解:(1) 设,连结EO ∵ O、M分别是AC、EF的中点, 四边形ACEF为矩形························ 2分 ∴ AM∥EO ∵ EO面BDE,AM面BDE ∴ AM∥面BDE·························· 4分 (2) 由已知有BD⊥面ACEF ∴ BD⊥AM···························· 5分 又,知四边形AOMF为正方形 ∴ FO⊥AM···························· 6分 又 ∴ AM⊥面BDF·························· 8分 (3) 令,作HG⊥DF于G,连结AG,由三垂线定理知AG⊥DF ∴ ∠AGH为所求的二面角的平面角················· 10分 易算得···················· 12分 ∴ ∴ 所求二面角的大小为60··················· 13分 |