9.7三角形按角度分类

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  三角形按角度分类

  

  [Figure 1]

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  Michael's mother bought a caddy that fits in a corner. She uses the caddy to store their brooms and mops in the garage. His mother's caddy had the appearance of a  right triangle  when viewed from above and fit perfectly in the corner. Michael liked the caddy, but he couldn't afford to buy one like hers, so he decided to make his own. He bought wood and nails and put it together, but when he tried to put it in the corner in his room, it wouldn't fit. He measures the  angles  of his caddy and realizes that the top is the  triangle  below:
  ::Michael的妈妈买了一个在角落的稻谷。 她用稻谷把扫帚和拖把存放在车库里。 他母亲的稻谷从上面看,完全适合角落里。 迈克尔喜欢稻谷, 但是他买不起像她那样的稻谷, 所以他决定自己去买。 他买了木头和钉子, 把它装在一起, 但是当他试图把木头和钉子放在他房间的角落里时, 它就不合适了。 他测量了他的稻谷的角, 并意识到顶部是下面的三角形:

  

  [Figure 2]

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  What is the classification of Michael's triangle?
  ::迈克尔三角形的分类是什么?

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  In this concept, you will learn how to use angles to  .
  
  ::在这个概念中,你会学会如何使用角度到 。

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  Classifying Triangles by Angles
  ::由角对三角进行分类

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  The prefix “tri” means three.  Triangle  means three angles.
  ::前缀“三”是指三个。三角形是指三个角度。

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  To classify a triangle according to its angles, you must look at the angles inside the triangle. Use the number of degrees in these angles to classify the triangle. Let’s look at a picture of a triangle to explain.
  ::要根据三角形的角度对三角形进行分类, 您必须查看三角形内的角度。 使用这些角度的度数来对三角形进行分类。 让我们查看三角形的图片来解释 。

   

  

  [Figure 3]

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  Look at the  measure  of each angle inside the triangle to figure out what kind of triangle it is. There are four types of triangles based on angle measures.
  ::查看三角形内每个角度的度量, 以了解三角形的种类。 有四种三角形以角度量为基础 。

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  A  right triangle  is a triangle that has one  right angle  and two  acute angles . One of the angles in the triangle measures  $\begin{array}{r}{90}^{\circ }\end{array}$  and the other two angles are less than 90. Here is a picture of a right triangle.
  ::右三角是指一个右角和两个急性角的三角形。三角形中的一个角是90°Z,其他两个角小于90°C。这里是右三角形的图片。

  

  [Figure 4]

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  You can see that the 90 degree angle is the one in the bottom left corner. You can even draw in the small box to identify it as a 90 degree angle. If you look at the other two angles you cans see that those angles are less than 90 degrees and are acute.
  ::您可以看到, 90 度角是左下角角的角。 您甚至可以在小框中画出, 将其识别为 90 度角。 如果您查看另外两个角, 您可以看到这些角小于 90 度, 并且是 急性 。

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  Let's look at an example of a right angle.
  ::让我们看看一个正确角度的例子。

   

  

  [Figure 5]

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  This triangle has one  $\begin{array}{r}{90}^{\circ }\end{array}$  angle and two  $\begin{array}{r}{45}^{\circ }\end{array}$  angles. Find the sum of the three angles.
  ::此三角形有一个 90 角度和两个 45 角度。 查找三个角度的总和 。

  $$\begin{array}{r}90+45+45={180}^{\circ }\end{array}$$

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  The sum of the three angles of a triangle is always equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::三角形的三个角度的总和总是等于 180 。

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  In an  equiangular triangle , all three of the angles are equal.
  ::在方形三角形中,所有三个角度都是相等的。

   

  

  [Figure 6]

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  The three angles of this triangle are equal. This is an equiangular triangle.
  ::此三角形的三个角度相等。 这是一个等角三角形的三角形 。

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  You know that the sum of the three angles is equal to  $\begin{array}{r}{180}^o\end{array}$ , therefore, for all three angles to be equal, each angle must be equal to  $\begin{array}{r}{60}^{\circ }\end{array}$ .
  ::你知道这三个角度的总和等于180°, 因此,所有三个角度要相等, 每个角度必须等于60° 。

  $$\begin{array}{r}60+60+60={180}^{\circ }\end{array}$$

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  The sum of the angles is equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::角度之和等于 180 。

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  In an  acute triangle , all three angles of the triangle are less than 90 degrees. Here is an example of an acute triangle.
  ::在急性三角形中,三角形的所有三个角度都低于90度。这里是急性三角形的例子。

   

  

  [Figure 7]

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  All three of these angles measure less than 90 degrees.
  ::所有三个角度的度量均低于90度。

  $$\begin{array}{r}33+80+67={180}^{\circ }\end{array}$$

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  The sum of the angles is equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::角度之和等于 180 。

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  An  obtuse triangle  has one angle that is greater than 90 and two angles that are less than 90.
  ::斜三角形有一个大于90的角,两个角小于90的角小于90。

  

  [Figure 8]

  $$\begin{array}{r}130+25+25={180}^{\circ }\end{array}$$

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  The sum of the angles is equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::角度之和等于 180 。

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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 Michael and his caddy.
  ::早些时候,你得到一个问题 关于迈克尔和 他的球童。

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  He tried to build a caddy like his mother's caddy. Her caddy looked like a right triangle from above but he ended up building one that had the following measures and appearance from above:
  ::他试图建造一个像他母亲的稻田一样的稻田。 她的稻田看起来像一个从上方直立的三角形,但他最终建造了一个具有以下措施和表象的稻田:

  

  [Figure 9]

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  What is the classification of Michael's triangle?
  ::迈克尔三角形的分类是什么?

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  First, list the angle measures.
  ::首先,列出角度度量。

  20, 20, 140

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  Next, determine if any of the angles are equal to 90 degrees or larger than 90 degrees.
  ::下一步,确定任何角是否等于90度或大于90度。

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  Yes, one angle is larger than 90 degrees
  ::是, 一个角度大于 90 度

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  Then, classify the triangle.
  ::然后,分类三角形。

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  Obtuse
  ::阻力

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  The answer is an obtuse triangle. Michael created an obtuse triangle instead of a right triangle.
  ::答案是一个隐形三角形。 迈克尔创建了一个隐形三角形, 而不是一个右三角形。

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  Example 2
  ::例2

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  Identify the type of triangle according to its angles.
  ::根据三角形的角度来识别三角形的类型 。

  

  [Figure 10]

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  First, list the angle measures.
  ::首先,列出角度度量。

  10, 75, 95

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  Next, determine if any of the angles are equal to 90 degrees or larger than 90 degrees.
  ::下一步,确定任何角是否等于90度或大于90度。

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  Yes, one angle is larger than 90 degrees
  ::是, 一个角度大于 90 度

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  Then, classify the triangle.
  ::然后,分类三角形。

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  Obtuse
  ::阻力

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  The answer is an obtuse triangle.
  ::答案是一个隐形三角形。

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  Example 3
  ::例3

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  Identify the type of triangle according to its angles.
  ::根据三角形的角度来识别三角形的类型 。

  

  [Figure 11]

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  First, list the angle measures.
  ::首先,列出角度度量。

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  30, 70 and 80
  ::30、70和80

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  Next, determine if any of the angles are equal to 90 degrees or larger than 90 degrees.
  ::下一步,确定任何角是否等于90度或大于90度。

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  No
  ::否 无

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  Then, classify the angle.
  ::然后,分类角度。

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  Acute
  ::急性

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  The answer is an acute triangle.
  ::答案是一个尖锐的三角形

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  Example 4
  ::例4

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  Identify the type of triangle according to its angles.
  ::根据三角形的角度来识别三角形的类型 。

  

  [Figure 12]

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  First, list the angle measures.
  ::首先,列出角度度量。

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  35, 55 and 90
  ::35、55和90

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  Next, determine if any of the angles are equal to 90 degrees or larger than 90 degrees.
  ::下一步,确定任何角是否等于90度或大于90度。

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  Yes, one of the angles is equal to 90 degrees
  ::是的,其中一个角度等于90度

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  Then, classify the angle.
  ::然后,分类角度。

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  Right
  ::右右右

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  The answer is a right triangle
  ::答案是一个右三角形

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  Example 5
  ::例5

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  Classify the triangle by looking at the sum of its angles.
  ::通过查看三角形角度的总和对三角形进行分类。

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  $\begin{array}{r}{40}^o+{60}^o+{80}^o={180}^o\end{array}$
  ::40 o + 60 o + 80 o = 180 o

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  First, list the angle measures.
  ::首先,列出角度度量。

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  40, 60 and 80
  ::40、60和80

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  Next, determine if any of the angles are equal to 90 degrees or larger than  90 degrees.
  ::下一步,确定任何角是否等于90度或大于90度。

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  No
  ::否 无

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  Then, classify the angle.
  ::然后,分类角度。

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  Acute
  ::急性

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  The answer is an acute triangle.
  
  ::答案是一个尖锐的三角形

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  Review
  ::回顾

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  Classify each triangle according to its angles.
  ::根据三角形的角度对三角形进行分类。

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  1. 

     [Figure 13]
     
     ::[图13]
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  2. 

     [Figure 14]
     
     ::[表14]
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  3. 

     [Figure 15]
     
     ::[图15]
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  4. 

     [Figure 16]
     
     ::[图16]
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  5. 

     [Figure 17]
     
     ::[图17]

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  Classify the following triangles by looking at the sum of the angle measures.
  ::通过查看角度量的和,对以下三角进行分类。

  6. 
     
     $$\begin{array}{r}40+55+45={180}^{\circ }\end{array}$$
  7. 
     
     $$\begin{array}{r}20+135+25={180}^{\circ }\end{array}$$
  8. 
     
     $$\begin{array}{r}30+90+60={180}^{\circ }\end{array}$$
  9. 
     
     $$\begin{array}{r}60+60+60={180}^{\circ }\end{array}$$
  10. 
      
      $$\begin{array}{r}110+15+55={180}^{\circ }\end{array}$$
  11. 
      
      $$\begin{array}{r}105+65+10={180}^{\circ }\end{array}$$
  12. 
      
      $$\begin{array}{r}80+55+45={180}^{\circ }\end{array}$$
  13. 
      
      $$\begin{array}{r}70+45+65={180}^{\circ }\end{array}$$
  14. 
      
      $$\begin{array}{r}145+20+15={180}^{\circ }\end{array}$$
  15. 
      
      $$\begin{array}{r}60+80+40={180}^{\circ }\end{array}$$

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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