6.1 Convex 多边形中的内角

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• 选择节Convex 多边形中的内部角

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  Convex 多边形中的内部角

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  Interior Angles in Convex Polygons
  ::Convex 多边形中的内部角

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  The interior angle of a polygon is one of the angles on the inside, as shown in the picture below. A polygon has the same number of interior angles as it does sides.
  ::多边形的内部角是内部角度之一,如下图所示。多边形的内部角与侧角的数目相同。

  

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  The sum of the interior angles in a polygon depends on the number of sides it has. The Polygon Sum Formula states that for any $\begin{array}{r}n-\end{array}$ gon, the interior angles add up to $\begin{array}{r}(n-2)\times {180}^{\circ }.\end{array}$
  ::多边形内角的总和取决于其边数。多边形总和表示,对于任何 n-gon,内角加起来等于(n-2)x180。

  

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  $$\begin{array}{rl}\to n=8& \\ (8-2)& \times {180}^{\circ }\\ 6& \times {180}^{\circ }\\ 1& {080}^{\circ }\end{array}$$

  
  ::=88(8- 2) ×180 ×6x180 1080

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  Once you know the sum of the interior angles in a polygon it is easy to find the measure of ONE interior angle if the polygon is regular : all sides are congruent and all angles are congruent. Just divide the sum of the angles by the number of sides.
  ::一旦您知道多边形内角的总和, 如果多边形是正态的, 很容易找到一个内部角的大小: 方形是正态的, 方形是正态的, 方形是正态的。 只要将角度之和除以边数即可 。

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  Regular Polygon Interior Angle Formula: For any equiangular $\begin{array}{r}n-\end{array}$ gon, the measure of each angle is $\begin{array}{r}\frac{(n-2)\times {180}^{\circ }}{n}\end{array}$ .
  ::常规多边形内部角公式:对于任何等角-角,每个角的度量为(n-2)x180n。

  

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  In the picture below, if all eight angles are congruent then each angle is $\begin{array}{r}\frac{(8-2)\times {180}^{\circ }}{8}=\frac{6\times {180}^{\circ }}{8}=\frac{{1080}^{\circ }}{8}={135}^{\circ }\end{array}$ .
  ::在下面的图片中,如果所有八个角度是相同的,那么每个角度是(8-2)x1808=6x1808=6x1808=10808=135}。

  

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  What if you were given an equiangular seven-sided convex polygon? How could you determine the measure of its interior angles?
  ::如果给您一个方形七面方形的方形多边形呢?您如何确定内角的大小 ?

   

   

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  Examples
  ::实例

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  Example 1
  ::例1

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  The interior angles of a pentagon are $\begin{array}{r}{x}^{\circ },x^{\circ },2x^{\circ },2x^{\circ },\end{array}$ and $\begin{array}{r}2x^{\circ }\end{array}$ . What is $\begin{array}{r}x\end{array}$ ?
  ::五角形的内角是 x, x , 2x , 2x , 和 2x 。 什么是 x?

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  From the Polygon Sum Formula we know that a pentagon has interior angles that sum to $\begin{array}{r}(5-2)\times {180}^{\circ }={540}^{\circ }\end{array}$ .
  ::从多边形和公式中,我们知道,一个五角形的内部角度等于(5-2)x180540。

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  Write an equation and solve for $\begin{array}{r}x\end{array}$ .
  ::写入公式并解析 x。

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  $$\begin{array}{rl}{x}^{\circ }+x^{\circ }+2x^{\circ }+2x^{\circ }+2x^{\circ }& ={540}^{\circ }\\ 8x& =540\\ x& =67.5\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不...

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  Example 2
  ::例2

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  What is the sum of the interior angles in a 100-gon?
  ::100公分内角的内角总和是多少?

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  Use the Polygon Sum Formula. $\begin{array}{r}(100-2)\times {180}^{\circ }=17,{640}^{\circ }\end{array}$ .
  ::使用多边形总和公式。 (100-2) ×18017, 640。

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  Example 3
  ::例3

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  The interior angles of a polygon add up to $\begin{array}{r}{1980}^{\circ }\end{array}$ . How many sides does it have?
  ::多边形的内角加到1980年 。它有多少边?

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  Use the Polygon Sum Formula and solve for $\begin{array}{r}n\end{array}$ .
  ::使用多边形总和公式并解决 n.

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  $$\begin{array}{rl}(n-2)\times {180}^{\circ }& ={1980}^{\circ }\\ {180}^{\circ }n-{360}^{\circ }& ={1980}^{\circ }\\ {180}^{\circ }n& ={2340}^{\circ }\\ n& =13\end{array}$$

  
  ::==============================================================================================

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  The polygon has 13 sides.
  ::多边形有13边。

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  Example 4
  ::例4

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  How many degrees does each angle in an equiangular nonagon have?
  ::等角非形中每个角有多少度?

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  First we need to find the sum of the interior angles; set $\begin{array}{r}n=9.\end{array}$
  ::首先我们需要找到内部角度的总和; 设定 n=9 。

  $$\begin{array}{r}(9-2)\times {180}^{\circ }=7\times {180}^{\circ }={1260}^{\circ }\end{array}$$

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  “Equiangular” tells us every angle is equal. So, each angle is $\begin{array}{r}\frac{{1260}^{\circ }}{9}={140}^{\circ }\end{array}$ .
  ::“ 等角 ” 告诉我们每个角度都是相等的。 所以, 每个角度是 1260\\\9=140\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Example 5
  ::例5

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  An interior angle in a regular polygon is $\begin{array}{r}{135}^{\circ }\end{array}$ . How many sides does this polygon have?
  ::普通多边形中的内角是 135 。 这个多边形有多少边 ?

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  Here, we will set the Regular Polygon Interior Angle Formula equal to $\begin{array}{r}{135}^{\circ }\end{array}$ and solve for $\begin{array}{r}n\end{array}$ .
  ::在此, 我们将设置常规多边形内部角公式, 等于 135 , 并解决 n 。

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  $$\begin{array}{rl}\frac{(n-2)\times {180}^{\circ }}{n}& ={135}^{\circ }\\ {180}^{\circ }n-{360}^{\circ }& ={135}^{\circ }n\\ -{360}^{\circ }& =-{45}^{\circ }n\\ n& =8\text{The polygon is an octagon}.\end{array}$$

  
  :n-2)x180n=13518n-360135n-36045nn=8)多边形是八边形。

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  Review
  ::回顾

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  1. Fill in the table. Round values to the nearest hundredths place if needed.
     ::填满表格。 如果需要, 圆形值为最接近的百位数 。

  # of sides	Sum of the Interior Angles	Measure of Each Interior Angle in a Regular $\begin{array}{r}n-\end{array}$ gon
  3		$\begin{array}{r}{60}^{\circ }\end{array}$
  4	$\begin{array}{r}{360}^{\circ }\end{array}$
  5	$\begin{array}{r}{540}^{\circ }\end{array}$	$\begin{array}{r}{108}^{\circ }\end{array}$
  6		$\begin{array}{r}{120}^{\circ }\end{array}$
  7
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  12

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  2. What is the sum of the angles in a 15-gon?
     ::15角角角度的总和是多少?
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  3. What is the sum of the angles in a 23-gon?
     ::23角角角度的总和是多少?
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  4. The sum of the interior angles of a polygon is $\begin{array}{r}{4320}^{\circ }\end{array}$ . How many sides does the polygon have?
     ::多边形的内部角总和是4320。多边形有多少边?
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  5. The sum of the interior angles of a polygon is $\begin{array}{r}{3240}^{\circ }\end{array}$ . How many sides does the polygon have?
     ::多边形的内部角总和是 3240。 多边形有多少边?
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  6. What is the measure of each angle in a regular 16-gon?
     ::在正常的 16 个角中,每个角度的量度是多少 ?
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  7. What is the measure of each angle in an equiangular 24-gon?
     ::方形 24 角 方形 24 角 每个角度的量度是多少 ?
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  8. Each interior angle in a regular polygon is $\begin{array}{r}{156}^{\circ }\end{array}$ . How many sides does it have?
     ::普通多边形中每个内部角为156。它有多少边?
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  9. Each interior angle in an equiangular polygon is $\begin{array}{r}{90}^{\circ }\end{array}$ . How many sides does it have?
     ::方形多边形内每个内角为 90 。 它有多少边 ?

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  For questions 10-18, find the value of the missing variable(s).
  ::对于问题10-18,请找到缺失变量的价值。

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  19. The interior angles of a hexagon are $\begin{array}{r}{x}^{\circ },(x+1{)}^{\circ },(x+2{)}^{\circ },(x+3{)}^{\circ },(x+4{)}^{\circ },\end{array}$ and $\begin{array}{r}(x+5{)}^{\circ }.\end{array}$ What is $\begin{array}{r}x\end{array}$ ?
      ::六边形的内角是 x 、 (x+1) 、 (x+2) 、 (x+3) 、 (x+4) 和 (x+5) 。 什么是 x?

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  Review (Answers)
  ::回顾(答复)

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  To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
  ::要查看本书的答案键, 请在“ ” 选项下点击答案键 。

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  Resources
  ::资源