三角形内角和180°证明方法
1. 如图,证明/ B+Z C+Z BAC=180 证明:过A点作DE// BC
??? DE// BC
???Z B=Z DAB Z C=Z EAC
(两直线平行,内错角相等)
??? D,A,E三点共线
?Z DAE=180
vZ DAE Z DAB Z BAC+Z CAE
?Z DAB Z BAC+Z CAE=180
?Z B+Z C+Z BAC=180
2. 如图,证明:Z B+Z A+Z ACB=180
证明:过C点作CD// AB,延长BC交CD于 C
v CD// AB
?Z A=Z ACD(两直线平行,内错角相等)
Z B=Z DCE(两直线平行,同位角相等)
v B,C,E三点共线
?Z BCE=180
vZ BCE Z ACB Z ACD Z DCE
?Z ACB Z ACD Z DCE=180
?Z A+Z B+Z ACB=180
3. 如图,证明:Z C+Z BAC Z B=180°
证明:过A点作AD// BC
v AD// BC
?Z C=Z ADC(两直线平行,内错角相等)Z DAC Z B=180°(两直线平行,同旁内角互补)vZ DAC Z DAC Z CAB
?Z DAC Z CAB Z B=180°
vZ C=Z ADC
?Z C+Z CAB Z B=180°
4. 如图,证明:Z BAC Z C+Z B=180°
证明:过A点作DE// BC,延长AC BC交DE于A点v DE// BC
?Z C=Z FDA Z B=Z GAE
(两直线平行,同位角相等)
v D,A,E三点共线
?Z DAE=180
vZ DAE Z DFA Z FAG Z GAE
?Z DFA+Z FAG Z GAE=180
v?Z GAE Z BAC(对顶角相等)E
E
???/ BAC V C+Z B=180(
5. 如图,证明:Z A+Z C+Z B=180°证明:作直线DE// AC, FE// AB 交BC于E
???DE// AC
???Z AFE+Z DEF=180 (两直线平行,同旁内角互补)
Z C=Z DEB(两直线平行,同位角相等)
?FE// AB
?Z AFE+Z A=180°(两直线平行,同旁内角互补)
Z B=Z FEC(两直线平行,同位角相等)
?Z A=Z DEF
?B,C,E三点共线
?Z BCE=180
?Z BCE Z DEB Z DEF Z FEC
?Z DEB Z DEF Z FEC =180°
?Z A+Z C+Z B=180°
6. 如图,证明:Z A+Z B+Z C=180 证明:作DE// AC, FG// AB MN/ BC,都交于点O
?DE// AC
?Z AFO Z FOD=180 (两直线平行,同旁内角互补)
?FG// AB
?Z AFO Z A=180°
(两直线平行,同旁内角互补)
?Z A=Z FOD
?MN/ BC
?Z C=Z FNO(两直线平行,同位角相等)
?DE// AC
?Z FNO Z DO(两直线平行,同位角相等)
?Z C=Z DOM
?MN/ BC
?Z B=Z DM(两直线平行,同位角相等)
?FG// AB
?Z DMO Z FON(两直线平行,同位角相等)
?Z B=Z FNO
?M,O,N三点共线
?Z MON=180
?Z MON Z DOM Z DOF Z FON
?Z DOF Z DOM Z FON=180
?Z A+Z B+Z C=180
7. 如图,证明:Z BAC Z CBA Z ACB=180
证明:作DE// AC, FG// AB MN/ BC,都交于点O
延长AC交FG于点K,延长AB到点L,延长BC交FG于点P
??? MN// BC
???/ ABC=/ AHN / ACB M ANM
(两直线平行,同位角相等)
??? AB // FG
???/ AHN M FON / BAC M AKO
(两直线平行,同位角相等)
:丄 ABC2 FON
??? DE// AC
???/ ANM M DOM
(两直线平行,同位角相等)
/ OKA M DOF
(两直线平行,内错角相等)
???/ ACB M DOM
??? FG// AB
/ BAC M OKA(两直线平行,同位角相等)/ BAC M DOF
M,O,N三点共线
/ MON=18°
/ MON M DOM M DOF M FON
/ DOM M DOF M FON=180
/ BAC M CBA M ACB=180