1. 5 角度测量

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  角度测量

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  Angles
  ::角度角

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  An angle is formed when two rays have the same endpoint . The vertex is the common endpoint of the two rays that form an angle. The sides are the two rays that form an angle.
  ::当两个射线的端点相同时,就会形成角。顶点是形成角的两个射线的共同端点。两边是形成角的两个射线。

  

  Label It	Say It
  $\begin{array}{r}\mathrm{\angle }ABC\end{array}$	Angle $\begin{array}{r}ABC\end{array}$
  $\begin{array}{r}\mathrm{\angle }CBA\end{array}$	Angle $\begin{array}{r}CBA\end{array}$

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  The vertex is $\begin{array}{r}B\end{array}$ and the sides are $\begin{array}{r}\stackrel{\to }{BA}\end{array}$ and $\begin{array}{r}\stackrel{\to }{BC}\end{array}$ . Always use three letters to name an angle, $\begin{array}{r}\mathrm{\angle }\end{array}$ SIDE-VERTEX-SIDE.
  ::顶端是 B , 侧面是 B 和 BB 和 BC 。 总是用三个字母来命名角度, 也就是 SEDE- VERTEX- SIDE 。

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  Angles are measured with something called a protractor . A protractor is a measuring device that measures how “open” an angle is. Angles are measured in degrees and are labeled with a $\begin{array}{r}{}^{\circ }\end{array}$ symbol. For now, angles are always positive.
  ::角度是用一个叫做减速器的物体测量的。 减速器是一个测量装置, 测量角度“ 打开” 的大小。 角度以度测量, 并用 {{{} 符号标注。 目前, 角度总是正的 。

  

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  There are two sets of measurements, one starting on the left and the other on the right side of the protractor. Both go around from $\begin{array}{r}{0}^{\circ }\end{array}$ to $\begin{array}{r}{180}^{\circ }\end{array}$ . When measuring angles, you can line up one side with $\begin{array}{r}{0}^{\circ }\end{array}$ , and see where the other side hits the protractor. The vertex lines up in the middle of the bottom line.
  ::有两组测量,一组从左侧开始,另一组从减速器右侧开始。两组都从0到180。当测量角度时,你可以用0排到一边,看看另一边击中减速器的位置。顶点在底线中间。

  

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  Note that if you don't line up one side with $\begin{array}{r}{0}^{\circ }\end{array}$ , the angle's measure will be the difference of the degrees where the sides of the angle intersect the protractor.
  ::注意,如果你不与 0 和 0 相排, 角度的测量将是角的两侧交叉点的度差。

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  Sometimes you will want to draw an angle that is a specific number of degrees. Follow the steps below to draw a $\begin{array}{r}{50}^{\circ }\end{array}$ angle with a protractor:
  ::有时您会想要绘制一个角度, 该角度是一定的度数。 跟随下面的步骤, 用一个减速器绘制一个 50 角度 :

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  1. Start by drawing a horizontal line across the page, 2 in long.
     ::开始在页面上画一条水平线, 2 长 。

  

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  2. Place an endpoint at the left side of your line.
     ::在行的左侧设置端点。
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  3. Place the protractor on this point , such that the line passes through the $\begin{array}{r}{0}^{\circ }\end{array}$ mark on the protractor and the endpoint is at the center . Mark $\begin{array}{r}{50}^{\circ }\end{array}$ on the appropriate scale.
     ::将减速器放在此点上, 以便线条通过减速器上的 0 标记, 终点在中间 。 马克 50 在适当的比例尺上 。

  

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  4. Remove the protractor and connect the vertex and the $\begin{array}{r}{50}^{\circ }\end{array}$ mark.
     ::删除减速器,连接顶点和50标记。

  

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  This process can be used to draw any angle between $\begin{array}{r}{0}^{\circ }\end{array}$ and $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::此进程可用于在 0 和 180 之间绘制角 。

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  When two smaller angles form to make a larger angle, the sum of the measures of the smaller angles will equal the measure of the larger angle. This is called the Angle Addition Postulate . So, if $\begin{array}{r}B\end{array}$ is on the interior of $\begin{array}{r}\mathrm{\angle }ADC\end{array}$ , then

  $$\begin{array}{r}m\mathrm{\angle }ADC=m\mathrm{\angle }ADB+m\mathrm{\angle }BDC\end{array}$$

  .
  ::当两个较小角度形成一个更大的角度时,较小角度的测量量和大小角度的测量量将等于较大角度的测量量。这称为“角添加假设”。因此,如果B位于 ADC 的内部,那么mADC=mADB+MBDC。

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  Examples
  ::实例

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  Example 1
  ::例1

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  How many angles are in the picture below? Label each one.
  ::下面的图片中有多少角度?

  

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  There are three angles with vertex $\begin{array}{r}U\end{array}$ . It might be easier to see them all if we separate them.
  ::有三个角度与 U 顶点。 如果我们分开它们, 可能更容易看到它们。

  

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  So, the three angles can be labeled, $\begin{array}{r}\mathrm{\angle }XUY\end{array}$ (or $\begin{array}{r}\mathrm{\angle }YUX\end{array}$ ), $\begin{array}{r}\mathrm{\angle }YUZ\end{array}$ (or $\begin{array}{r}\mathrm{\angle }ZUY\end{array}$ ), and $\begin{array}{r}\mathrm{\angle }XUZ\end{array}$ (or $\begin{array}{r}\mathrm{\angle }ZUX\end{array}$ ).
  ::因此,三个角度可以贴上标签,即XUY(或YUX)、YUZ(或UY)和XUZ(或UX)。

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  Example 2
  ::例2

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  Measure the three angles from Example 1, using a protractor.
  ::使用减速器从例1中测量三个角度。

  

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  Just like in Example 1, it might be easier to measure these three angles if we separate them.
  ::与例1一样,如果我们将这三个角度分开,衡量这三个角度可能比较容易。

  

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  With measurement, we put an $\begin{array}{r}m\end{array}$ in front of the $\begin{array}{r}\mathrm{\angle }\end{array}$ sign to indicate measure. So, $\begin{array}{r}m\mathrm{\angle }XUY={84}^{\circ },m\mathrm{\angle }YUZ={42}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }XUZ={126}^{\circ }\end{array}$ .
  ::用测量方法,我们把一米放在 符号前表示测量。 所以, mXUY=84, mY42, 和 mXU126。

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  Example 3
  ::例3

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  What is the measure of the angle shown below?
  ::下面所显示的角度的量度是多少?

  

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  This angle is not lined up with $\begin{array}{r}{0}^{\circ }\end{array}$ , so use subtraction to find its measure. It does not matter which scale you use, as long as you are consistent.
  ::此角度不与 0\\\ 相排, 所以使用减法来找到它的量度。 只要您是一致的, 您使用哪个比例并不重要 。

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  Inner scale: $\begin{array}{r}{140}^{\circ }-{15}^{\circ }={125}^{\circ }\end{array}$
  ::内部规模:14015125

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  Outer scale: $\begin{array}{r}{165}^{\circ }-{40}^{\circ }={125}^{\circ }\end{array}$
  ::外部规模:16540125

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  Example 4
  ::例4

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  Use a protractor to measure $\begin{array}{r}\mathrm{\angle }RST\end{array}$ below.
  ::使用减量器测量以下的 RST 。

  

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  Lining up one side with $\begin{array}{r}{0}^{\circ }\end{array}$ on the protractor, the other side hits $\begin{array}{r}{100}^{\circ }\end{array}$ .
  ::一边靠着减速器,另一边点击100

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  Example 5
  ::例5

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  What is $\begin{array}{r}m\mathrm{\angle }QRT\end{array}$ in the diagram below?
  ::下面的图中是什么?

  

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  Using the Angle Addition Postulate, $\begin{array}{r}m\mathrm{\angle }QRT={15}^{\circ }+{30}^{\circ }={45}^{\circ }\end{array}$ .
  ::使用 angle 附加 假设, mRT= 153045。

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  Review
  ::回顾

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  1. What is $\begin{array}{r}m\mathrm{\angle }LMN\end{array}$ if $\begin{array}{r}m\mathrm{\angle }LMO={85}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }NMO={53}^{\circ }\end{array}$ ?
  ::1. 如果MLMO=85和MNMO=53,什么是MLMN?

  

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  2. If $\begin{array}{r}m\mathrm{\angle }ABD={100}^{\circ }\end{array}$ , find $\begin{array}{r}x\end{array}$ .
  ::2. 如果 mABD=100,则找x。

  

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  For questions 3-6, determine if the statement is true or false.
  ::问题3-6,确定声明是真实的还是虚假的。

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  3. For an angle $\begin{array}{r}\mathrm{\angle }ABC,C\end{array}$ is the vertex.
     ::对于角度 ABC, C 是顶点 。
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  4. For an angle $\begin{array}{r}\mathrm{\angle }ABC,\overline{AB}\end{array}$ and $\begin{array}{r}\overline{BC}\end{array}$ are the sides.
     ::以ABC为角度,AB和BC是两边的
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  5. The $\begin{array}{r}m\end{array}$ in front of $\begin{array}{r}m\mathrm{\angle }ABC\end{array}$ means measure.
     ::M在MABC前面意味着量度。
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  6. The Angle Addition Postulate says that an angle is equal to the sum of the smaller angles around it.
     ::Agle Apple Apple Postresult 表示角等于其周围较小角度的和。

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  For 7-12, draw the angle with the given degree, using a protractor and a ruler.
  ::对于 7-12, 用给定度绘制角度, 使用减速器和标尺 。

  7. $\begin{array}{r}{55}^{\circ }\end{array}$
  8. $\begin{array}{r}{92}^{\circ }\end{array}$
  9. $\begin{array}{r}{178}^{\circ }\end{array}$
  10. $\begin{array}{r}{5}^{\circ }\end{array}$
  11. $\begin{array}{r}{120}^{\circ }\end{array}$
  12. $\begin{array}{r}{73}^{\circ }\end{array}$

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  For 13-16, use a protractor to determine the measure of each angle.
  ::对于13-16,使用减号来确定每个角度的量度。

  13. 
  14. 
  15. 
  16. 

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  Solve for $\begin{array}{r}x\end{array}$ .
  ::解决x。

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  17. $\begin{array}{r}m\mathrm{\angle }ADC={56}^{\circ }\end{array}$
      ::mADC=56

  

   

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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 。