一、 等角定理：一个角的两边分别与另一个角的两边平行，则两个角相等或互补。
二：异面直线夹角
（1）意义：（2）![](https://wkrtcs.bdimg.com/rtcs/image?w=50.67&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_3","w":"50.67","h":"34.67","dataType":"gif","c":"word/media/image1.gif"})
0，]
注：两异面直线夹角为![](https://wkrtcs.bdimg.com/rtcs/image?w=38.67&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_3","w":"38.67","h":"30.67","dataType":"gif","c":"word/media/image3.gif"})
时，也叫做两直线互相垂直。
三、异面直线夹角的求法：
1、平移不改变线段长度[主要适用于柱体]{直接法}
![](https://wkrtcs.bdimg.com/rtcs/image?w=642.53&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_4","w":"642.53","h":"101.59","dataType":"png","c":"word/media/image4.png","imgOriW":913,"imgOriH":144})
2 .A1B1C1—ABC是直三棱柱，∠BCA=90°，点D1、F1 分别是A1B1、A1C1的中点
若BC=CA=CC1，求BD1与AF1所成角的余弦值.![](https://wkrtcs.bdimg.com/rtcs/image?w=148.00&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_7","w":"148.00","h":"143.00","dataType":"gif","c":"word/media/image5.gif"})
![](https://wkrtcs.bdimg.com/rtcs/image?w=123.00&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","w":"123.00","h":"133.00","dataType":"gif","c":"word/media/image6.gif"})
3.在棱长为1的正方体ABCD—A1B1C1D1中，M和N分别为A1B1和BB1的中点，
求直线A![](https://wkrtcs.bdimg.com/rtcs/image?w=32.00&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_7","w":"32.00","h":"16.00","dataType":"gif","c":"word/media/image7.gif"})
与CN所成角的余弦值
二、平移改变线段长度[主要适用于锥体]
注：选择平移方向的法则：在两条异面直线上，各选择一个点形成线段，则该线段的中点就是平移的目标位置。
![](https://wkrtcs.bdimg.com/rtcs/image?w=642.53&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_4","w":"642.53","h":"73.21","dataType":"png","c":"word/media/image8.png","imgOriW":916,"imgOriH":104})
![](https://wkrtcs.bdimg.com/rtcs/image?w=642.53&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_4","w":"642.53","h":"288.32","dataType":"png","c":"word/media/image9.png","imgOriW":919,"imgOriH":412})
注：正三棱锥对棱垂直。[性质]
三、补形[主要适用于线段的位置不易发生移动，如体对角线，同时要求在规则的柱体中如正方体、长方体中和一些正棱柱中]
例：正方体ABCD－![](https://wkrtcs.bdimg.com/rtcs/image?w=65.33&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_3","w":"65.33","h":"16.00","dataType":"gif","c":"word/media/image10.gif"})
中，求异面直线所成的角.
![](https://wkrtcs.bdimg.com/rtcs/image?w=642.53&md5sum=584dc0e0d60005116306d36dd20fb7fb&sign=ad76f1f3b9&rtcs_flag=2&rtcs_ver=3.2&l=webapp&bucketNum=78&ipr={"t":"img","r":"r_4","w":"642.53","h":"362.93","dataType":"png","c":"word/media/image12.png","imgOriW":921,"imgOriH":520})
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