Question

如图,菱形ABCD中,〔1〕若∠ABC=60°,点E,F分别在BC,CD上,且∠AEF=60°

Answer 1

连结AC,交EF于G

在△AEG和△FCG中

∠AEF = ∠ACF = 60°

∠AGE = ∠FGC

∴△AEG∽△FGC（紫色）

∴EG：CG = AG：FG

∴EG：AG = CG：FG

∴△EGC∽△AGF（绿色）

∴∠AFG = ∠ACB

∵∠ACB = 60°

∴∠AFG = 60°

∴△AEF是等边三角形

2）

∵菱形ABCD,

∴AB = AD,∠B = ∠D

∵AB = AE,△AEF是正三角形

∴AB = AE,AD = AF

∴△ABE≌△ADF

设∠B = x = ∠D

∵AB = AE,AD = AF

∴∠AEB = ∠AFD = x

∴∠BAE = ∠DAF = 180°-2x

∵菱形ABCD

∴∠B = ∠BAD = 180°

∴ x + 2（180°-2x）+60° = 180°

∴x = 80°

∴∠BAD = 100°