Question

1．   (本小题满分13分)

如右图所示，已知正方形和矩形所在的平面互相垂直，，AF = 1，M是线段的中点．

(1) 求证：平面；

(2) 求证：平面；

(3) 求二面角的大小．

 

Answer

60

【解析】【解析】
(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