grandmother's glasses", "this is not green book"," This furniture is not made of wood.

When does "not" play the role of a prefix and is put together with the word?

The rules for continuous writing are also quite easy to remember.

• Some words can only be used with this prefix. For example, a person can be “nondescript”, but the word “impressive” does not exist in nature. The day may be "rainy", but if you remove the prefix "not", then the word will cease to exist again.
• The phrase in which the adjective is used does not contain opposition, and at the same time the adjective itself can be replaced by a synonym without the prefix “not”. For example - "a low tree grew on a hillock." If the phrase does not contain additional mention that the tree was "low", then there is no opposition - however, the word "low" remains a synonym. In this case, "not" would be a prefix.
• The adjective with the prefix "not" is involved in the listing of any features without opposition. For example, "the tree was low, but sprawling."
• The adjective is used together with such adverbs as "very", "extremely", "extremely". For example - "the house was very uncomfortable", "the weather is extremely unfriendly."

Conveniently, these rules are the same for both full and short forms of adjectives.

1. Adjectives are written together with non-names, which are not used without, for example: careless, nondescript, hostile.

2. They are written together with non-adjectives, which, in combination with, do not acquire the opposite meaning; usually such words can be replaced by synonyms without not. For example: small (cf.: small), unmarried (cf.: single), fake (cf.: false, feigned).

Note. It is not always possible to find a similar synonym, but the affirmative shade of meaning contained in the adjective serves as the basis for a continuous spelling, for example: Someone from outside is sitting on a stone in the chapel (Zhukovsky); Gerwig had some kind of unmanly effeminacy (Herzen); His youthful pranks ... were determined by childish thoughtfulness (Leonov).

3. They are written separately from non-adjectives if opposition is present or implied, for example: the problem is not simple, but complex; the relationship is not hostile, the light is not harsh, the milk is not sour, the meat is not fresh, the look is not kind, the task is not urgent; not many of those present supported the speaker (it is thought: ... but some).

4. As a rule, it is not written separately from relative adjectives, giving a negation to the sign they express, for example: the clock is not golden, the honey is not linden, the sky is not southern here. Of the qualitative adjectives, these include adjectives that denote color and which, in combination with do not form words with opposite meaning, for example: the ink is not blue, the binding is not yellow, the tint is not grey.

This takes into account syntactic function adjective: the rule usually applies to adjectives in the role of a predicate, since the proposed opposition gives the statement the character of a general negative judgment expressed by the particle not, but may not apply to adjectives in the role of a definition. Wed: these people are not from here - to shine with unearthly beauty; non-white paper - non-white citizens (colored); logic is not female - the girl reasoned with non-female logic; the shape is not round - an account for a non-round amount of 119 rubles ( figurative meaning words).

Note. In some cases, a double interpretation is possible and, as a result, a double spelling; cf .: this task is not difficult (the “ease” is affirmed) - this task is not difficult (the “difficulty” is denied); not in front of us common occurrence(i.e. rare) - before us is not an ordinary phenomenon (the opposition is thought: ... but exceptional, out of the ordinary). When written separately logical stress falls on a particle no. When addressing such issues, a broader context is needed.

5. There is a difference between the opposition expressed by the union a, and the opposition expressed by the union but. In the first case, one of the two signs opposite to each other is denied, and the other is affirmed, for example: the river is not deep, but shallow; in the second case, there are no concepts that are opposite to each other, they are completely compatible, i.e. two attributes are simultaneously attributed to the object without denying one of them, for example: the river is shallow, but cold. In the first case, it is not written separately, in the second - together.

6. The presence of explanatory words, as a rule, does not affect the continuous spelling not with adjectives (cf .: spelling not with participles, § 70, paragraph 2), for example: an author unknown to us, facts unknown to science, a remark inappropriate in these conditions, an error imperceptible at first glance, words incomprehensible to the student, details unnecessary for the case, incorrect conclusions in many respects; a case unlike any other; behavior unworthy of a decent person; site unsuitable for construction; a lake invisible behind the forest (everywhere a negative sign is affirmed, and a positive sign is not denied).

Note 1. Separate spelling not with an adjective that has explanatory words with it occurs:

1) with adjectives that in full and short form have different meaning(see paragraph 8 below), for example: an actor not ready to go out, a child not prone to colds;

2) in the presence of negative pronouns and adverbs (beginning with neither) or combinations as explanatory words far not, not at all, not at all, for example: unknown address, innocent people, not at all understandable expression, not at all harmful drink, far from being an easy solution, not at all a fruitless search, by no means new plot; cf .: ways unknown to me - ways unknown to anyone; but.: Nothing is impossible in life; There is nothing illegal in what my neighbor is accused of (the negative pronoun does not depend on adjectives, but is itself explained by them);

3) sometimes when placing an adjective with dependent words after the noun being defined, for example: enterprises that are not subordinate to the trust (under conditions of isolation, the construction with the adjective approaches in meaning to participle turnover); cf .: the chess player played in a style unusual for him - features that are not characteristic of our youth.

Note 2. If an adverb of measure and degree acts as an explanatory word (very, extremely, very, almost, adverbial expression in the highest degree etc.), then it is not written together with an adjective, for example: a very ugly act, an extremely inappropriate attack, a very unsuccessful performance, an almost unfamiliar text, an extremely illegible handwriting.

Note 3. If there is an adverb as an explanatory word, it is quite possible both continuous and separate spelling not with adjectives, which is associated with two meanings in which the indicated adverb is used:

1) “absolutely, very much”;

2) “by no means”, “by no means”;

cf .: a completely unnecessary meeting (completely unnecessary, superfluous) - not at all chance meeting(by no means accidental, the logical stress does not fall on it). In some cases, both interpretations are possible and, as a result, both spellings are possible, for example: very small achievements (small, modest) - not great achievements at all (not great at all). In these cases, a broader context is not needed to resolve the issue of spelling (see also note to item 4). Consolidated writing is typical for informative texts, separate - for polemical ones.

A double interpretation allows the adverb at all:

1) "by no means";

2) “absolutely, completely” - in colloquial style speech;

cf .: Against revival, he spoke inconsistently, got confused, and did not give convincing arguments at all. – The authors of these works are less popular or completely unknown scientists. As is the case with absolutely, final decision depends on the nature of the text.

7. C short adjectives negation is not written basically in the same way as with complete ones: together in the absence of opposition and separately if there is one, for example: a low room - a low room, an incomprehensible question - the question is incomprehensible, invalid transaction - a transaction in violation of the law is invalid; the novel is not interesting, but boring; the bay is not deep, but it is convenient for sailing on a motor boat.

Wed writing in the presence of various explanatory words: Searching for materials on the chosen topic is not difficult for him; Too uncertain were the people of the future; He is completely unfamiliar with latest achievements in the field of animal science. – No control is possible here; They are in no way similar to each other. Wed See also: This river is always restless. This river is never calm.

8. Written separately from non-short adjectives that are not used in full form or have a different meaning in full form, for example: not ready to leave, should not do this, do not intend to be silent, not obliged to help, not glad to meet, not inclined to believe not in the mood for conversation.

Note. Depending on the meaning, not with short adjectives, as with full ones, it is written either together or separately; cf .: our family is not rich (about the same as poor) - our family is not rich (i.e., average income); this girl is ugly (a negative feature is affirmed) - this girl is not beautiful (a positive feature is denied), the address is unknown (the "uncertainty" is affirmed) - the address is not known (the "fame" is denied). Wed See also: No big deal. - It would seem that this distance is not great for stayers.

Separate spelling is more common in pairs: not needed - unnecessary, not right - wrong, disagree - disagree, not capable - incapable (usually in these cases, the denial of a positive sign is felt more than the affirmation of a negative one).

9. Double spelling is also found in combinations not with a comparative degree of adjectives, for example: This screensaver is uglier than that one (more ugly). – This screen saver is not more beautiful than that one (does not have more beauty); That night the patient's sleep was more restless than the previous one (it was even more restless). - That night, the patient's sleep was not calmer than the last (was not calmer).

It is written separately: not lower, not higher, not better, not worse, not closer, not poorer, etc. ( private form spellings not with the comparative degree of adjectives).

It is written separately not with the forms greater, lesser, better, worse, for example: with no less success, with no better chances.

10. Spelling differs not from verbal adjectives on -my and with participles on -my; in the presence of explanatory words, the first ones are written together (like denominative adjectives), the second ones - separately, for example:

a) an island uninhabited for a long time, crystals insoluble in water, figures of people indistinguishable in the dark;

b) reserves not visited by hunters, magazines not read by non-specialists, a child not loved by the mother.

Adjectives in -my include words formed from not transitive verbs(for example: independent, waterproof, fireproof) or from verbs perfect look(for example: incorrigible, infeasible, indestructible). These words apply general rules spellings not with adjectives, i.e. they are written together and in the presence of explanatory words (see examples above), as well as in short form (for example: the island is uninhabited, the disease is incurable, these countries are economically independent). However, the rule remains separate writing adjectives with not, if the explanatory words are pronouns and adverbs starting with neither, or combinations far from, not at all, not at all (see above, paragraph 6, note 1. subparagraph 2), for example: with nothing an incomparable impression, independent countries, by no means soluble crystals; this phenomenon cannot be eliminated either from life or from art. The exception is words that are not used without, for example: no one invincible army, for no one an incomprehensible case, under no circumstances a unique experiment.

Note. It is necessary to distinguish between writing not with words in -my, formed from transitive verbs imperfect form: such words can be like passive participles present tense, and adjectives (in the first case, the spelling with is not separate, in the second it is continuous). They are participles if they use as an explanatory word instrumental case actor, less often creative tools (the so-called instrumental); in the presence of other explanatory words, they become adjectives (lose the meaning of passiveness and the meaning of time and acquire qualitative value). Compare: a child not loved by the mother - games unloved in childhood (in the second case, the word unloved indicates constant feature, means approximately the same as "unpleasant", "undesirable"); motion not inhibited by air - the side of the moon invisible from the Earth.

Adjectives of this type include: invisible, insane, inflammable, inextinguishable, immovable, indivisible, unforgettable, invisible, immutable, unloved, unthinkable, tax-free, inalienable, untranslatable, incommunicable, unknowable, unverifiable, unconjugated, intolerant, etc. Cf. their writing in the presence of explanatory words: a number indivisible by three, unforgettable meetings for us, through tears invisible to the world, records unthinkable in the recent past, indescribable in simple words feelings, accounts that have not been verified for a long time, mud that is impenetrable in the springtime, nouns that are indeclinable in Russian, behavior that is intolerable in our society, etc.

11. Separately spelled not with an adjective in interrogative sentence, if the negation is logically underlined, for example: Is this position clear without any evidence? Who does not know the names of our astronauts?

Note. If the negation is not emphasized, then continuous spelling is used, for example: Is this provision unclear? Is this statement false! (possible replacement: Is this statement erroneous?)

Continuous and separate spelling of the particle and the prefix "not" with different parts speech is always difficult. Someone relies on the principle: “If you can insert a word, write separately,” others try to avoid controversial points, preferring to use synonyms. Spelling “not” with adjectives is one of the simplest topics if you understand it. The main thing is to correctly explain the material.

Rule Structure

Spelling "not" with adjectives is a rule that completely coincides with the spelling of the same particle and prefix with nouns and adverbs ending in "o". You will have to remember only eight points of the rule, six of which are about separate, and the remaining two are about continuous spelling negative particle prefix.

"Not" merged

In no case should we forget that there are words with which “not” is always written together - it is part of the root, which cannot be separated from the word. So, “not” with adjectives that are not used without it is always written together. For example: careless, unlucky, clumsy - the words "careless", "traveling" and "clumsy" do not exist in Russian, so in no case can you tear off "not" from them. Usually, no one has any difficulties with this part of the rule.

The second case when “not” is written together with adjectives is the ability to replace the adjective with a synonym without a negative prefix. For example: shallow - shallow, bad - bad, rather big - big. It is important to remember that this rule only works with quality adjectives(it is they who have synonyms and antonyms and form degrees of comparison).

"Not" separately - common points

Separate "not" with adjectives - the rule is more voluminous, but not at all more complicated. Of course, you can go from the opposite: if it is not replaced by a synonym without “not” and is used without “not”, write separately, but such universal tricks do not always work, besides, sometimes you need to be able to explain why it is written that way and not otherwise.

Firstly, the particle “not” is written separately from the adjective if there is a contrast with the conjunction “a”. For example: not big, but small; not wide, but narrow. It is important to remember that "not" with adjectives, after which there is a union "but", is written together (if the word fits the rule of continuous spelling "not"): a narrow, but deep river; ugly but a kind person; irrational but correct decision.

Secondly, the Russian language is one of the few where there is a double negation. It is with him that “not” is written separately: not interesting to anyone, not at all funny, not secret to anyone.

Thirdly, there are some points of the rules for the continuous and separate spelling of "not", which are universal for all parts of speech. This is the point with “not at all”, “not at all”, “far from”, with which this particle is always written separately. For example: by no means cheerful, not at all spacious, far from happy. Here, by the way, there is also a difficulty: very often they are confused with adverbs of measure and degree, such as “perfectly” and “absolutely”. When they are present, "not" is spelled together, for example: absolutely not cold, completely narrow-minded.

"Not" separately - features

A separate part of the rule is the points on which "not" and adjectives are always written separately. There are no exceptions here, you just need to remember three short statements:

First. Words that are written with a hyphen and "not" are always separated: not in a comradely way, not in Russian.

Second. Relative (denoting a sign that cannot be more or less) and possessive (belonging to someone) adjectives, adjectives taste (sour) and color (pink) are written separately from "not": not winter, not mother's, not bitter, not black. By the way, in order to distinguish the categories of adjectives, one can argue from the contrary: if an adjective does not have degrees of comparison, synonyms and antonyms, it is definitely relative or possessive and fits this rule.

Third. Adjectives in the comparative degree are also written separately: no better, no freer, no more original.

Consolidation with exercises

In a number of cases, in textbooks about writing the particle-prefix "not" with adjectives together and separately, examples are given very poorly. Most often rules are given, but there are few exercises that can really help in remembering these rules. In lessons, teachers usually just give a series of words, and the student must decide whether to become a "not" particle or a prefix for each case. Familiarization with the rule and subsequent practice will definitely help to better consolidate the material covered.

The first exercise is to determine the spelling of "not" with adjectives. Examples: (un)beautiful; absolutely (not) funny; (not) wide, but long; (not) onboard; far (not) cheerful; (not better; (not) blue; (un)friendly; (un)interesting to anyone; (not) deep, but shallow.

This simple task will make it possible to either remember the rules or test your intuition - in any case, such exercises are considered the most common in working out this rule.

The second exercise is the explanation of "not" with adjectives. Examples of "separate": not blue; not prettier; by no means soft; not wooden; not fresh, but stale; not sweet; not Turkish; not at all big; not mother's; not attractive at all. Examples of "smooth": not rough; unlucky; not wide, but deep; sad; careless; uninteresting; ugly; difficult, but fascinating; not angry; not cold.

The answers to the exercises can be found in the rules themselves.

Conclusion

Spelling "not" with adjectives is a very simple topic. However, it is very important to pay due attention to it. The main thing here is to remember the points of the rules responsible for writing "not" with adjectives together and separately. Examples and practice will help you understand better and consolidate the material covered.

Theorem

Proof

Consider triangles ABC and A 1 B 1 C 1, in which AB \u003d A 1 B 1, ∠A \u003d ∠A 1, ∠B \u003d ∠B 1 (Fig. 68). Let us prove that Δ ABC = Δ A 1 B 1 C 1 .

Rice. 68

We impose triangle ABC on triangle A 1 B 1 C 1 so that vertex A is aligned with vertex A 1, side AB is equal to its side AjBj, and vertices C and C 1 are on the same side of the line A 1 B 1.

Since ∠A \u003d ∠A 1 and ∠B \u003d ∠B 1, then the AC side will overlap the beam A 1 C 1, and the BC side will overlap the beam B 1 C 1. Therefore, the vertex C - the common point of the sides AC and BC - will lie both on the ray A 1 C 1 and on the ray B 1 C 1 and, therefore, will coincide with common point of these rays - the top C 1. Hence, the sides AC and A 1 C 1, BC and B 1 C 1 will be combined.

So, the triangles ABC and A 1 B 1 C 1 are completely compatible, therefore they are equal. The theorem has been proven.

The third sign of equality of triangles

Theorem

Proof

Consider triangles ABC and A 1 B 1 C 1, in which AB \u003d A 1 B 1, BC \u003d B 1 C 1, CA \u003d C 1 A 1 (Fig. 69).

Rice. 69

Let us prove that Δ ABC = Δ A 1 B 1 C 1 . We apply triangle ABC to triangle A 1 B 1 C 1 so that vertex A is aligned with vertex A 1, vertex B is aligned with vertex B 1, and vertices C and C 1 are on different sides from the straight line A 1 B 1 (Fig. 70).

Rice. 70

Three cases are possible: the beam C 1 C passes inside the angle A 1 C 1 B 1 (Fig. 70, a); the beam C 1 C coincides with one of the sides of this angle (Fig. 70, b); the beam C 1 C passes outside the angle A 1 C 1 B 1 (Fig. 70, c). Consider the first case (consider the rest of the cases yourself).

Since, according to the condition of the theorem, the sides AC and A 1 C 1, BC and B 1 C 1 are equal, then the triangles A 1 C 1 C and B 1 C 1 C are isosceles (see Fig. 70, a). By the theorem on the property of angles of an isosceles triangle ∠1 = ∠2, ∠3 = ∠4, therefore ∠A 1 CB 1 = ∠A 1 C 1 B 1 . So, AC \u003d A 1 C 1, BC \u003d B 1 C 1, ∠C \u003d ∠C 1.

Therefore, the triangles ABC and A 1 B 1 C 1 are equal according to the first criterion for the equality of triangles. The theorem has been proven.

From the third criterion for the equality of triangles, it follows that triangle - rigid figure. Let's explain what this means.

Imagine two slats, in which two ends are fastened with a nail (Fig. 71, a). This design is not rigid: by shifting or pushing the free ends of the rails, we can change the angle between them. Now let's take another rail and fasten its ends with the free ends of the first two rails (Fig. 71, b).

Rice. 71

The resulting construction - a triangle - will already be rigid. No two sides can be moved or moved apart in it, that is, not a single corner can be changed. Indeed, if this succeeded, then we would get a new triangle, not equal to the original one. But this is impossible, since the new triangle must be equal to the original one according to the third criterion for the equality of triangles.

This property - the rigidity of a triangle - is widely used in practice. So, in order to fix the post in a vertical position, a support is placed on it (Fig. 72, a); the same principle is used when installing the bracket (Fig. 72, b).

Rice. 72

Tasks

121. Segments AB and CD intersect at the middle O of segment AB, ∠OAD = ∠OBC.

a) Prove that ∆CBO = ∆DAO;
b) find BC and CO if CD = 26 cm, AD = 15 cm.

122. In figure 53 (see p. 31) ∠1 = ∠2, ∠3 = ∠4.

a) Prove that Δ ABC = Δ CDA;
b) find AB and BC if AO = 19 cm, CD = 11 cm.

123. Point D is taken on the bisector of angle A, and points B and C are taken on the sides of this angle such that ∠ADB = ∠ADC. Prove that BD = CD.

124. According to Figure 73, prove that OP = OT, ∠P = ∠T.

Rice. 73

125. In figure 74 ∠DAC = ∠DBC, AO = BO. Prove that ∠C = ∠D and AC = BD.

Rice. 74

126. In Figure 74, ∠DAB = ∠CBA, ∠CAB = ∠DBA, AC = 13 cm. Find BD.

127. In triangles ABC and A 1 B 1 C 1 AB \u003d A 1 B 1, BC \u003d B 1 C 1, ∠B - ∠B 1. Points D and D 1 are marked on sides AB and A 1 B 1 so that ∠ACO = ∠A 1 C 1 D 1 . Prove that ∆BCD = ∆B 1 C 1 D 1 .

128. Prove that in equal triangles bisectors drawn to respectively equal parties, are equal.

129. Segments AC and BD intersect in the middle O of segment AC, ∠BCO = ∠DAO. Prove that ∆BOA = ∆DOC.

130. In triangles ABC and A 1 B 1 C 1, segments CO and C 1 O 1 are medians, BC \u003d B 1 C 1, ∠B - ∠B 1 and ∠C \u003d ∠C 1. Prove that:

a) Δ ACO \u003d Δ A 1 C 1 O 1;
b) Δ BCO \u003d Δ B 1 C 1 O.

131. In triangles DEF and MNP, EF - NP, DF = MP and ∠F = ∠P. The bisectors of angles E and D intersect at point O, and the bisectors of angles M and N intersect at point K. Prove that ∠DOE = ∠MKN.

132. A line perpendicular to the bisector of angle A intersects the sides of the angle at points M and N. Prove that triangle AMN is isosceles.

133. Prove that if the bisector of a triangle is its height, then the triangle is isosceles.

134. Prove that isosceles triangles are congruent if the base and adjacent angle of one triangle are respectively equal to the base and adjacent angle of another triangle.

135. Prove that if the side of one equilateral triangle equal to the side of another equilateral triangle, then the triangles are congruent.

136. In figure 52 (see p. 31) AB-AC, BD = DC and ∠BAC = 50°. Find ∠CAD.

137. In figure 53 (see p. 31) BC = AD, AB = CD. Prove that ∠B = ∠D.

138. In figure 75 AB = CD and BD = AC. Prove that: a) ∠CAD = ∠ADB; b) ∠BAC = ∠CDB.

Rice. 75

139. In Figure 76, AB = CD, AD = BC, BE is the bisector of angle ABC, and DF is the bisector of angle ADC. Prove that:

a) ∠ABE = ∠ADF;
b) Δ ABE = Δ CDF.

Rice. 76

140. In triangles ABC and A 1 B 1 C 1, the medians BM and B 1 M 1 are equal, AB \u003d A 1 B 1 AC \u003d A 1 C 1. Prove that Δ ABC = Δ A 1 B 1 C 1 .

141. In triangles ABC and A 1 B 1 C 1, segments AD and A 1 D 1 are bisectors, AB \u003d A 1 B 1, BD \u003d B 1 D 1 and AD \u003d A 1 D 1. Prove that Δ ABC = Δ A 1 B 1 C 1 .

142. Isosceles triangles ADC and BCD have common ground DC. Line AB intersects segment CD at point O. Prove that: a) ∠ADB = ∠ACB; b) DO = OC.

Answers to tasks

121. b) BC = 15 cm, CO = 13 cm.

122. b) AB = 11 cm, BC = 19 cm.

142. Instruction. Consider two cases. Point B lies: a) on the ray AO; b) on the continuation of the ray AO.