3.1 平行线和扭曲线

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  平行线和扭曲线

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  Parallel and Skew Lines
  ::平行线和扭曲线

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  Parallel lines are two or more lines that lie in the same plane and never intersect. To show that lines are parallel, arrows are used.
  ::平行线是两条或多条线, 位于同一平面上, 而不是交叉。 要显示线线是平行的, 则使用箭头 。

  

  Label It	Say It
  $\begin{align*}\overleftrightarrow{AB} || \overleftrightarrow{MN}\end{align*}$	Line $\begin{align*}AB\end{align*}$ is parallel to line $\begin{align*}MN\end{align*}$
  $\begin{align*}l || m\end{align*}$	Line $\begin{align*}l\end{align*}$ is parallel to line $\begin{align*}m\end{align*}$ .

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  In the definition of parallel the word “line” is used. However, line segments, rays and planes can also be parallel. The image below shows two parallel planes, with a third blue plane that is perpendicular to both of them.
  ::在“线性”一词的平行定义中,使用了“线性”一词,但线性部分、射线和平面也可以是平行的。 下面的图像显示两架平行的平面,第三架蓝色平面与这两架平行。

  

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  Skew lines are lines that are in different planes and never intersect. They are different from parallel lines because parallel lines lie in the SAME plane. In the cube below, $\begin{align*}\overline{AB}\end{align*}$ and $\begin{align*}\overline{FH}\end{align*}$ are skew and $\begin{align*}\overline{AC}\end{align*}$ and $\begin{align*}\overline{EF}\end{align*}$ are skew.
  ::扭曲线是不同平面的线条,从不交叉。 它们与平行线条不同, 因为平行线线在 SAME 平面上。 在下面的立方体中, AB 和 FH 是 skew, AC 和 EF 是 skew 。

  

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  Basic Facts About Parallel Lines
  ::《平行线基本事实》

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  Property : If lines $\begin{align*}l || m\end{align*}$ and $\begin{align*}m || n\end{align*}$ , then $\begin{align*}l || n\end{align*}$ .
  ::属性:如果行 lm 和 mn, 那么ln 。

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  If then
  ::如果当时

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  Postulate : For any line and a point not on the line, there is one line parallel to this line through the point. There are infinitely many lines that go through $\begin{align*}A\end{align*}$ , but only one that is parallel to $\begin{align*}l\end{align*}$ .
  ::假设 : 对于任何直线和不在直线上的点, 有一条直线与这条直线平行穿过点。 有无限多的直线通过 A 线, 但只有一条直线与 I 线平行 。

  

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  A transversal is a line that intersects two other lines. The area between $\begin{align*}l\end{align*}$ and $\begin{align*}m\end{align*}$ is the interior . The area outside $\begin{align*}l\end{align*}$ and $\begin{align*}m\end{align*}$ is the exterior .
  ::横横线是横交另外两条线的线条。介于l和m之间的区域是内部。外部和m之间的区域是外部。

  

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  What if you were given a pair of lines that never intersect and were asked to describe them? What terminology would you use?
  ::如果给你们一对永不交叉的线条,而且要你们描述这些线条呢?你们要用什么术语?

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  Examples
  ::实例

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  Use the figure below for Examples 1 and 2. The two pentagons are parallel and all of the rectangular sides are perpendicular to both of them.
  ::实例1和2使用下图。 两个五边是平行的,所有长方形的两边都与这两边相连接。

  

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  Example 1
  ::例1

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  Find two pairs of skew lines .
  ::找到两对斜线线

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  $\begin{align*}\overline{ZV}\end{align*}$ and $\begin{align*}\overline{WB}\end{align*}$ . $\begin{align*}\overline{YD}\end{align*}$ and $\begin{align*}\overline{VW}\end{align*}$
  ::ZV和WW 你和VW

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  Example 2
  ::例2

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  For $\begin{align*}\overline{XY}\end{align*}$ , how many parallel lines would pass through point $\begin{align*}D\end{align*}$ ? Name this/these line(s).
  ::对于 XY 来说, 有多少平行线将通过 D 点? 请说出这条线/ 这几条线 。

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  One line, $\begin{align*}\overline{CD}\end{align*}$
  ::一行,CD

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  Example 3
  ::例3

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  True or false: some pairs of skew lines are also parallel.
  ::真实的或假的:有些斜线线也平行。

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  This is false, by definition skew lines are in different planes and parallel lines are in the same plane. Two lines could be skew or parallel (or neither), but never both.
  ::这是虚假的,根据定义,斜线线位于不同的平面上,平行线位于同一个平面上。 两条线可能是斜线或平行线(或两者都不是),但绝非两者兼而有之。

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  Example 4
  ::例4

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  Using the cube below, list a pair of parallel lines.
  ::使用下面的立方体,列出一对平行线条。

  

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  One possible answer is lines $\begin{align*}\overline{AB}\end{align*}$ and $\begin{align*}\overline{EF}\end{align*}$ .
  ::一个可能的答案是线AB和EF。

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  Example 5
  ::例5

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  Using the cube below, list a pair of skew lines.
  ::使用下面的立方体,列出一对斜线。

  

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  One possible answer is $\begin{align*}\overline{BD}\end{align*}$ and $\begin{align*}\overline{CG}\end{align*}$ .
  ::一个可能的答案是BD和CG

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  Review
  ::回顾

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  1. Which of the following is the best example of parallel lines?
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     1. Railroad Tracks
        ::铁路铁路轨道
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     2. Lamp Post and a Sidewalk
        ::灯光邮报和人行道
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     3. Longitude on a Globe
        ::环球经度
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     4. Stonehenge (the stone structure in Scotland)
        ::巨石柱(苏格兰的石块结构)
     
     ::以下哪些是平行线的最好例子?铁路路路路路灯站和环球巨石柱上的人行道经(苏格兰的石头结构)
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  2. Which of the following is the best example of skew lines?
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     1. Roof of a Home
        ::家园屋顶
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     2. Northbound Freeway and an Eastbound Overpass
        ::北向高速公路和东向过桥
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     3. Longitude on a Globe
        ::环球经度
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     4. The Golden Gate Bridge
        ::金门大桥
     
     ::下面哪个是斜线的最好例子?

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  Use the picture below for questions 3-5.
  ::问题3-5请使用下面的图片。

  

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  3. If $\begin{align*}m\angle 2 = 55^\circ\end{align*}$ , what other angles do you know?
     ::如果m2=55, 你知道什么其他角度吗?
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  4. If $\begin{align*}m\angle 5 = 123^\circ\end{align*}$ , what other angles do you know?
     ::如果m5=123, 你知道什么其他角度吗?
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  5. Is $\begin{align*}l || m\end{align*}$ ? Why or why not?
     ::为什么,为什么?

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  For 6-10, determine whether the statement is true or false.
  ::6-10时,确定声明是真实还是虚假。

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  6. If $\begin{align*}p || q\end{align*}$ and $\begin{align*} q || r\end{align*}$ , then $\begin{align*} p || r\end{align*}$ .
     ::如果pQQ和qQr,那么pQr。
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  7. Skew lines are never in the same plane.
     ::扭曲线从不在同一架飞机上。
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  8. Skew lines can be perpendicular.
     ::扭曲线可以是垂直的。
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  9. Planes can be parallel.
     ::计划可以平行进行。
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  10. Parallel lines are never in the same plane.
      ::平行线从不在同一平面上。

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  Review (Answers)
  ::回顾(答复)

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  Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
  ::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键 。

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  Resources
  ::资源