如图 解题?
2020-05-30
解:如图 1,以 BC 为一边构造等边 △BCF,连接DF、AF.
∠BAD = ∠BDA = 70° ⇒ ∠ABD = 40°,∠BAC = ∠ABC = 70° ⇒ ∠ACB = 40°.
∴ ∠DBC = ∠ABC - ∠ABD = 70° - 40° = 30°,
∴ ∠DBF = ∠CBF - ∠DBC = 60° - 30° = 30° = ∠DBC,
又 ∵ BD = BD,BF = BC,
∴ △FDB ≌ △CDB (SAS).
∴ ∠DFB = ∠DCB = 40°,DF = DC .
DF = DC ⇒ ∠DFC = ∠DCF = ∠BCF - ∠BCA = 60° - 40° = 20°.
∴∠FDA =∠DFC + ∠DCF = 20° + 20° = 40°.
∠BAC = ∠ABC = 70°⇒AC = BC,
又∵ FC = BC,
∴ FC = BC = AC.
∴ ∠CFA =∠CAF = (180°-∠ACF/2) = (180°-20°/2) = 80°.
AC = BC ,CD = BE. ⇒ CE = DA,
又 ∵∠DCE = 40° = ∠FDA,CD = DF .
∴ △CDE ≌ △DFA(SAS).
∴ ∠CDE = ∠DFA = ∠CFA - ∠CFD = 80° - 20° = 60°.
∴ ∠BDE = 180° - ∠CDE - ∠ADB = 180° - 60° - 70° = 50°.