梯形辅助线的常见做法和例题

发布网友发布时间：2023-04-26 03:01

共1个回答

热心网友时间：2023-10-17 20:18

规律56.从梯形的一个顶点作一腰的平行线,把梯形分成一个平行四边形和一个三角形.例：已知,如图,等腰梯形ABCD中,AD∥BC,AD = 3,AB = 4,BC = 7求∠B的度数过A作AE∥CD交BC于E,则四边形AECD为平行四边形∴AD = EC, CD = AE∵AB = CD = 4,AD = 3, BC = 7 ∴BE = AE = AB = 4∴△ABE为等边三角形∴∠B = 60o 规律57.从梯形同一底的两端作另一底所在直线的垂线,把梯形转化成一个矩形和两个三角形.例：已知,如图,在梯形ABCD中,AD∥BC,AB = AC,∠BAC = 90o,BD = BC,BD交AC于O求证：CO = CD证明：过A、D分别作AE⊥BC,DF⊥BC,垂足分别为E、F则四边形AEFD为矩形∴AE = DF∵AB = AC,AE⊥BC,∠BAC = 90o,∴AE = BE = CE = BC,∠ACB = 45o ∵BC = BD∴AE = DF =BD又∵DF⊥BC∴∠DBC = 30o∵BD = BC∴∠BDC =∠BCD=(180o－∠DBC)= 75o∵∠DOC =∠DBC＋∠ACB = 30o＋45o = 75o∴∠BDC =∠DOC∴CO = CD规律58.从梯形的一个顶点作一条对角线的平行线,把梯形转化成平行四边形和三角形.例：已知,如图,等腰梯形ABCD中,AD∥BC,AC⊥BD,AD＋BC = 10,DE⊥BC于E求DE的长.过D作DF∥AC,交BC的延长线于F,则四边形ACFD为平行四边形∴AC = DF, AD = CF∵四边形ABCD为等腰梯形∴AC = DB∴BD = FD∵DE⊥BC ∴BE = EF = BF= (BC＋CF) = (BC＋AD)= ×10 = 5∵AC∥DF,BD⊥AC∴BD⊥DF∵BE = FE∴DE = BE = EF =BF = 5答：DE的长为5.规律59.延长梯形两腰使它们交于一点,把梯形转化成三角形.例：已知,如图,在四边形ABCD中,有AB = DC,∠B =∠C,AD＜BC求证：四边形ABCD等腰梯形证明：延长BA、CD,它们交于点E∵∠B =∠C∴EB = EC又∵AB = DC∴AE =DE ∴∠EAD =∠EDA∵∠E＋∠EAD＋∠EDA = 180o∠B＋∠C＋∠E = 180o ∴∠EAD =∠B∴AD∥BC∵AD≠BC,∠B =∠C∴四边形ABCD等腰梯形（此题还可以过一顶点作AB或CD的平行线；也可以过A、D作BC的垂线）规律60.有梯形一腰中点时,常过此中点作另一腰的平行线,把梯形转化成平行四边形.例：已知,如图,梯形ABCD中,AD∥BC,E为CD中点,EF⊥AB于F求证：S梯形ABCD = EF•AB证明：过E作MN∥AB,交AD的延长线于M,交BC于N,则四边形ABNM为平行四边形∵EF⊥AB∴S□ABNM = AB•EF∵AD∥BC∴∠M =∠MNC 又∵DE = CE∠1 =∠2∴△CEN≌△DEM∴S△CEN= S△DEM∴S梯形ABCD = S五边形ABNED＋S△CEN = S五边形ABNED＋S△DEM= S梯形ABCD = EF•AB规律61. 有梯形一腰中点时,也常把一底的端点与中点连结并延长与另一底的延长线相交,把梯形转换成三角形.例：已知,如图,直角梯形ABCD中,AD∥BC,AB⊥AD于A,DE = EC = BC求证：∠AEC = 3∠DAE证明：连结BE并延长交AD的延长线于N∵AD∥BC∴∠3 =∠N又∵∠1 =∠2ED = EC∴△DEN≌△CEB∴BE = ENDN = BC∵AB⊥AD∴AE = EN = BE∴∠N =∠DAE∴∠AEB =∠N＋∠DAE = 2∠DAE∵DE = BCBC = DN∴DE = DN∴∠N =∠1∵∠1 =∠2∠N =∠DAE∴∠2 =∠DAE∴∠AEB＋∠2 = 2∠DAE＋∠DAE即∠AEC = 3∠DAE规律62.梯形有底的中点时,常过中点做两腰的平行线.例：已知,如图,梯形ABCD中,AD∥BC,AD＜BC,E、F分别是AD、BC的中点,且EF⊥BC求证：∠B =∠C证明：过E作EM∥AB, EN∥CD,交BC于M、N,则得□ABME,□NCDE∴AE = BM,AB∥= EM,DE = CN,CD = NE∵AE = DE∴BM = CN又∵BF = CF∴FM = FN又∵EF⊥BC∴EM = EN∴∠1 =∠2∵AB∥EM, CD∥EN∴∠1 =∠B∠2 =∠C∴∠B = ∠C