题干

如图,在四棱锥P﹣ABCD中,PA⊥底面ABCD,AB⊥AD,AC⊥CD,∠ABC=60°,PA=AB=BC,E是PC的中点.

(1)求PB和平面PAD所成的角的大小;

(2)证明AE⊥平面PCD.

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答案(点此获取答案解析)

解:(1)在四棱锥P﹣ABCD中,

因PA⊥底面ABCD,AB⊂平面ABCD,

故PA⊥AB.

又AB⊥AD,PA∩AD=A,

从而AB⊥平面PAD,

故PB在平面PAD内的射影为PA,从而∠APB为PB和平面PAD所成的角.

在Rt△PAB中,AB=PA,故∠APB=45°.

所以PB和平面PAD所成的角的大小为45°.

(2)证明:在四棱锥P﹣ABCD中,

因为PA⊥底面ABCD,CD⊂平

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