题目
题型:不详难度:来源:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045951-47775.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045951-96647.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045951-86392.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045951-65763.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-64422.png)
(1)求直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-14496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-47640.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-44578.png)
答案
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-64223.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-38111.png)
解析
试题分析:(1)解法一:建立坐标系
平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-36234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-27595.png)
因为
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-75056.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045954-34395.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045954-14389.png)
可知直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-14496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045954-51119.png)
设直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-14496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045954-10616.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045955-74723.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045955-46355.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045955-10664.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045955-82419.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045955-90042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045956-32456.png)
解法二:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045956-15330.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-36234.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045956-53637.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-14496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-36234.png)
故
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045956-35307.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-14496.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045954-10616.png)
在
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045956-40878.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045957-60904.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045957-50042.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045957-92004.png)
(2)解法一:建立坐标系如图.平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045957-87031.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045958-87817.png)
设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045958-88651.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045958-73423.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045958-53580.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045958-53074.png)
所以
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045959-77920.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045959-91221.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045959-87652.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045959-42143.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050000-29332.png)
由图知二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-44578.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050000-23886.png)
解法二:过
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050000-59739.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050000-94389.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-48282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-64129.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-45508.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-48282.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-48898.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050001-38852.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050002-81864.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050002-49562.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050002-21271.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050002-71906.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050003-72018.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022050003-26601.png)
所以 二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045952-44578.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045953-38111.png)
点评:解决的关键是利用角的定义作图来结合几何中的性质定理和判定定理来得到,解三角形得到,或者建立空间直角坐标系,运用向量法来求解。属于中档题。
核心考点
举一反三
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045944-79945.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045944-67305.png)
形,
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045944-67312.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045944-67305.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045944-14440.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-61402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-92517.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-92517.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-91822.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-68397.png)
(1)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-61402.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045945-92517.png)
(2)求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045946-53565.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045931-23860.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045932-99453.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045932-10467.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045932-70154.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045932-38424.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045932-66584.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045933-60946.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045933-53697.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045933-62076.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045933-48086.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045934-94109.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045934-47950.png)
(Ⅱ)若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045934-82486.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045934-95084.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045934-54713.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045918-91692.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045918-58718.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045919-17867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045919-32512.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045919-62831.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045919-66770.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045919-55915.png)
(Ⅰ)求证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045920-18654.png)
(Ⅱ)设平面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045920-81530.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045920-26362.png)
①试证:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045920-15660.png)
②若
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045921-26520.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045921-79292.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045904-48954.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045905-40592.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045905-48607.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045905-62148.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045905-35771.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045906-43655.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045906-99437.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045906-52263.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045906-26413.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045906-97311.png)
(Ⅰ)证明:
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045907-70884.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045907-17895.png)
(Ⅱ)在线段
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045907-29532.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045907-15050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045908-47836.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045908-28849.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045908-14909.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045907-15050.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-71447.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-13737.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-76021.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-85493.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-75867.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-12426.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045847-28798.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-55769.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-51914.png)
(1)求证:侧面
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-33354.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-46508.png)
(2)若异面直线
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-60527.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-89958.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-16599.png)
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045848-87981.png)
求二面角
![](http://img.shitiku.com.cn/uploads/allimg/20191022/20191022045849-66820.png)
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