9.6 补角和余角

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  补角和余角

  

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  Paul is fixing a house. The wood frames have already been placed and he is almost done with one of the rooms. He looks around the room and sees that along each wall, the boards are placed at a 90 degree  angle . There are pairs of 90 degree angles next to each other. He knows that there is a special name for the angle pair, but he can't remember the name. What type of angle pair is formed by two 90 degree angles?
  ::保罗正在修房子。木框已经布置好了, 他几乎把其中一个房间搞定了。 他环顾房间四周, 并看到每面墙边的板被放置在90度角上。 双边有90度角。 他知道角对有一个特殊的名称, 但他不记得名字了。 什么样的角对以2 90度角度组成?

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  In this concept, you will learn about complementary and  .
  
  ::在这一概念中,你们将了解互补和互补。

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  Supplementary and Complementary Angles
  ::补充和补充角

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  Sometimes, two angles are a part of each other or are connected to each other. These angles are called  angle pairs .
  ::有时,两个角度是彼此的一部分,或者互相连接。这些角度被称为角度对。

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  Let's look at two special types of angle pairs,  supplementary angles  or  complementary angles .
  ::让我们来看看两种特殊的角对, 补充角度或互补角度。

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  Supplementary angles are two angles whose sum is equal to  $\begin{array}{r}{180}^{\circ }\end{array}$ . In other words when you add the  measure  of one angle in the pair with the other angle in the pair, they equal 180 degrees.
  ::补充角度是两个角度,其总和等于 180 。换句话说,当您在对配中加上一个角度的量度和对中另一个角度时,它们等于 180 度。

  

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  These two angles are supplementary because together they form a straight  line . You can also tell that they are supplementary because when you add their angle measures the sum is equal to 180 degrees.
  ::这两个角度是补充性的,因为它们共同形成一条直线。您也可以告诉它们是补充性的,因为当您添加角度时,其和数等于180度。

  $\begin{array}{r}120+60={180}^{\circ }\end{array}$

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  Complementary angles are a pair of angles whose sum is  $\begin{array}{r}{90}^{\circ }\end{array}$ . Here is an example of complementary angles.
  ::补充角度是一对角度,其总和为 90\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\Q\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  

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  If you add up the two angle measures, the sum is equal to 90 degrees. Therefore, the two angles are complementary.
  ::如果将两个角度相加,该数值等于90度。因此,这两个角度是互补的。

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  You can find missing angle measures by using information about supplementary and complementary angles.
  ::使用补充和互补角度的信息,您可以找到缺失的角度量。

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  Let's look at an example.
  ::让我们举个例子。

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  Find the measure of  $\begin{array}{r}x\end{array}$ .
  ::查找 x 的度量。

  

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  First, identify that these two angles are supplementary. They form a straight line. The total number of degrees in a straight line is 180. Therefore, you can write the following equation to solve.
  ::首先, 确定这两个角度是补充性的。 它们构成一条直线。 直线的总度为 180。 因此, 您可以写入下面的方程来解析 。

  $$\begin{array}{rl}130+x& =180\\ x& ={50}^{\circ }\end{array}$$

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  The missing angle is equal to  $\begin{array}{r}{50}^{\circ }\end{array}$ .
  ::缺失角度等于 50 。

  

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  Here the two angles are complementary. Therefore the sum of the two angles is equal to 90 degrees. Write an equation and solve for the missing angle measure.
  ::在这里,两个角度是互补的。 因此, 两个角度的总和等于 90 度。 写一个方程式, 并解决丢失角度的测量 。

  $$\begin{array}{rl}75+x& =90\\ x& ={15}^{\circ }\end{array}$$

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  The missing angle measure is equal to  $\begin{array}{r}{50}^{\circ }\end{array}$ .
  ::缺失的角度量等于 50 。

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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 Paul and the house.
  ::早些时候,你得到一个问题 关于保罗和房子。

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  He has an angle pair that is formed by two 90 degree angles. What type of angle pair is formed when two 90 degree angles are combined?
  ::他有一个角度对, 由两个 90 度角度组成。 当两个 90 度角度结合时, 哪种角度对会形成 ?

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  First,add the two angles to calculate their sum.
  ::首先,添加两个角度来计算其总和。

  90 + 90 = 180

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  Then, identify which angle pair adds up to 90 degrees.
  ::然后,确定哪个角度对加起来等于90度。

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  Supplementary angles
  ::补充角度

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  The answer is that the pair are supplementary angles.
  ::答案是,对子是补充角度。

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  Example 2
  ::例2

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  Identify the angle pair as complementary or supplementary.
  ::将角对确定为补充或补充。

  

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  First, add the two angles to calculate their sum.
  ::首先,添加两个角度来计算其总和。

  75 + 105 = 180

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  Then, identify which angle pair adds up to 180 degrees.
  ::然后,确定哪个角度对加起来等于180度。

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  Supplementary angles
  ::补充角度

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  The answer is supplementary angles.
  ::答案是补充角度。

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  Write whether each pair is complementary or supplementary.
  ::写下每一对是补充性的还是补充性的。

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  Example 3
  ::例3

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  Identify the angle pair as complementary or supplementary.
  ::将角对确定为补充或补充。

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  If the sum of the angles is equal to 180 degrees.
  ::如果角度之和等于180度,则该角度之和等于180度。

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  First, identify which angle pair adds up to 180 degrees.
  ::首先,确定哪个角度对加起来等于180度。

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  Supplementary angles
  ::补充角度

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  The answer is supplementary angles.
  ::答案是补充角度。

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  Example 4
  ::例4

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  Identify the angle pair as complementary or supplementary.
  ::将角对确定为补充或补充。

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  If one angle is 60 degrees and the other angle is 120 degrees.
  ::如果一个角度为60度,另一个角度为120度。

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  First, add the two angles together to find the sum.
  ::首先,将两个角度放在一起寻找总和。

  60 + 120 = 180

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  Then, identify which angle pair adds up to 180 degrees.
  ::然后,确定哪个角度对加起来等于180度。

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  Supplementary angles
  ::补充角度

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  The answer is supplementary angles.
  ::答案是补充角度。

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  Example 5
  ::例5

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  Identify the angle pair as complementary or supplementary.
  ::将角对确定为补充或补充。

  

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  First, add the two angles to calculate their sum.
  ::首先,添加两个角度来计算其总和。

  30 + 60 = 90

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  Then, identify which angle pair adds up to 90 degrees.
  ::然后,确定哪个角度对加起来等于90度。

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  Complementary angles
  ::补充角度

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  The answer is that the pair are complementary angles.
  ::答案是,对子是互补的角度。

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  Review
  ::回顾

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  Identify each angle pair as supplementary or complementary angles.
  ::将每个角对标为补充或补充角度。

  1. 
  2. 
  3. 
  4. 

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  Use what you have learned about complementary and supplementary angles to answer the following questions.
  ::利用你学到的关于补充和补充角度的知识回答下列问题。

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  5. If two angles are complementary, then their sum is equal to _________ degrees.
     ::如果两个角度是互补的,那么它们的总和等于°。
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  6. If two angles are supplementary, then their sum is equal to ________ degrees.
     ::如果两个角度是补充的,则其总和等于°。
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  7. True or false. If one angle is  $\begin{array}{r}{120}^{\circ }\end{array}$ , then the second angle must be equal to  $\begin{array}{r}{90}^{\circ }\end{array}$  for the angles to be supplementary.
     ::如果一个角度是 120 ,那么第二个角度必须等于 90 角才能补充角度。
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  8. True or false. If the angles are supplementary and one angle is equal to  $\begin{array}{r}{100}^{\circ }\end{array}$ , then the other angle must be equal to  $\begin{array}{r}{80}^{\circ }\end{array}$ .
     ::如果角度是补充的,而一个角度等于100 ,那么另一个角度必须等于80 。
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  9. True or false. The sum of complementary angles is  $\begin{array}{r}{180}^{\circ }\end{array}$ .
     ::无论真假,互补角度的总和是180
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  10. True or false. The sum of supplementary angles is  $\begin{array}{r}{90}^{\circ }\end{array}$ .
      ::补充角度的总和是90 + + + + + + + = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = ==== === === ======= =======================================================================================================================================================================================================================================================================================================================================================================================================

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  Identify whether the angles are supplementary, complementary or neither based on the angle measures.
  ::确定角度是否是补充性的、补充性的,还是根据角度计量。

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  11. One angle is 50 degrees. The other angle is 130 degrees.
      ::一个角度是50度 另一个角度是130度
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  12. One angle is 30 degrees. The other angle is 60 degrees.
      ::一个角度是30度 另一个角度是60度
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  13. One angle is 112 degrees. The other angle is 70 degrees.
      ::一个角为112度,另一个角为70度。
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  14. One angle is 110 degrees. The other angle is 50 degrees.
      ::一个角度为110度,另一个角度为50度。
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  15. One angle is 35 degrees. The other angle is 55 degrees.
      ::一个角度是35度 另一个角度是55度

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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