Umdibaniso engile kanxantathu. Le theorem phezu sum of engile kanxantathu

Nxantathu yi buyimilo kokuba amacala ezintathu (engile ezintathu). Amaninzi, inxalenye luchazwa ngegama oonobumba abancinane engqinelanayo oonobumba abakhulu, nto leyo emele eziphezulu esahlukileyo. Kweli nqaku sijonga ezi ntlobo iimilo zejiyometri, theorem, nto leyo ichaza into elingana udibaniso engile kanxantathu.

Types angles ezinkulu

Ezi ndidi zilandelayo yepholigoni kunye eziphezulu ezintathu:

• Okubuhlungu-engile egqithe, apho zonke engile zitsolile;
• engunxantathu ukuba engile ekunene, icala iyilwe nguye, wabhekisela imilenze, kwaye icala zilahlwa malunga akwiengile elungileyo ibizwa hypotenuse;
• obtuse xa elinye engela obtuse ;
• isosceles, ogama macala emibini iyalingana, yaye ibizwa ngokuba osecaleni, kwaye eyesithathu - unxantathu kwisiseko;
• alinganayo ukuba amacala ezintathu ngokulinganayo.

izakhiwo

Nika iimpawu ezisisiseko uphawu uhlobo ngalunye kanxantathu:

• malunga icala kakhulu-engile lisoloko lingaphezulu, and vice versa;
• are angles lingana malunga alinganayo ngobukhulu iqela, and vice versa;
• kuyo nayiphi na unxantathu uye engile ezimbini oyingozi;
• angle engaphandle mkhulu kunabo bonke engile lwangaphakathi noko ongagudlanga;
• udibaniso naziphi engile ezimbini usoloko izidanga ngaphantsi kwe 180;
• angle Ingaphandle lilingana sum of ezinye ezimbombeni zaso ezimbini, ezo nto mezhuyut naye.

Le theorem phezu sum of engile kanxantathu

Le theorem ithi xa edityanisiwe zonke iimbombo imilo zejiyometri, nto leyo ifumaneka kwi-moya Euclidean, ngoko mali yabo iya kuba degrees 180. Masizame ukungqina le theorem.

Makhe siye sibe unxantathu kubomi eziphezulu KMN. Ngaphesheya encotsheni M uya kubamba onxusene ngqo kumgca KN (nokuba lo mgca kuthiwa Euclid). Kufuneka kuqatshelwe indawo A ukuze iingongoma K no A zicwangciswe evela kumacala ahlukeneyo umgca MN. Sifumana engela efanayo AMS kunye MUF, leyo, efana embindini, ulale crosswise ukwenza ezidibanayo MN ngokusebenzisana ngqo CN kunye MA, leyo ngaxeshanye. Kulo kulandela ukuba udibaniso engile lo nxantathu, ingabekwa eziphezulu ze M N lilingana ubungakanani koMmandla weNqila yeKapa ekujikeni. Zonke angles amathathu luqulathe lodibaniso elilingana isixa engile ze KMA kunye MCS. Ekubeni ze data engile zangaphakathi isalamane imigca enamehlo parallel CL kunye CM MA kwi ezidibanayo, sum yabo degrees 180. Oku kubonisa theorem.

ngenxa

Kubo ngaphezu theorem ngasentla kuthetha le ecacileyo ilandelayo: yonke unxantathu uye engile ezimbini bukhali. Ukungqina oku, makhe sicinge ukuba eli nani zejometri kuye omnye kuphela engile oyingozi. Unako kwakhona zicinge ukuba akukho nanye ezimbombeni azikho elibukhali. Kule meko kufuneka iiengile ubuncinane ezimbini, ubukhulu leyo ilingana no okanye inkulu kuna degrees 90. Kodwa ke inani engile mkhulu 180 degrees. Kodwa ayikwazi kuba oku, njengoko nenze engile theorem udibaniso unxantathu ilingana 180 ° - hayi ngaphezulu, kungekho ngaphantsi. Yiloo nto kwafuneka kwaba oko.

Property amagumbi ngaphandle

Yintoni na udibaniso engile kanxantathu, leyo yangaphandle? Impendulo yalo mbuzo inokufunyanwa ngokusebenzisa enye ngeendlela ezimbini. Eyokuqala kukuba kufuneka ukuba ufumane inani engile, ezo ezithathwe omnye enekona ngalinye, oko kukuthi, i-engile ezintathu. Eyesibini ibonisa ukuba kufuneka ukuba ufumane inani engile ezintandathu eziphezulu. Ukujongana ekuqaleni lwalo lokuqala. Ngoko ke, lo nxantathu iqulethe iikona outer ezintandathu - phezulu nganye ezimbini. Isibini ngalinye iiengile ngokulinganayo phakathi kwabo, ekubeni nkqo:

∟1 = ∟4, ∟2 = ∟5, ∟3 = ∟6.

Ukongeza, kwakusazeka ukuba kwikona engaphandle kanxantathu lilingana sum of Kumbindi ezimbini, izinto ezo mezhuyutsya kunye naye. ke,

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

Kulo kubonakala ukuba inani engile umphandle, leyo zithathwa ngabanye kufuphi enekona nganye iya kulingana:

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

Ngenxa yokuba umdibaniso engile ilingana degrees 180, oko Kungaphikiswa ukuba ∟A + ∟V ∟S = + 180 °. Oku kuthetha ukuba ∟1 + ∟2 + ∟3 = 2 x 180 ° = 360 °. Xa kusetyenziswa ukhetho lwesibili, udibaniso engile zithandathu ziya kuba olunge ngakumbi kabini. Okt udibaniso engile kanxantathu ngaphandle iya kuba:

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

unxantathu ilungelo

Yintoni elingana udibaniso engile kanxantathu ilungelo, nguye siqithi? Impendulo ke, kwakhona, ukusuka theorem, ocacisa ukuba engile kanxantathu zidibanisa degrees 180. A isandi ngoluvo yethu (ipropati) ngolu hlobo lulandelayo: xa unxantathu ilungelo iiengile abukhali zidibanisa degrees 90. Sibonisa bayaphika yayo. Makubekho unxantathu wanikwa KMN, leyo ∟N = 90 °. Kuyimfuneko ukungqina ukuba ∟K ∟M = + 90 °.

Ngoko ke, ngokutsho theorem phezu sum of the engile ∟K + ∟M ∟N + = 180 °. Kule meko kuthiwa ∟N = ° 90. Kubonakala ∟K ∟M + + 90 ° = 180 °. Oko kukuthi ∟K ∟M + = 180 ° - 90 ° = 90 °. Yiloo nto esimelwe ubungqina.

Ukongeza kule propati zingentla unxantathu ekunene, ungadibanisa ezi:

• angles, leyo ukuyiqhayisela nokuyixoka imilenze zitsolile;
• i hypotenuse kwe elingunxantathu ngaphezu imilenze;
• udibaniso imilenze ngaphezu hypotenuse;
• Umlenze unxantathu, edulusele malunga ekujikeni degrees 30, nesiqingatha hypotenuse, oko kukuthi ilingana isiqingatha sayo.

Njengoko enye impahla shape zejometri zingahlukaniswa theorem kaPythagoras. Yena uthi ukuba unxantathu-engile degrees 90 (uxande), udibaniso lwezikweri imilenze ilingana isikweri hypotenuse.

Umdibaniso engile ze unxantathu isosceles

Ngaphambilana siye sathi ukuba unxantathu isosceles yi buyimilo kunye eziphezulu ezintathu, equlethe macala amabini alinganayo. Le propati owaziwayo zejometri: i engile kwi kwisiseko layo alinganayo. Masibe oku.

Thatha unxantathu KMN, nto leyo isosceles, SC - noseko lwalo. Nathi kufuneka ukubonisa ukuba ∟K = ∟N. Ngoko ke, makhe kucinga ukuba MA - KMN ke bisector kanxantathu yethu. ICA unxantathu uphawu lokuqala lokulingana unxantathu mna. Oko kukuthi, ngenxa hypothesis enikwe ukuba CM = NM, MA na icala eqhelekileyo, ∟1 = ∟2, ngenxa yokuba MA - le bisector. Ukusebenzisa yokulingana oonxantathu amabini, umntu athi ∟K = ∟N. Ngenxa yoko, le theorem lawo.

Kodwa nomdla na, yintoni na udibaniso engile kanxantathu (isosceles). Ngenxa malunga oku ayinayo iimpawu zayo, siya luzakuqala ukusuka theorem ekuxoxwe ngazo ngaphambili. Oko kukuthi, sinokuthi ∟K + ∟M ∟N + = 180 °, okanye 2 x ∟K ∟M + = 180 ° (njengoko ∟K = ∟N). Oku akuyi kuba ipropati, njengoko kwaba theorem phezu udibaniso engile kanxantathu ngaphambili.

Ngaphandle kokuba iimpawu kuqwalaselwa saso ezimbombeni zaso kanxantathu, kukho kwakhona kukho iingxelo ezibalulekileyo ezifana:

• e an ukuphakama triangle alinganayo, leyo ithotywe kwisiseko, kuba ngaxeshanye lo bisector udibaniso le angle leyo phakathi amacala ngokulinganayo esqwini yolingano kwesiseko salo;
• udibaniso (bisector, ephakame), leyo babambelela emacaleni umzobo zejometri, bayalingana.

triangle alinganayo

Kwakhona ngokuba ilungelo, nguye lo nxantathu, leyo esilingana onke amaqela. Kwaye ngoko ke alinganayo engile. Ngamnye kubo degrees 60. Masibe kule propati.

Makhe sicinge ukuba sibe unxantathu KMN. Siyazi ukuba KM = HM = KH. Oku kuthetha ukuba, ngokutsho yipropathi engile kubekwe isiseko unxantathu alinganayo ∟K = ∟M = ∟N. Ekubeni, ngokutsho isixa engile ye theorem triangle ∟K + ∟M ∟N + = 180 °, ngoko x 3 = 180 ° ∟K okanye ∟K = 60 °, ∟M = 60 °, ∟N = 60 °. Ngoko ke, lo ngoluvo lawo. Njengoko kubonakala ubungqina ngentla esekelwe theorem ngasentla, umdibaniso engile ze unxantathu alinganayo, njengoko udibaniso engile nawuphi na kanxantathu na izinyuko 180. Kwakhona ebonakalalisa le theorem ayikho imfuneko.

Kusekho ezinye iimpawu uphawu unxantathu alinganayo:

• median bisector ukuphakama ngokomzekeliso zejometri ziyafana, kunye nobude yabo zibalwe (a x √3): 2;
• ukuba eli polygon circumscribing kwisangqa, ngoko ke radius iya kulingana (a x √3): 3;
• ukuba ubhalwe isangqa triangle alinganayo, radius luza kuba (a x √3): 6;
• indawo ye mzobo zejometri ibalwa wokubala: (A2 x √3): 4.

unxantathu Obtuse

Xa inkcazelo, unxantathu obtuse-engile egqithe, omnye iikona zayo phakathi degrees 90 ukuya 180. Kodwa ngenxa yokuba ezinye engile ezimbini shape zejometri ezibukhali, kunokwenziwa isigqibo ukuba akukho ngaphezulu degrees 90. Ngoko ke, lo udibaniso engile ye theorem unxathathu isebenza ibale inani engile e unxantathu obtuse. Ngoko ke, singakwazi ngokukhuselekileyo sithi, ezisekelwe phezu theorem ngasentla ukuba udibaniso engile obtuse kanxantathu na izinyuko 180. Kwakhona, le theorem akuyomfuneko ukuba ukuphinda-proof.