<prompt> Pembetatu ABC ina viambata viwili vya pembe overlineAD, overlineBE, na overlineCF. Kwa kuzingatia kwamba angle EDF 90circ, ingiza thamani zote zinazowezekana za angle BAC katika digrii, ikitenganishwa na koma.  <response>: Kama kawaida, acha a BC, b AC, na c AB.asyukubwa wa kitengo sentimita 0.8;jozi A, B, C, D, E, F;A 0,0;B 8,0;C 3dir120;D uganiA, kitovuA,B,C, B, C;E uganiB, kitovuA,B,C, C, A;F kiendeleziC, kitovuA,B,C, A, B;kuchoraA--B--C--mzunguko;kuchoraA--D;kuchoraB--E;kuchoraC--F;kuchoraE--D--F;leboA, A, SW;leboB, B, SE;leboC, C, NW;leboD, D, N;leboE, E, SW;leboF, F, S;asyKwa Nadharia ya Angle Bisector, BD:DC c:b, hivyooverrightarrowD fracbb c overrightarrowB fraccb c overrightarrowC fracb overrightarrowB c overrightarrowCb c.Vile vile,anzaalignoverrightarrowE fraca overrightarrowA c overrightarrowCa c, overrightarrowF fraca overrightarrowA b overrightarrowBa b.endalignTukiruhusu A iwe asili, basi tutapataoverrightarrowE fracc overrightarrowCa c, quad overrightarrowF fracb overrightarrowBa b. Kwa hiyo,anzaalignmshale wa kulia zaidiDE ushale wa kulia zaidiE - ushale uliopitilizaD fracc overrightarrowCa c - fracb overrightarrowB c overrightarrowCb c frac- ba c mshale wa kulia kupita kiasiB cb - a overrightarrowCa cb c,mwishoalignnaanzaalignmshale wa kulia zaidiDF ushale wa kulia zaidiF - ushale uliopitilizaD fracb overrightarrowBa b - fracb overrightarrowB c overrightarrowCb c fracbc - a overrightarrowB - ca b overrightarrowCa bb c.endalignKwa vile A ndio asili, overrightarrowB c, overrightarrowC b, na kwa Sheria ya Cosines,arrow overrightarrowB cdot overrightarrowC overrightarrowB arrowoverrightC cos A bc cdot fracb2 c2 - a22bc fracb2 c2 - a22.Tuna hiyo angle EDF 90circ ikiwa na tu ikiwa overrightarrowDE cdot overrightarrowDF 0, au kwa usawa,anzaalign-ba c mshale wa kulia kupita kiasiB cb - a mshale wa kulia kupita kiasiC cdot bc - a mshale uliopitilizaB - ca b mshale uliopitiliza C -b2 a cc - a arrowoverrightarrowB2 bca ca b overrightarrowB cdot overrightarrowC quad bcb - ac - a mshale wa kulia kupita kiasiB cdot mshale ulio sawaC - c2 b - aa b arrowoverrightarrowC2 -b2 c2 c2 - a2 2bca2 bc cdot fracb2 c2 - a22 - b 2 c2 b2 - a2 a2 bcb2 bc c2 - a2 0,mwishoalignhivyo a2 b2 bc c2. Kisha kwa Sheria ya Cosines,cos A fracb2 c2 - a22bc frac-bc2bc -frac12.Kwa hiyo, A 120circ. 