One possible versionThen I wondered whether the man would still be at the same spot. I walked out the doorand back to the street where I had met him. There the man was, at the same spot, knees curledup against his chest, with the same cardboard sign shading his head. I walked down the street,and before I could think twice, I stood before him. He cautiously glanced in my direction.Ismiled and held up the lunch bag. "I brought you some lunch, "I saidI handed him the bag, and he looked closely at it. " Thank you for this. I was praying thismorning, asking for someone,s heart to open. We talked for a little while. As he shared hisstory, there was an, open, honest tone in his voice. There was something about giving thatmade me feel like I was talking with a brother. "I hope you have a nice day, "I told him<"Thank you! It's already gotten a lot better, " he said. As I walked by, his thank fulness hadbecome mind. My day, too, had already gotten a lot better. Anything helps, indeed

19.解析:(1)如图,取AC的中点E,连接A1C1与B1D1交于点O1,连接ECL, AO1则M是CE的中点又N是CC1的中点,则MN∥CE,(2分)因为AE=C1O1,AE∥CO1,所以AEC1O1是平行四边形,所以AO1∥C1E,(4分)所以MN∥AO(5分)又MN平面AB1D1,AOC平面AB1D1,所以MN∥平面AB1D1(6分)(2)分别以A1D1,A1B1,A1A所在的直线为x,y,z轴建立如图所示的空间直角坐标系不妨设AA1=A1B1=A1D1=4,则A(0,0,4),D1(4,0,0),M(3,3,4),N(4,4,2)所以MN=(1,1,-2),AD1=(4,0,-4),AM=(3,3,0)D(8分)设平面AD1M的一个法向量为n=(x,y,x),n·AD1=(x,y,4,0,-4)=0,4x-4z=0得由3x+3y=0n. AM(x, y,2)令x=1,得平面AD1M的一个法向量为n=(1,-1,1).…(10分)设直线MN与平面AD1M所成角为0所以sn=1os(m,M)|=|m:MInl/MN√3×√6即直线MN与平面AD1M所成角的正弦值为3(12分)

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