Question

在ΔABC中,AB=AC,∠BAC=90°,∠C=45°,D为AC的中点,AE⊥BD于F交BC于E,AM⊥BC于N交BD

Answer 1

⑴∵∠BAC=90°,∴∠BAF+∠DAF=90°,

∵AE⊥BD,∴∠ABF+∠BAF=90°,

∴∠ABF=∠DAF.

⑵∵AB=AC,∠BAC=90°,∴∠ABC=∠C=45°,

∵AM⊥BC,∴∠BAM=45°=∠C,

∵∠DAF=∠ABF,AB=AC,

∴ΔBNA≌ΔAEC.

⑶∠CDE=∠ADB.

证明：过C作CG⊥AC交AE延长线于G,

∵AB=AC,∠BAD=∠ACG=90°,∠ABF=∠DAF,

∴ΔBAD≌ΔACG,∴AD=CG,∠ADB=∠G,

∵AD=CD,∴CG=CD,

∵∠ECG=90°-∠ACE=45°=∠ACE,CE=CE,

∴ΔCED≌ΔCEG,∴∠CDE=∠G,

∴∠ADB=∠CDE.