Isamba engele calantsatfu. Theorem ku isamba engeli unxantathu

Unxantathu kuyinto ipholigoni kokuba izinhlangothi ezintathu (engeli ezintathu). Ngokuvamile, ingxenye okhonjiswe izinhlamvu ezincane elihambisana bofeleba, amelela vertices okuphambene. Kulesi sihloko thina sibheke lezi zinhlobo ngamajamo weJiyomethri, theorem, okuyinto ichaza ukuthi ilingana isamba engeli calantsatfu.

Izinhlobo engeli ngobukhulu

Lezi zinhlobo ezilandelayo ipholigoni nge vertices ezintathu:

• acute-angled, lapho bonke engele ibukhali
• unxande kokuba omunye engela kwesokudla, ohlangothini obumba ke, okukhulunywe imilenze ohlangothini ukuthi ithambekele okuphambene kuya engela kwesokudla ubizwa ngokuthi hypotenuse;
• obtuse lapho omunye engela obtuse ;
• isosceles, ogama izinhlangothi ababili bayalingana, futhi babizwa lateral, kanti eyesithathu - calantsatfu base;
• equilateral kokuba izinhlangothi ezintathu alinganayo.

izakhiwo

Ukwabela sezindawo eziyisisekelo nobuntu uhlobo unxantathu ngamunye:

• okuphambene ohlangothini esikhulu kunazo engela njalo mkhulu, futhi okuphambene nalokho;
• kukhona engeli alinganayo malungana alinganayo ngobukhulu party, futhi okuphambene nalokho;
• kunoma iyiphi unxantathu has engeli amabili acute;
• engela yangaphandle mkhulu kakhulu kunanoma yimaphi engela yangaphakathi nalokho nendawo;
• isamba noma iyiphi engeli amabili njalo degrees ngaphansi kuka-180;
• engela Ingaphandle lilingana isamba nezinye namakhona amabili, ezingalotshwanga mezhuyut naye.

Theorem ku isamba engeli unxantathu

Theorem uthi uma tente emagumbini onke ukuma weJiyomethri, esemgwaqweni indiza Euclidean ke sum yabo iyoba 180 degrees. Ake sizame ukufakazela lokhu ifayela echaza ifomu.

Ake esinalo unxantathu ngokungenasizathu nge vertices KMN. Yonkana phezulu M izobamba kufana ngokuphawulekayo nalokho okwenzeka ngqo umugqa KN (ngisho lo mugqa libizwa Euclid). Kufanele kuqashelwe iphuzu A ukuze amaphuzu K no ahlelwe kusukela izinhlangothi ezahlukene emgqeni MN. Sithola engela efanayo AMS futhi MUF, okuyinto, efana wezangaphakathi, amanga crosswise ukwakha intersecting MN ngokuhlanganyela CN eqondile MA, okuyizinto ngokulinganisa. Kule kusobala ukuthi isamba engeli unxantathu, sise vertices of M N ilingana ubukhulu CMA engela. Zonke engeli ezintathu aqukethe isamba ilingana nenani lezinombolo of engele of KMA futhi i-MCS. Njengoba idatha engele zangaphakathi isihlobo emaceleni imigqa parallel CL kanye CM MA at intersecting, sum yabo 180 degrees. Lokhu kufakazela theorem.

yi

Okungenhla theorem ngenhla lisikisela eliphezulu ezilandelayo: zonke unxantathu has engeli amabili oyingozi. Ukuze afakazele lokhu, ake sithi lesi sibalo Jomethri has engela oyingozi eyodwa kuphela. Ungase futhi uphethe ngokuthi ayikho into esiyibonisa ngezikhathi akuzona abukhali. Kulokhu-ke kumelwe abe engeli okungenani ezimbili, ubukhulu okuyinto ilingana noma enkulu kuno-90 degrees. Kodwa-ke isamba engeli mkhulu kunezinhliziyo 180 degrees. Kodwa lokhu ngeke kube, njengoba ngokuvumelana engeli theorem isamba unxantathu ilingana 180 ° - kusaba khona, akukho kancane. Yilokho kwadingeka ukuba luhlolwe.

Impahla emakhoneni ngaphandle

Kuyini isamba engeli unxantathu, okuyizinto zangaphandle? Impendulo yalo mbuzo ingatholakala ngokusebenzisa enye yezindlela ezimbili. Esokuqala siwukuthi udinga ukuthola isamba engeli, okuyinto zithathwa esisodwa vertex ngamunye, okungukuthi, ama-engeli ezintathu. Eyesibili isikisela ukuthi udinga ukuthola isamba engeli eziyisithupha ngesikhathi vertices. Ukubhekana ekuqaleni samuntu kuqala. Ngakho, unxantathu iqukethe emakhoneni eziyisithupha yangaphandle - phezulu ngamunye amabili. pair ngamunye has engeli elilinganayo phakathi ngokwabo, ngoba mpo:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Ngaphezu kwalokho, kuyaziwa ukuthi ekhoneni yangaphandle unxantathu lilingana isamba ezimbili ingaphakathi, engewona nokho mezhuyutsya naye. Ngakho-ke,

∟1 = ∟A + ∟S, ∟2 = ∟A + ∟V, ∟3 = ∟V + ∟S.

Kule kubonakala sengathi isamba engeli Ingaphandle, okuyinto zithathwa ngamunye ngamunye eduze vertex ngamunye kuyoba ulingana:

∟1 + ∟2 + ∟3 = ∟A + ∟S ∟A ∟V + + + ∟V ∟S = 2 x (∟A + ∟V ∟S +).

Sikhishwe yokuthi isamba engeli kulingana 180 degrees, kungashiwo wagomela ukuthi ∟A + ∟V ∟S = + 180 °. Lokhu kusho ukuthi ∟1 + ∟2 + ∟3 = 2 x 180 ° = 360 °. Uma ukhetho lwesibili isetshenziswa, isamba engeli eziyisithupha kuyoba esihambelana enkulu kabili. Okungukuthi isamba engeli unxantathu ngaphandle kuyoba:

∟1 + ∟2 + ∟3 + ∟4 + ∟5 + ∟6 = 2 x (∟1 + ∟2 + ∟2) = 720 °.

unxantathu wesokudla

Kuyini ilingana nenani lezinombolo engele unxantathu kwesokudla, iyisiqhingi? Impendulo, futhi, kusukela Theorem, othi engeli unxantathu tente 180 degrees. A umsindo ukugomela yethu (impahla) kanje: in a unxantathu wesokudla engeli abukhali tente 90 degrees. Sihlola ukunemba yayo. Makube khona unxantathu unikezwa KMN, okuyinto ∟N = 90 °. Kuyinto luyadingeka ukuze kufakazelwe ukuthi ∟K ∟M = + 90 °.

Ngakho, ngokuvumelana theorem ku isamba engeli ∟K + ∟M ∟N + = 180 °. Kulesi simo kuthiwa ∟N = 90 °. It kuvela ∟K ∟M + 90 ° = 180 °. Lokho ∟K ∟M + = 180 ° - 90 ° = 90 °. Yilokho nathi kufanele ukufakazela.

Ngaphezu sezindawo ngenhla unxantathu kwesokudla, ungakwazi ukwengeza lezi:

• engeli, okuyinto amanga ngokumelene imilenze ibukhali
• the hypotenuse we esingunxantathu mkhulu kakhulu kunanoma yimaphi of the imilenze;
• isamba imilenze aminingi ukudlula hypotenuse;
• umlenze unxantathu elisemandleni obuhlukile engela degrees 30, isigamu se-hypotenuse, ukuthi ilingana yingxenye yokuphakama kwalo.

Njengoba enye impahla ukuma weJiyomethri ungabahlukanisa theorem kaPythagoras. Yena uthi e calantsatfu engela 90 degrees (unxande), isamba sezikwele imilenze lilingana esigcawini hypotenuse.

Isamba engeli ye-unxantathu isosceles

Ngasekuqaleni sithi unxantathu isosceles kuyinto ipholigoni nge vertices ezintathu, equkethe izinhlangothi ezimbili zilingana. Le mpahla is sibalo Jomethri eyaziwa: engele phansi layo alinganayo. Masibe lokhu.

Thatha unxantathu KMN, okuyinto isosceles, SC - isisekelo sayo. Siyadingeka ukufakazela ukuthi ∟K = ∟N. Ngakho, ake sithi MA - KMN iyona bisector ka unxantathu yethu. ICA calantsatfu kuqubuka ukulingana calantsatfu MNA. Okungukuthi, ngo umbono kulokho CM = NM, MA is a side ezivamile, ∟1 = ∟2, ngoba MA - lokhu bisector. Ukusebenzisa yokulingana onxantathu ezimbili, eyodwa ingaphikisana ngokuthi ∟K = ∟N. Ngakho, theorem kubonakaliswa.

Kodwa sinesithakazelo kwesikwenzayo, yini isamba engeli unxantathu (isosceles). Ngoba kulesi sici ayinayo izici zayo, sizoqala kusukela theorem okuxoxwe ngaphambili. Okungukuthi, singasho ukuthi ∟K + ∟M ∟N + = 180 °, noma 2 x ∟K ∟M + = 180 ° (njengoba ∟K = ∟N). Lokhu ngeke azibonakalise impahla, njengoba theorem ku isamba engeli unxantathu kwafakazelani ngaphambili.

Ngaphandle izakhiwo kubhekwe we kanxantathu, kukhona izinkulumo ezibalulekile ezifana:

• e i equilateral unxantathu ukuphakama, eyayihunyushwe sehliselwe base, kuyinto kanyekanye le bisector imidiyeni engela esiphakathi amacala alingane futhi eksisi wesimethri kwesisekelo salo;
• lesemkhatsini (bisector, engamamitha) eqhutshwa ezinhlangothini sibalo Jomethri, bayalingana.

unxantathu equilateral

Ibizwa nangokuthi ilungelo, kuyinto unxantathu, okuyizinto ngokulinganayo kubo bonke abathintekayo. Futhi-ke futhi alinganayo nama-engeli. Ngamunye wabo degrees 60. Masibe lokhu impahla.

Ake ucabange ukuthi sinawo unxantathu KMN. Siyazi ukuthi KM = HM = KH. Lokhu kusho ukuthi, ngokusho impahla engele sise ayehlala kuwo lapho ekuleso unxantathu equilateral ∟K = ∟M = ∟N. Njengoba, ngokuvumelana isamba engeli we theorem unxantathu ∟K + ∟M ∟N + = 180 ° ke x 3 = 180 ° ∟K noma ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Ngakho, egomela kubonakaliswa. Njengoba ungase ubone ubufakazi obungenhla esekelwe theorem ngenhla, isamba engeli ye-unxantathu equilateral, njengoba isamba engeli yanoma yisiphi esinye unxantathu kuyinto 180 degrees. Nalapha efakazela lokhu theorem akudingekile.

Kusekhona ezinye izakhiwo nobuntu unxantathu equilateral:

• lesemkhatsini bisector ubude sibalo Jomethri ezifanayo, futhi ubude bawo esibaliwe (a x √3): 2;
• uma lokhu ipholigoni onciphisa umbuthano ke engaba kuyoba ulingana (a x √3): 3;
• uma alotshiwe umbuthano unxantathu equilateral, engaba yayo kungaba (a x √3): 6;
• indawo sibalo Jomethri ibalwa ifomula: (A2 x √3): 4.

unxantathu Obtuse

Ngu definition, unxantathu obtuse-angled, omunye emakhoneni salo phakathi 90 degrees kuya ku-180. Kodwa uma ucabangela iqiniso lokuthi abanye ababili engeli ijamo weJiyomethri abukhali, kungashiwo baphetha ngokuthi akudingeki idlule 90 degrees. Ngakho-ke, isamba engeli we theorem unxantathu usebenza ekubaleni isamba engeli e unxantathu obtuse. Ngakho, singasho ngokuphepha, esekelwe theorem ngenhla ukuthi isamba engeli obtuse unxantathu kuyinto 180 degrees. Nalapha futhi, lesi theorem ayidingi kabusha ubufakazi.