3.5 备选外部角

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  替代的外部角

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  Alternate Exterior Angles
  ::替代的外部角

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  Alternate exterior angles are two angles that are on the exterior of $\begin{array}{r}l\end{array}$ and $\begin{array}{r}m\end{array}$ , but on opposite sides of the transversal .
  ::不同的外部角度是介于I和M外部的两个角度,但位于横向的对面。

  

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  Alternate Exterior Angles Theorem : If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent .
  ::其它外角定理 : 如果两条平行线由横切线切除, 则不同的外部角度是相似的 。

  

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

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  Converse of the Alternate Exterior Angles Theorem: If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel .
  ::对应的外部外角理论:如果两条线被横切切开,而不同的外部角度是相似的,那么两条线是平行的。

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

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  What if you were presented with two angles that are on the exterior of two parallel lines cut by a transversal but on opposite sides of the transversal? How would you describe these angles and what could you conclude about their measures? 
  ::如果有人向你们展示两条平行线的外侧,两条平行线由横跨的但横跨的两条线截取,两条平行线的外侧截取的两个角度呢?你们如何描述这些角度,以及你们能得出什么结论呢?

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  Examples
  ::实例

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  For Examples 1 and 2, use the following diagram : 
  ::关于例1和例2,请使用下图:

  

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  Example 1
  ::例1

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  Give an example of a pair of alternate exterior angles.
  ::举个例子来说明一对不同的外部角度。

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  $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }14\end{array}$  (many other possibilities)
  ::1和14(许多其他可能性)

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  Example 2
  ::例2

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  Give another example of a pair of alternate exterior angles.
  ::举另一个不同的外部角度的例子 。

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  $\begin{array}{r}\mathrm{\angle }2\end{array}$ and $\begin{array}{r}\mathrm{\angle }13\end{array}$  (many other possibilities, must be different than answer to Example 1)
  ::2和13(许多其他可能性必须不同于对例1的答复)

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  Example 3
  ::例3

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  Find the measure of each angle and the value of $\begin{array}{r}y\end{array}$ .
  ::查找每个角度的度量和y的值。

  

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  The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.
  ::角度是不同的外部角度。 由于线条是平行的, 角度是相等的 。

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  $$\begin{array}{rl}(3y+53{)}^{\circ }& =(7y-55{)}^{\circ }\\ 108=4y\\ 27=y\end{array}$$

  
  :3y+53)(7y-55)108=4y27=y

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  If $\begin{array}{r}y=27\end{array}$ , then each angle is $\begin{array}{r}[3(27)+53{]}^{\circ }={134}^{\circ }\end{array}$ .
  ::如果y=27,那么每个角度是[3(27)+53]134。

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  Example 4
  ::例4

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  The map below shows three roads in Julio’s town.
  ::以下地图显示胡利奥镇的三条道路。

  

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  Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.
  ::Julio用一个测量工具测量他绘制的相交点的两个角度。 Julio想知道Franklin Way是否与Chavez大道平行。

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  The $\begin{array}{r}{130}^{\circ }\end{array}$ angle and $\begin{array}{r}\mathrm{\angle }a\end{array}$ are alternate exterior angles. If $\begin{array}{r}m\mathrm{\angle }a={130}^{\circ }\end{array}$ , then the lines are parallel.
  ::130 角度和 a 是不同的外部角度。 如果 ma=130, 那么线条是平行的 。

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  $$\begin{array}{rlrl}\mathrm{\angle }a+{40}^{\circ }& ={180}^{\circ }& & \text{by the Linear Pair Postulate}\\ \mathrm{\angle }a& ={140}^{\circ }\end{array}$$

  
  ::a+40180 by the Linear pair Pair Postaudulea=140

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  $\begin{array}{r}{140}^{\circ }\ne {130}^{\circ }\end{array}$ , so Franklin Way and Chavez Avenue are not parallel streets.
  ::因此富兰克林大道与查韦斯大道并非平行街道。

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  Example 5
  ::例5

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  Which lines are parallel if $\begin{array}{r}\mathrm{\angle }AFG\cong \mathrm{\angle }IJM\end{array}$ ?
  ::如果“AFGIJM”是平行的, 那么哪条线是平行的 ?

  

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  These two angles are alternate exterior angles so if they are congruent it means that $\begin{array}{r}\stackrel{\leftrightarrow }{CG}||\stackrel{\leftrightarrow }{HK}\end{array}$ .
  ::这两个角度是不同的外部角度, 所以如果它们相近, 这意味着 CGHK。

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  Review
  ::回顾

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  1. Find the value of $\begin{array}{r}x\end{array}$ if $\begin{array}{r}m\mathrm{\angle }1=(4x+35{)}^{\circ },m\mathrm{\angle }8=(7x-40{)}^{\circ }\end{array}$ :
     ::查找 x 的值, 如果 m1 =( 4x+35) , m8 =( 7x- 40) :
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  2. Are lines 1 and 2 parallel? Why or why not?
     ::第一行和第二行是平行的吗?

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  For 3-6, what does the value of $\begin{array}{r}x\end{array}$ have to be to make the lines parallel?
  ::就3-6而言,x的值是多少才能使线平行?

  

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  3. $\begin{array}{r}m\mathrm{\angle }2=(8x{)}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }7=(11x-36{)}^{\circ }\end{array}$
     ::m2=( 8x) (11x-36)
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  4. $\begin{array}{r}m\mathrm{\angle }1=(3x+5{)}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }8=(4x-3{)}^{\circ }\end{array}$
     ::m*1=( 3x+5) 8=( 4x- 3)
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  5. $\begin{array}{r}m\mathrm{\angle }2=(6x-4{)}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }7=(5x+10{)}^{\circ }\end{array}$
     ::m2=( 6x- 4) 和 m7=( 5x+10)
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  6. $\begin{array}{r}m\mathrm{\angle }1=(2x-5{)}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }8=(x{)}^{\circ }\end{array}$
     ::m1=( 2x-5) 8=( x)

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  For 7-10, determine whether the statement is true or false.
  ::7-10时,确定声明是真实还是虚假。

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  7. Alternate exterior angles are always congruent.
     ::不同的外部角度总是相似的。
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  8. If alternate exterior angles are congruent then lines are parallel.
     ::如果其它外部角度一致,则线条平行。
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  9. Alternate exterior angles are on the interior of two lines.
     ::两条线内侧有其他外角。
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  10. Alternate exterior angles are on opposite sides of the transversal.
      ::不同的外部角度在横向的对面。

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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