多边形边与角和对角线的关系

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Bradley is helping design an amusement park. He sketches his designs by using regular polygons. He also uses regular polygons to sketch the tables that he wants to place in the picnic  area . He sketches a table that has 6 sides. What is the sum of the  interior angles ?
::Bradley正在帮助设计一个游乐公园。 他使用普通多边形绘制他的设计图。 他还使用普通多边形绘制他想要在野餐区放置的桌子。 他绘制了一张有六边的桌子。 内部角度的总和是多少?

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In this concept, you will learn how to relate the sides of a  polygon  to  angle  measures.

::在这个概念中,你们将学会如何将多边形的两面与角度测量联系起来。

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Relating the Sides of a Polygon to Angles and Diagonals
::将多边形的侧面与角和对角相对

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Polygons can be divided into triangles using  diagonals . This becomes very helpful when you try to figure out the sum of the interior angles of a polygon other than a  triangle  or a  quadrilateral .
::多边形可以使用对角形分为三角形。当您试图找出除三角形或四边形以外的多边形内部角的总和时,这非常有用。

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The sum of the interior angles of a quadrilateral is  $\begin{array}{r}{360}^{\circ }\end{array}$ .  You can divide a quadrilateral into two triangles using diagonals. Each triangle is  $\begin{array}{r}{180}^{\circ }\end{array}$ , so the sum of the interior angles of a quadrilateral is  $\begin{array}{r}{360}^{\circ }\end{array}$ .
::四边形的内部角之和为 360 。 您可以使用对角形将四边形分为两个三角形。 每个三角形为 180 , 因此四边形的内部角之和为 360 。

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Here is one  diagonal  in the quadrilateral. 
::这里有一个四边形的对角。

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A  diagonal  is a  line segment  in a polygon that joins two  nonconsecutive   vertices . Divide up the polygon using diagonals and figure out the sum of the interior angles.
::对角线是一个多边形的线条段,它与两个非连续的脊椎相连接。使用对角线将多边形分开,并找出内角的总和。

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Here is a  hexagon  that has been divided into triangles by the diagonals. You can see here that there are four triangles formed. If the sum of the interior angles of each triangle is equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ , and you have four triangles, then the sum of the interior angles of a hexagon is:
::这里有一个六边形, 由对角线分为三角形。 您可以在此看到有四个三角形组成。 如果每个三角形的内角加起来等于 180 , 并且您有四个三角形, 那么六边形的内角加起来是 :

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You can follow this same procedure with any other polygon.
::您可以对其它多边形遵循同样的程序 。

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If you do not have a picture of the polygon, you can use a formula that involves the number of sides in a polygon.
::如果您没有多边形的图片,可以使用包含多边形边数的公式。

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To better understand how this works, let’s look at a table that shows the number of triangles related to the number of sides in a polygon.
::为了更好地了解这是怎么回事,让我们看看一个表格,显示与多边形边数有关的三角数。

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The biggest pattern to notice is that the number of triangles is 2 less than the number of sides. If you know that the sum of the interior angles of one triangle is equal to 180 degrees and if you know that there are three triangles in a polygon, then you can multiply the number of triangles by 180 and that will give you the sum of the interior angles.
::注意的最大模式是三角形的数目比边数少2个。如果您知道一个三角形的内部角和角等于180度,如果您知道多边形中有三个三角形,那么您可以将三角形的数目乘以180,这将为您提供内部角的总和。

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Here is the formula.
::这是公式。

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$\begin{array}{r}x=\end{array}$  number of sides
::x = 边数

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$\begin{array}{r}(x-2)180=\end{array}$  sum of the interior angles
:x- 2) 180 = 内角总和

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You can take the number of sides and use that as  $\begin{array}{r}x\end{array}$ .
::您可以将边数乘以 x 。

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Then solve for the sum of the interior angles.
::然后解决内部角度的总和 。

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Let’s look at an example.
::让我们举个例子。

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What is the sum of the interior angles of a  decagon ?
::十角形内角的总和是多少?

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A decagon has ten sides. That is your  $\begin{array}{r}x\end{array}$  measurement. Now let’s use the formula.
::十角形有十面。这是你的x测量。现在让我们使用公式。

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The answer is that there are  $\begin{array}{r}{1440}^{\circ }\end{array}$  in a decagon.
::答案是,在十进制中是1440年。

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Examples
::实例

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Example 1
::例1

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Earlier, you were given a problem about Bradley and his picnic table designs.
::之前有人给你一个问题 关于布拉德利和他的野餐桌设计

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He sketches a table that has 6 sides. What is the sum of the interior angles?
::他绘制了一张有六边的桌子。内侧角度的总和是多少?

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First, write the number of sides that are in the polygon.
::首先,写入多边形中的边数。

6

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Next, plug the number of sides in to the formula.
::下一步,在公式中插入边数。

$\begin{array}{r}(6-2)(180)\end{array}$ $\begin{array}{r}(5-2)(180)\end{array}$

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Then, solve for the sum of the interior angles.
::然后,解决内部角度的总和。

$\begin{array}{r}4(180)=720\end{array}$

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The answer is  $\begin{array}{r}{720}^o\end{array}$ . The sum of the interior angles is  $\begin{array}{r}{720}^o\end{array}$ .
::答案是720o。内部角度的总和是720o。

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Example 2
::例2

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Calculate the sum of the interior angles of a regular  nonagon .
::计算正则非角的内角的总和。

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First, write the number of sides that are in a nonagon.
::首先,写出非角的边数 。

9

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Next, plug the number of sides in to the formula.

::下一步,在公式中插入边数。

$\begin{array}{r}(9-2)180\end{array}$

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Then, solve for the sum of the interior angles.
::然后,解决内部角度的总和。

  $\begin{array}{r}(7)(180)=1260\end{array}$

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The answer is  $\begin{array}{r}{1260}^{\circ }\end{array}$ .
::答案是1260年。

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Calculate the sums for each example.
::计算每个示例的金额。

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Example 3
::例3

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Calculate the sum of the interior angles of a  pentagon .
::计算五角形内角的总和。

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First, write the number of sides that are in a pentagon.
::首先,写下五角形中的边数。

5

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Next, plug the number of sides in to the formula.
::下一步,在公式中插入边数。

  $\begin{array}{r}(5-2)(180)\end{array}$

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Then, solve for the sum of the interior angles.
::然后,解决内部角度的总和。

  $\begin{array}{r}(3)(180)=540\end{array}$

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The answer is  $\begin{array}{r}{540}^o\end{array}$ .
::答案是540 o。

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Example 4
::例4

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Calculate the sum of the interior angles of a triangle.
::计算三角形内角的总和。

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First, write the number of sides that are in a triangle.
::首先,写入三角形中的边数。

3

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Next, plug the number of sides in to the formula.
::下一步,在公式中插入边数。

  $\begin{array}{r}(3-2)(180)\end{array}$

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Then, solve for the sum of the interior angles.
::然后,解决内部角度的总和。

  $\begin{array}{r}1(180)=180\end{array}$

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The answer is  $\begin{array}{r}{180}^o\end{array}$ .
::答案是180o。

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Example 5
::例5

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Calculate the sum of the interior angles of an  octagon .
::计算八边形内角的总和。

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First, write the number of sides that are in an octagon.
::首先,写出八角的边数。

8

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Next, plug the number of sides in to the formula?'
::下一步,在公式中插入边数?”

  $\begin{array}{r}(8-2)(180)\end{array}$

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Then, solve for the sum of the interior angles.
::然后,解决内部角度的总和。

  $\begin{array}{r}6(180)=1080\end{array}$

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The answer is  $\begin{array}{r}{1080}^o\end{array}$ .
::答案是1080 o。

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Review
::回顾

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Look at each image and name the type of polygon pictured.
::查看每张图像并命名所拍摄的多边形类型 。

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1. 

   [Figure 5]
   
   ::[图5]
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2. 

   [Figure 6]
   
   ::[图6]
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3. 

   [Figure 7]
   
   ::[图7]
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4. 

   [Figure 8]
   
   ::[表8]
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5. 

   [Figure 9]
   
   ::[图9]
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6. 

   [Figure 10]
   
   ::[图10]

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How many diagonals are needed to divide the following polygons into triangles?
::要将以下多边形分为三角形,需要多少对角线?

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7. 

   [Figure 11]
   
   ::[表11]
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8. 

   [Figure 12]
   
   ::[图12]
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9. 

   [Figure 13]
   
   ::[图13]

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Use the formula to name the sum of the interior angles of each polygon.
::使用公式来命名每个多边形的内部角度的总和。

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10. Hexagon
    ::六边形
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11. Pentagon
    ::五角大楼
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12. Decagon
    ::十角
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13. Heptagon
    ::七边形
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14. Octagon
    ::八角
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15. Square
    ::广场广场

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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