6.2 Convex 多边形中的外部角

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  在 Convex 多边形中的外侧角

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  Exterior Angle Sum Theorem
  ::外部外外角集合定理

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  An exterior angle is an angle that is formed by extending a side of the polygon .
  ::外部角是一个通过延伸多边形侧面而形成的角度。

  

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  As you can see, there are two sets of exterior angles for any vertex on a polygon, one going around clockwise ( $\begin{array}{r}{1}^{st}\end{array}$ hexagon), and the other going around counter-clockwise ( $\begin{array}{r}{2}^{nd}\end{array}$ hexagon). The angles with the same colors are vertical and congruent .
  ::正如您所看到的,多边形上的任何顶点都有两组外部角度,其中一组围绕时针(第1个六边形),另一组则围绕逆时针(第2个六边形)运行。同一颜色的角是垂直和相近的。

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  The Exterior Angle Sum Theorem states that the sum of the exterior angles of ANY convex polygon is $\begin{array}{r}{360}^{\circ }\end{array}$ . If the polygon is regular with $\begin{array}{r}n\end{array}$ sides, this means that each exterior angle is $\begin{array}{r}{\frac{360}{n}}^{\circ }\end{array}$ .
  ::外部角集合理论称,任何锥形多边形的外部角之和为360 。如果多边形是正向正向,这意味着每个外部角为360n 。

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  What if you were given a seven-sided regular polygon ? How could you determine the measure of each of its exterior angles?
  ::如果给你们一个七面正多边形呢?你怎么确定每个外角的尺寸呢?

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  Examples
  ::实例

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  Example 1
  ::例1

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  What is the measure of each exterior angle of a regular 12-gon?
  ::正常的 12 边形的每个外部角度的量度是多少 ?

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  Divide  $\begin{array}{r}{360}^{\circ }\end{array}$  by the given number of sides.
  ::以给定的边数除以 360 。

  $\begin{array}{r}{30}^{\circ }\end{array}$

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  Example 2
  ::例2

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  What is the measure of each exterior angle of a regular 100-gon?
  ::普通的100角外角的量度是多少?

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  Divide  $\begin{array}{r}{360}^{\circ }\end{array}$  by the given number of sides.
  ::以给定的边数除以 360 。

  $\begin{array}{r}{3.6}^{\circ }\end{array}$

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  Example 3
  ::例3

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  What is $\begin{array}{r}y\end{array}$ ?
  ::什么是Y? What is y?

  

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  $\begin{array}{r}y\end{array}$ is an exterior angle and all the given angles add up to $\begin{array}{r}{360}^{\circ }\end{array}$ . Set up an equation.
  ::y 是一个外部角度,所有给定角度加起来为360。设置一个方程。

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  $$\begin{array}{rl}{70}^{\circ }+{60}^{\circ }+{65}^{\circ }+{40}^{\circ }+y& ={360}^{\circ }\\ y& ={125}^{\circ }\end{array}$$

  
  ::70606540y=360y=125

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  Example 4
  ::例4

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  What is the measure of each exterior angle of a regular heptagon ?
  ::普通七边形的每个外部角度的量度是多少?

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  Because the polygon is regular, the interior angles are equal. It also means the exterior angles are equal. $\begin{array}{r}\frac{{360}^{\circ }}{7}\approx {51.43}^{\circ }\end{array}$
  ::多边形是普通的, 内部角度是相等的 。 这还意味着外角是相等的 。 360\\\\\\\\\\\\\\51.43\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Example 5
  ::例5

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  What is the sum of the exterior angles in a regular 15-gon?
  ::正常的15角外角的外角总和是多少?

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  The sum of the exterior angles in any convex polygon, including a regular 15-gon, is $\begin{array}{r}{360}^{\circ }\end{array}$ .
  ::包括正常的 15 个方形在内的任何方形多边形的外部角总和为360 。

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  Review
  ::回顾

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  1. What is the measure of each exterior angle of a regular decagon?
     ::普通十角形的每个外部角度的量度是多少?
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  2. What is the measure of each exterior angle of a regular 30-gon?
     ::普通的30角外角的每个外角的量度是多少?
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  3. What is the sum of the exterior angles of a regular 27-gon?
     ::普通的27角外角的外角总和是多少?

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  Find the measure of the missing variables:
  ::查找缺失变量的度量 :

  4. 
  5. 

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  6. The exterior angles of a quadrilateral are $\begin{array}{r}{x}^{\circ },2x^{\circ },3x^{\circ },\end{array}$ and $\begin{array}{r}4x^{\circ }.\end{array}$ What is $\begin{array}{r}x\end{array}$ ?
     ::四边形的外角是 x, 2x, 3x, 和 4x。 什么是 x?

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  Find the measure of each exterior angle for each regular polygon below:
  ::查找以下每个常规多边形的每个外部角的测量值:

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  7. octagon
     ::八边
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  8. nonagon
     ::符号
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  9. triangle
     ::三角形
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  10. pentagon
      ::五边形

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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