3.3对应的角

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  对应的角

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  Corresponding Angles
  ::对应的角

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  Corresponding angles are two angles that are in the "same place" with respect to the transversal but on different lines. Imagine sliding the four angles formed with line $\begin{align*}l\end{align*}$ down to line $\begin{align*}m\end{align*}$ . The angles which match up are corresponding .
  ::相应的角度是两个在“ 相同位置” 的横向角度, 但是在不同的线条上 。 想象一下滑动四个角度, 由 l 线向下形成到 m 线 。 匹配的角度是对应的 。

  

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  Corresponding Angles Postulate : If two parallel lines are cut by a transversal, then the corresponding angles are congruent .
  ::对应的角 假设 : 如果两个平行的线条被横切, 那么相应的角是相同的 。

  

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  If $\begin{align*}l || m\end{align*}$ , then $\begin{align*}\angle 1 \cong \angle 2\end{align*}$ .
  ::如果我要的话,那就1点2分

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  Converse of Corresponding Angles Postulate: If corresponding angles are congruent when two lines are cut by a transversal, then the lines are parallel .
  ::对应的角的对等假设 : 如果对应的角度在两条线被横切时是相同的, 那么线是平行的 。

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  If then $\begin{align*}l || m\end{align*}$ .
  ::如果那样的话

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  What if you were presented with two angles that are in the same place with respect to the transversal but on different lines? How would you describe these angles and what could you conclude about their measures?
  ::如果有人向你们展示了两个角度,这两个角度位于同一地点,与横贯不同,但线不同呢?你将如何描述这些角度,以及你能得出什么结论来衡量这些角度?

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  Examples
  ::实例

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  Example 1
  ::例1

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  If $\begin{align*}m\angle 2 = 76^\circ\end{align*}$ , what is $\begin{align*}m\angle 6\end{align*}$ ?
  ::如果m2=76,什么是m6?

  

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  $\begin{align*}\angle 2\end{align*}$ and $\begin{align*}\angle 6\end{align*}$ are corresponding angles and $\begin{align*}l || m\end{align*}$ from the arrows in the figure. $\begin{align*}\angle 2 \cong \angle 6\end{align*}$ by the Corresponding Angles Postulate, which means that $\begin{align*}m\angle 6 = 76^\circ\end{align*}$ .
  ::======================================================================================================================================================================================================================================================

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  Example 2
  ::例2

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  Using the measures of $\begin{align*}\angle 2\end{align*}$ and $\begin{align*}\angle 6\end{align*}$ from Example 1, find all the other angle measures.
  ::使用例1中的2和6的计量方法,找到所有其他角度的计量方法。

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  If $\begin{align*}m\angle 2 = 76^\circ\end{align*}$ , then $\begin{align*}m\angle 1 = 180^\circ - 76^\circ =104^\circ\end{align*}$ (linear pair). $\begin{align*}\angle 3 \cong \angle 2\end{align*}$ (vertical angles), so $\begin{align*}m\angle 3 = 76^\circ. \ m\angle 4 = 104^\circ\end{align*}$ (vertical angle with $\begin{align*}\angle 1\end{align*}$ ).
  ::如果 m2=76, 那么m1=18076104(线性对子)。 32( 垂直角度), 所以 m3=76. m4=104( 垂直角度与 1) 。

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  By the Corresponding Angles Postulate, we know $\begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}$ , and $\begin{align*}\angle 4 \cong \angle 8\end{align*}$ , so $\begin{align*}m\angle 5 = 104^\circ, \ m\angle 6 = 76^\circ, \ m\angle 7 = 76^\circ\end{align*}$ , and $\begin{align*}m\angle 8 = 104^\circ\end{align*}$ .
  ::根据相应的角图,我们知道15,26,37,和48, 所以m5=104, m6=76, m7=76,和m8=104。

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  Example 3
  ::例3

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  Find the value of $\begin{align*}y\end{align*}$ :
  ::查找 y 值:

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  The horizontal lines are marked parallel and the angle marked $\begin{align*}2y\end{align*}$ is corresponding to the angle marked $\begin{align*}80\end{align*}$ so these two angles are congruent. This means that $\begin{align*}2y=80\end{align*}$ and therefore $\begin{align*}y=40\end{align*}$ .
  ::水平线是平行的标记,而标为 2y 的角与标为 80 的角相对应,所以这两个角是相似的。这意味着 2y=80, 因此y=40。

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  Example 4
  ::例4

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  If $\begin{align*}a || b\end{align*}$ , which pairs of angles are congruent by the Corresponding Angles Postulate?
  ::如果 ab , 对应的角假设对应的哪一对角度是相同的 ?

  

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  There are 4 pairs of congruent corresponding angles:
  ::有四对一致的对应角度:

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  $\begin{align*}\angle 1 \cong \angle 5, \ \angle 2 \cong \angle 6, \ \angle 3 \cong \angle 7\end{align*}$ , and $\begin{align*}\angle 4 \cong \angle 8\end{align*}$ .
  ::15,26,37和48。

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  Example 5
  ::例5

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  If $\begin{align*}m\angle 8 = 110^\circ\end{align*}$ and $\begin{align*}m\angle 4 = 110^\circ\end{align*}$ , then what do we know about lines $\begin{align*}l\end{align*}$ and $\begin{align*}m\end{align*}$ ?
  ::如果 m8=110和 m4=110, 那么关于l线和m线我们了解多少?

  

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  $\begin{align*}\angle 8\end{align*}$ and $\begin{align*}\angle 4\end{align*}$ are corresponding angles. Since $\begin{align*}m\angle 8 = m\angle 4\end{align*}$ , we can conclude that $\begin{align*}l || m\end{align*}$ .
  ::8和4是相应的角度。从 m8=m4开始,我们可以得出 lm的结论。

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  Review
  ::回顾

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  1. Determine if the angle pair $\begin{align*}\angle 4\end{align*}$ and $\begin{align*}\angle 2\end{align*}$ is congruent, supplementary or neither:
     ::确定角度对对% 4 和% 2 是否一致,是否是补充的,或两者兼而有之:
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  2. Give two examples of corresponding angles in the diagram:
     ::给出图表中相应角度的两个示例 :
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  3. Find the value of $\begin{align*}x\end{align*}$ :
     ::查找 x 值:
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  4. Are the lines parallel? Why or why not?
     
     
     ::线条是平行的吗?
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  5. Are the lines parallel? Justify your answer.
     ::线条平行吗?

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  For 6-10, what does the value of $\begin{align*}x\end{align*}$ have to be to make the lines parallel?
  ::对于6-10,x的值是多少才能使线平行?

  

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  6. If $\begin{align*}m\angle 1 = (6x-5)^\circ\end{align*}$ and $\begin{align*}m\angle 5 = (5x+7)^\circ\end{align*}$ .
     ::如果 m1 =( 6x- 5) 和 m5 =( 5x+7) * 。
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  7. If $\begin{align*}m\angle 2 = (3x-4)^\circ\end{align*}$ and $\begin{align*}m\angle 6 = (4x-10)^\circ\end{align*}$ .
     ::如果 m2 =( 3x- 4) 和 m6 =( 4x- 10) * 。
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  8. If $\begin{align*}m\angle 3 = (7x-5)^\circ\end{align*}$ and $\begin{align*}m\angle 7 = (5x+11)^\circ\end{align*}$ .
     ::如果 m*3 =( 7x- 5) =( 7x- 5) =( 5x+ 11) =( 7x- 5) =( 7x+ 11) * =( 7x- 5) =( 7x+ 11) =( 5x+ 11) = 。
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  9. If $\begin{align*}m\angle 4 = (5x-5)^\circ\end{align*}$ and $\begin{align*}m\angle 8 = (3x+15)^\circ\end{align*}$ .
     ::如果 m4=( 5x- 5) 和 m8=( 3x+15) 。
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  10. If $\begin{align*}m\angle 2 = (2x+4)^\circ\end{align*}$ and $\begin{align*}m\angle 6 = (5x-2)^\circ\end{align*}$ .
      ::如果 m2 = (2x+4) 和 m6 = (5x-2) 。

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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