天天资源网 / 高中数学 / 教学同步 / 人教A版 / 必修2 / 第二章 点、直线、平面之间的位置关系 / 2.3.1 直线与平面垂直的判定 / 直线_平面垂直的判定及其性质》试题
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第1题.已知直线，和平面，且，，则与的位置关系是 ．答案：或．第2题.已知两个平面垂直，下列命题一个平面内已知直线必垂直于另一个平面内的任意一条直线．一个平面内的已知直线必垂直于另一个平面的无数条直线．一个平面内的任一条直线必垂直于另一个平面．过一个平面内任意一点作交线的垂线，则此垂线必垂直于另一个平面．其中正确的个数是（ ）A．３Ｂ．２Ｃ．１Ｄ．０答案：Ｂ．第3题.已知平面，，且，，求证．答案：证明：设，在平面内作直线．因为，所以．过作一个平面与平面相交于直线，由，得．又，所以．因为，所以．共12页 第12页
第4题.已知平面，，满足，，，求证：．答案：在平面内做两条相交直线分别垂直于平面，与平面的交线，再利用面面垂直的性质定理证直线．第5题.如图，已知平面，，直线满足，，，试判断直线与平面的位置关系．答案：解：在内作垂直于与交线的直线，因为，所以．因为，所以．又因为，所以．即直线与平面平行．第6题.如图所示，为正方形，平面，过且垂直于的平面分别交，，于，，．求证：．共12页 第12页
答案：证明：平面，．又，．，，，，．同理．第7题.已知直线，有以下几个判断：若，则；若，则；若，则；若，则．上述判断中正确的是（ ）Ａ．Ｂ．Ｃ．Ｄ．答案：Ｂ．第8题.是两个不同的平面，是平面及之外的两条不同的直线，给出四个论断：；；；．以其中三个论断作为条件，余下的一个论断作为结论，写出你认为正确的一个命题 ．答案：．第9题.如图所示，四棱锥的底面是正方形，底面，，，．求证：是异面直线与的公垂线．答案：证明：底面，．已知，面．．又，且．是矩形，．又，，平面．又，平面．．共12页 第12页
是异面直线与的公垂线．第10题.设为平行四边形对角线的交点，为平面外一点且有，，则与平面的关系是 ．答案：垂直第11题.如图，直角所在平面外一点，且，点为斜边的中点．（１）求证：平面；（２）若，求证：面．答案：证明：（１），为的中点，．连结．在中，则．，．又，面．（２），为的中点，．又由（１）知面，．于是垂直于平面内的两条相交直线．面．第12题.在三棱锥中，侧面与面垂直，．（１）求证：；（２）设，求与平面所成角的大小．答案：证明：如图（１）所示，取中点，连结，．共12页 第12页
，．又平面平面，面．，．可知为的外接圆直径．．图（１）（２）解：如图（２），作于，连结，．，，．平面．面面，交线为．直线在平面内的射影为直线．为与平面所成的角．在中，，．在中，，．在中，．在中，．．即与平面所成角为．图（２）共12页 第12页
第13题.在正方形中，，分别是及的中点，是的中点，沿，及把，，折起使，，三点重合，重合后的点记作，那么在四面体中必有（ ）Ａ．面Ｂ．面Ｃ．面Ｄ．面答案：Ａ．第14题.直线不垂直于平面，则内与垂直的直线有（ ）Ａ．条Ｂ．条Ｃ．无数条Ｄ．内所有直线答案：Ｃ．第15题.已知三条直线，，，三个平面，，．下面四个命题中，正确的是（ ）Ａ．Ｂ．Ｃ．Ｄ．答案：Ｄ．第16题.在空间四边形中，若，，为对角线的中点，下列判断正确的是（ ）Ａ．平面平面Ｂ．平面平面Ｃ．平面平面Ｄ．平面平面答案：Ｄ．第17题.，，，是四个不同平面，若，，，，则（ ）共12页 第12页
Ａ．且Ｂ．或Ｃ．这四个平面中可能任意两个都不平行Ｄ．这四个平面中至多有一对平面平行答案：Ｂ．第18题.设，是异面直线，下列命题正确的是（ ）Ａ．过不在，上的一点一定可以作一条直线和，都相交Ｂ．过不在，上的一点一定可以作一个平面和，垂直Ｃ．过一定可以作一个平面与垂直Ｄ．过一定可以作一个平面与平行答案：Ｄ．第19题.已知，是异面直线，，，，是，的公垂线，求证：．答案：证明：过作，则．，．又，，设，确定平面，．又，，．同理．．．共12页 第12页
第20题.下面四个命题：①若直线平面，则内任何直线都与平行；②若直线平面，则内任何直线都与垂直；③若平面平面，则内任何直线都与平行；④若平面平面，则内任何直线都与垂直．其中正确的两个命题是（ ）Ａ．①与②Ｂ．②与③Ｃ．③与④Ｄ．②与④答案：Ｂ．第21题.设平面平面，且，直线，直线，且不与垂直，不与垂直，那么与（ ）Ａ．可能垂直，不可能平行Ｂ．可能平行，不可能垂直Ｃ．可能垂直，也可能平行Ｄ．不可能垂直，也不能垂直答案：Ｂ．第22题.已知：如图所示，平面平面，，在上取线段，，分别在平面和平面内，且，，，，求长．共12页 第12页
答案：解：连结．，，．，，．是直角三角形．在中，，在中，．长为．第23题.在正三棱柱中，若．求证：．答案：证明：取中点，中点，连结，，，，由正三棱柱性质知，，．又正三棱柱侧面与底面垂直，面，面，，分别为与在面上的射影．，．又，．．．共12页 第12页
第24题.设三棱锥的顶点在底面内射影（在内部，即过作底面，交于），且到三个侧面的距离相等，则是的（ ）Ａ．外心Ｂ．垂心Ｃ．内心Ｄ．重心答案：Ｃ．第25题.如图所示，是圆的直径，是异于，两点的圆周上的任意一点，垂直于圆所在的平面，则，，，中，直角三角形的个数是（ ）Ａ．Ｂ．Ｃ．Ｄ．答案：Ｄ．第26题.已知直线，和平面，有以下四个命题：共12页 第12页
①若，，则；②若，，则与异面；③若，，则；④若，，则．其中真命题的个数是（ 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