ࡱ> ac`7 GbjbjUU "7|7|*l   , RIRIRIRII, 2J^4K4K4K:O$* JF Ֆ* 4K:OG H44K :O   6 (H :OJ XRN, &>RIGP 0 v , ,  Punctul A % A Dreapta d sau dreapta AB d A B  Semidreapta OA, notata [OA O A  Segmentul AB, notat [AB] A B   Definitie : Punctul, dreapta si planul sunt multimi de puncte, deci sunt figuri geometrice. - M EMBED Equation.3 N puncte distincte sau diferite - E=F puncte identice sau confundate Definitie : D A B % C a    Axioma dreptei : Definitii :                Lungimea unui segment este numarul care exprima de cte ori o unitate de masura se cuprinde in segmental respective. Distanta dintre doua puncte A si B, notata AB, este lungimea lui [AB].  - Mijlocul unui segment este acel punt al segmentului care-l imparte in doua segmente congruente.  Definitii:        Unghiul cu laturile in prelungire are  EMBED Equation.3 . Unghiul nul are  EMBED Equation.3 . Doua unghiuri cu masuri egale sunt congruente si reciproc, doua unghiuri congruente au masuri egale. Un grad are 60 de minute Un minut are 60 de secunde.     Definitii:        Daca laturile necomune a doua unghiuri adiacente sunt semidrepte opuse, atunci unghiurile sunt suplementare. A B O A   Definitii:        Definitii:  Teoreme:    Definitii:       Demonstratie :  Definitii:   B  A   C O A Demonstratie : Teorema:   EMBED Equation.3   2 1  EMBED Equation.3  3 4 Demonstratie :   Definitii:                         Congruenta triunghiurilor              Orice multime nevida de puncte este o figura geometrica M N % % E F % Mai multe puncte care apartin aceleiasi drepte se numesc puncte coliniare. Prin doua puncte distincte trece o dreapta si numai una Pentru doua puncte A si B, segmental AB este multimea ale carui elemente sunt A,B, impreuna cu toate punctele care sunt intre A si B. Punctele A si B se numesc capetele lui [AB]. A B Fie A si B doua puncte diferite. Semidreapta AB este multimea : [AB] EMBED Equation.3 este intre A si B} C A B M Punctul A se numeste originea lui [AB Daca A este intre B si C, atunci [ AB si [ AC se numesc semidrepte opuse. C A B Orice dreapta d dintr-un plan il imparte in doua semiplane, numite semiplane opuse. % Dreapta d nu este inclusa in nici unul din semiplane. % Daca 2 puncte sunt in acelasi semiplan, atunci seg. care le uneste este in acelasi semiplan, > seg. care le uneste este in acel semiplan si deci nu inters. dreapta d. semiplan A d B M semiplan N Doua drepte care au un singur punct comun se numesc drepte concurente. O a  EMBED Equation.3 ; O este punctual de intersectie b Doua drepte a si b din acelasi plan care nu au nici un punct comun se numesc drepte paralele a Q% b a  EMBED Equation.3  b Doua drepte nesituate in in acelasi plan se numesc drepte necoplanare. a  EMBED Equation.3 =  b Doua figuri geometrice se numesc congruente daca prin suprapunere coincid. %Punctul M este intre A si B daca A, M si B sunt puncte diferite doua cte doua pe aceeasi dreapta si AM+MB=AB. %Doua segmente care au lungimi egale sunt segmente congruente si reciproc, doua segmente congruente au lungimi egale. Daca [AB] este congruent cu [CD] scriem [AB]a"[CD] " Daca cele doua semidrepte care formeaza un unghi sunt semidrepte opuse, atunci unghiul se numeste unghi alungit sau unghi cu laturile in prelungire. B O A  EMBED Equation.3  EMBED Equation.3 este unghi alungit Un unghi format din doua semidrepte identice se numeste unghi nul. O M N  EMBED Equation.3  este unghi nul Un unghi care nu este nici alungit si nici nul se numeste unghi propriu. Interiorul unui unghi propriu AOB este multimea punctelor M din planul unghiului AOB a.i. M si B sunt de aceeasi parte a dreptei OA si M si A sunt de aceeasi parte a dreptei OB. Exteriorul unghiului propriu AOB este multimea punctelor din planul unghiului AOB care nu este nici pe laturi , nici in interiorul sau. exterior B interior O M Exterior exterior A Numarul de grade ale unui unghi se numeste masura sa. Daca  EMBED Equation.3  are n grade, scriem  EMBED Equation.3  Daca M este in interiorul unghiului AOB atunci  EMBED Equation.3  B M A Pentru a aduna masurile a doua unghiuri exprimate in grade, minute si secunde se aduna numerele care reprezinta unitati de acelasi fel (grade, minute, secunde). Daca numarul minutelor sau secundelor obtinute este m. mare de 60 se transforma in unitati mai mari. Exemplu:  EMBED Equation.3  Pentru a scadea masurile a doua unghiuri expr. in grade, minute si secunde se scad numerele care reprezinta unitati de acelasi fel. Daca nr. de min. sau sec. de la descazut este m.mic dect cel de la scazator, se transforma un grad in minute sau un minut in secunde si se adauga la cele existente, apoi se efectueaza scaderea. Exemplu: a)  EMBED Equation.3  b)  EMBED Equation.3   Doua unghiuri proprii care au vrful comun , o latura comuna, iar celelalte doua sunt situate de o parte si de alta a dreptei care contine latura comuna, se numesc unghiuri adiacente.  Se numeste bisectoarea unui unghi propriu semidreapta cu originea in vrful unghiului, situata in interiorul lui, a.i. cele doua unghiuri formate de ea cu laturile unghiului initial sa fie congruente. C A B Doua unghiuri proprii pentru care suma masurilor este  EMBED Equation.3 , se numesc unghiuri suplementare. Fiecare dintre cele doua unghiuri se numeste suplementul celuilalt. C M  EMBED Equation.3  A P Unghiurile ABC si MNP sunt suplementare  EMBED Equation.3 este suplementul  EMBED Equation.3  si invers. B N TRIUNGHIUL Teorema: Daca doua unghiuri sunt congruente, atunci si suplementale lor sunt congruente Ipoteza:  EMBED Equation.3  si  EMBED Equation.3  suplementul  EMBED Equation.3  si  EMBED Equation.3  suplementul  EMBED Equation.3  Concluzie:  EMBED Equation.3  Demonstratie AFIRMATIIEXPLICATII1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  7.  EMBED Equation.3 Dat in ipoteza Unghiurile congruente au masuri egale Definitia unghiurilor suplementare Definitia unghiurilor suplementare Simetria si tranzitivitatea Scaderea egalitatilor 5. si 2. Unghiurile cu masuri egale sunt congruente. Se numeste unghi drept orice unghi care este congruent cu suplementul sau. Daca suma masurilor a doua unghiuri proprii este  EMBED Equation.3  atunci ele se numesc complementare, iar fiecare dintre ele se numeste complement al celuilalt. Un unghi propriu cu masura m.mica dect  EMBED Equation.3  se numeste unghi ascutit Un unghi propriu cu masura m.mare dect  EMBED Equation.3  se numeste unghi obtuz. obtuz ascutit Teorema complementului Daca doua unghiuri sunt congruente, atunci complementele lor sunt congruente. Daca [AB si [AC formeaza un unghi drept, atunci ele se numesc drepte perpendiculare si se noteaza [AB  EMBED Equation.3 [AC. Daca [AB  EMBED Equation.3 [AC., dreptele AB si AC se numesc drepte perpendiculare si se noteaza AB EMBED Equation.3 AC. Daca [AB  EMBED Equation.3 [AC. determina dreptele perpendiculare AB si AC, atunci ele se numesc segmente perpendiculare si se noteaza [AB]  EMBED Equation.3 [AC]. Axioma de adunare a unghiurilor Daca duoa unghiuri sunt complementare, atunci amndoua sunt ascutite Orice doua unghiuri drepte sunt congruente. Daca doua unghiuri sunt congruente si suplementare, ! fiecare dintre ele este drept. Doua unghiuri proprii se numesc opuse la vrf daca laturile lor formeaza doua perechi de semidrepte opuse. Teorema unghiurilor opuse la vrf Unghiurile opuse la vrf sunt congruente C O B Ipoteza :  EMBED Equation.3 si  EMBED Equation.3  sunt opuse la vrf. Concluzie :  EMBED Equation.3  D A Trei sau m.multe unghiuri care au vrful comun, nu au puncte interioare commune si care, impreuna cu interioarele lor, acopera intreg planul, se numesc unghiuri in jurul unui punct. Teorema unghiurilor in jurul unui punct Suma masurilor unghiurilor in jurul unui punct este 360 Daca la intersectia a doua drepte distincte si concurente se formeaza un unghi drept, atunci toate unghiurile care se formeaza sunt unghiuri drepte. < 1 este drept 5.  EMBED Equation.3  <1 si < 3 sunt opuse la vrf 6.  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