6.1 识别角对角

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  识别角对角

  

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  Joey is a member of the Search and Rescue team. They receive a lot of calls to find people who get lost while hiking on the mountain trails. His German shepherd dog has a keen sense of smell so Joey wants to train him to follow the scent trail left by a human being. He plans out the following tracking trail to use for training Rufus.
  ::Joey是搜救队的成员。他们接到许多电话,寻找在山路上徒步旅行时迷路的人。他的德国牧狗嗅觉敏锐,所以Joey想训练他跟踪一个人留下的气味。他计划用下面的跟踪路径来训练Rufus。

  

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  Joey looks at the tracking trail and wonders what kind of angle pairs he has created in the trail. How can he figure out the relationship between the angle pairs indicated on the trail?
  ::Joey看着追踪线索,并想知道他在追踪线索中创造了什么样的角度对。他如何能找出追踪线索上显示的角度对之间的关系呢?

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  In this concept, you will learn to identify angle pairs as being complementary, supplementary or neither.
  ::在这一概念中,你们将学会确定角对是互补的、补充的或两者兼而有之。

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  Angle Pairs
  ::角对角

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  An angle is the intersection of two non-collinear rays at a common endpoint . The rays are called the sides or arms of the angle and the common endpoint is called the vertex . Here is an angle with its parts labelled.
  ::一个角度是共同端点上两个非双向线的交叉点。 光线被称为角的侧面或臂, 共同端点被称为顶点。 这里有一个角, 其部分被贴上标签 。

  

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  A single angle that measures 180° is called a straight angle . The following diagram shows a straight angle with its vertex B.
  ::测量 180 ° 的单角称为直角。下图显示直角及其顶点 B 。

  

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  If sides BA and BC were extended such that they intersected at B, then the diagram would look like this:
  ::如果BA和BC两边的延伸使其在B交叉,那么图表将看起来是这样的:

  

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  The intersecting lines have created four angles with each angle having the intersection point its vertex. The angles are numbered $\begin{array}{r}\mathrm{\angle }1\end{array}$ , $\begin{array}{r}\mathrm{\angle }2\end{array}$ , $\begin{array}{r}\mathrm{\angle }3\end{array}$ , $\begin{array}{r}\mathrm{\angle }4\end{array}$ . The intersection of the two lines has created angle pairs. Angle pairs are two angles that share a unique relationship. The angle pairs in this diagram have a measure that equals 180°  which is the measure of a straight angle. Angle pairs that have a sum of 180° are called .
  ::交叉线创建了四个角度, 每个角度都有交叉点的顶点。 角编号为 @ 1, @ 2, @ 3, @ 4。 两行的交叉点创建了角度对齐。 角度对齐是两个共享独特关系的角度。 本图表中的角度对齐具有等于180°的测量尺度, 即直角的测量尺度。 有180°和角的角对被调用 。

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  In the above diagram the supplementary angles are:
  ::在上述图表中,补充角度如下:

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  $\begin{array}{r}\mathrm{\angle }1\end{array}$  and $\begin{array}{r}\mathrm{\angle }3\end{array}$ , $\begin{array}{r}\mathrm{\angle }1\end{array}$  and $\begin{array}{r}\mathrm{\angle }4\end{array}$ , $\begin{array}{r}\mathrm{\angle }2\end{array}$  and $\begin{array}{r}\mathrm{\angle }3\end{array}$ , $\begin{array}{r}\mathrm{\angle }2\end{array}$  and $\begin{array}{r}\mathrm{\angle }4\end{array}$ . Therefore, 
  ::第1和第3、第1和第4、第2和第3、第2和第4节。

  $\begin{array}{r}\mathrm{\angle }1+\mathrm{\angle }3={180}^{\circ },\mathrm{\angle }1+\mathrm{\angle }4={180}^{\circ },\mathrm{\angle }2+\mathrm{\angle }3={180}^{\circ },\mathrm{\angle }2+\mathrm{\angle }4={180}^{\circ }\end{array}$

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  The following diagram shows supplementary angles:
  ::下图显示补充角度:

  

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  A single angle that measures 90° is called a right angle . The following diagram shows a right angle with its vertex B.
  ::测量 90 度的单个角度被称为右角度。下图显示右角度及其顶点 B 。

  

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  . $\begin{array}{r}\mathrm{\angle }1+\mathrm{\angle }2={90}^{\circ }\end{array}$ . Therefore, $\begin{array}{r}\mathrm{\angle }2\end{array}$  and $\begin{array}{r}\mathrm{\angle }1\end{array}$ . In the following diagram the are complementary angles The angle pair shown in the following diagram have a measure equal to 90° which is the measure of a right angle. Angle pairs that have a sum of 90° are called
  ::12=90。因此,2和1。在下图中,是互补角度。下图中显示的角对的测量度等于90°,即右角的测量度。有90°和90°之和的角对被称为。

  

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  The following diagram shows complementary angles:
  ::下图显示了互补角度:

  

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  Now you can identify supplementary or complementary angles based on the sum of the angle pairs. Identify the following angle pairs as either complementary or supplementary.
  ::现在,您可以根据角度对数之和来识别补充或互补角度。将以下角度对数指定为补充或补充。

  

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  In the first diagram the sum of the angle pair is $\begin{array}{r}{30}^{\circ }+{60}^{\circ }={90}^{\circ }\end{array}$ . Therefore, the diagram shows complementary angles.
  ::在第一个图表中,角度对数的总和是 306090。因此,图表显示了互补角度。

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  In the second diagram the sum of the angle pair is $\begin{array}{r}{125}^{\circ }+{55}^{\circ }={180}^{\circ }\end{array}$ . Therefore, the diagram shows supplementary angles.
  ::在第二个图表中,角度对数的总和是12555180。因此,该图表显示了补充角度。

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  In the third diagram the sum of the angle pair is $\begin{array}{r}{34}^{\circ }+{87}^{\circ }={121}^{\circ }\end{array}$ . Therefore, the diagram shows neither supplementary nor complementary angles.
  ::在第三个图表中,角度对数的总和是3487121。因此,该图表既不显示补充角度,也不显示互补角度。

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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  Joey who was training his dog. He has marked three pairs of angles on his tracking trail and is wondering what the relationship between the marked angle pairs is.
  ::早些时候,你曾听说乔伊在训练他的狗时遇到一个问题。他在他的追踪路径上标记了三对角度,并想知道这些角度之间有什么关系。

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  First, remember if the sum of the measures of the angles is 90° then the angles are complementary. If the sum of the measures of the angles is 180° then the angles are supplementary.
  ::首先,记住角的测量总和是否为90°,角是互补的。如果角度的测量总和为180°,角是补充的。

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  Next, identify the first angle pair marked on the tracking trail to the right of the ‘Start’ point.
  ::下一点, 确定在跟踪路径上标记的第一个角度对到 ` 启动 ' 点右侧的角对 。

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  This angle pair has a sum of 90° since the right angle is indicated by the square . These angles are complementary.
  ::由于正方形显示右角,此角对的宽度为90°。这些角是互补的。

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  Next, identify the second angle pair marked on the first downward trail of the training track. This angle pair has a sum of 180° since the angles are on a straight line . These angles are supplementary.
  ::下一步, 标记训练轨迹第一个下行的第二个角度对 。 此角度对的总和为 180 °, 因为角度在直线上。 这些角度是补充的 。

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  Next, identify the third angle pair marked on the third downward trail of the training track. This angle pair has a sum of 180° since the angles are on a straight line. These angles are supplementary.
  ::下一步, 标记训练轨迹第三向下行的第三个角度对 。 此角度对的总和为 180 °, 因为角度在直线上。 这些角度是补充的 。

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  Example 2
  ::例2

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  Identify the following angles as being complementary, supplementary or neither.
  ::确定以下角度为互补、补充或两者兼而有之。

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  #1 - The sum of the angle pair is $\begin{array}{r}{23}^{\circ }+{67}^{\circ }={90}^{\circ }\end{array}$ . The angles are complementary.
  ::# 1 - 角度对对的总和是 236790。 角度是互补的 。

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  #2 - The sum of the angle pair is $\begin{array}{r}{115}^{\circ }+{112}^{\circ }={227}^{\circ }\end{array}$ . The sum of the angle pair $\begin{array}{r}\ne {90}^{\circ }\end{array}$  and $\begin{array}{r}\ne {180}^{\circ }\end{array}$ . The angle pair is neither complementary nor supplementary.
  ::# 2 - 角度对对的和值是 115112227。 角度对对的和值是 @ 90和180。 角度对既不是补充也不是补充 。

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  #3 - The sum of the angle pair is $\begin{array}{r}{126}^{\circ }+{54}^{\circ }={180}^{\circ }\end{array}$ . The angles are supplementary.
  ::# 3 - 角度对数的总和是 12654180。 角度是补充的 。

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  Example 3
  ::例3

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  Find the measure of two complementary angles, $\begin{array}{r}\mathrm{\angle }A\end{array}$  and $\begin{array}{r}\mathrm{\angle }B\end{array}$ , if the difference in the measure of the angle pair is 18°.
  ::查找两个互补角度(A和B)的度量,如果角度对的度量差为18°。

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  First, let ‘ $\begin{array}{r}x\end{array}$ ’ represent $\begin{array}{r}m\mathrm{\angle }A\end{array}$ .
  ::首先,让“x”代表mA。

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  Next, write an equation to model the measure of the complementary angles.
  ::接下来,写一个方程来模拟对互补角度的测量。

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  $$\begin{array}{r}m\mathrm{\angle }B+x=90\end{array}$$

  
  ::mB+x=90

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  Next, solve the equation for $\begin{array}{r}m\mathrm{\angle }B\end{array}$  by subtracting ‘ $\begin{array}{r}x\end{array}$ ’ from both sides of the equation.
  ::接下来,通过从方程式两侧减去 'x ' 来解析 mB 的方程式 。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B+x& =& 90\\ m\mathrm{\angle }B+x-x& =& 90-x\\ m\mathrm{\angle }B& =& 90-x\end{array}\end{array}$$

  
  ::mB+x=90mB+x-x=90-xmB=90-x

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  Next, write an equation to model the difference between the measures of the complementary angles.
  ::接下来,写一个方程来模拟 互补角度的测量之间的差别。

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  $$\begin{array}{r}m\mathrm{\angle }B-m\mathrm{\angle }A=18\end{array}$$

  
  ::mB-mA=18

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  Next, substitute in the expression for $\begin{array}{r}m\mathrm{\angle }B\end{array}$  and the variable for $\begin{array}{r}m\mathrm{\angle }A\end{array}$ .
  ::接下来,在 mB 和 mA 变量的表达式中替换 。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B-m\mathrm{\angle }A& =& 18\\ (90-x)-x& =& 18\end{array}\end{array}$$

  
  ::mB-mA=18(90-x)-x=18

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  Next, multiply $\begin{array}{r}1(90-x)\end{array}$  to clear the parenthesis.
  ::下一步,乘1(90-x)来清除括号。

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  $$\begin{array}{r}\begin{array}{rcl}1(90-x)-x& =& 18\\ 90-x-x& =& 18\end{array}\end{array}$$

  
  ::1,90-x)-x=1890-x-x=18

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  Next, simplify the equation.
  ::接下来,简化方程

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  $$\begin{array}{r}\begin{array}{rcl}90-x-x& =& 18\\ 90-2x& =& 18\end{array}\end{array}$$

  
  ::90-x-x=1890-2x=18

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  Next, isolate the variable by subtracting 90 from both sides of the equation. Simplify the equation.
  ::下一步, 从方程两侧减去 90 , 分离变量。 简化方程 。

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  $$\begin{array}{r}\begin{array}{rcl}90-2x& =& 18\\ 90-90-2x& =& 18-90\\ -2x& =& -72\end{array}\end{array}$$

  
  ::90-2x=1890-90-2x=18-90-2x=90-2x=72

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  Then, solve the equation for ‘ $\begin{array}{r}x\end{array}$ ’ by dividing both sides of the equation by -2.
  ::然后,通过将方程式两侧除以-2, 解决`x ' 方程式的方程式。

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  $$\begin{array}{r}\begin{array}{rcl}-2x& =& -72\\ \frac{\stackrel{1}{\cancel{-2}}x}{\cancel{-2}}& =& \frac{-72}{-2}\\ x& =& 36\end{array}\end{array}$$

  
  ::− 2x________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  The answer is $\begin{array}{r}x=36\end{array}$ .
  ::答案是 x=36。

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  The $\begin{array}{r}m\mathrm{\angle }A={36}^{\circ }\end{array}$ .
  ::3 6 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

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  Then, use the value to determine $\begin{array}{r}m\mathrm{\angle }B\end{array}$ .
  ::然后,用这个值来确定 mB。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B& =& 90-x\\ m\mathrm{\angle }B& =& 90-36\\ m\mathrm{\angle }B& =& {54}^{\circ }\end{array}\end{array}$$

  
  ::mB=90-xmB=90-36mB=54

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  Example 4
  ::例4

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  Find the measure of two supplementary angles, $\begin{array}{r}\mathrm{\angle }A\end{array}$  and $\begin{array}{r}\mathrm{\angle }B\end{array}$ , if the difference in the measure of the angle pair is 32°.
  ::查找两个补充角度(A和B)的度量,如果角对的度量差为32°。

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  First, let ‘ $\begin{array}{r}x\end{array}$ ’ represent $\begin{array}{r}m\mathrm{\angle }A\end{array}$ .
  ::首先,让“x”代表mA。

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  Next, write an equation to model the measure of the complementary angles.
  ::接下来,写一个方程来模拟对互补角度的测量。

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  $$\begin{array}{r}m\mathrm{\angle }B+x=180\end{array}$$

  
  ::mB+x=180

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  Next, solve the equation for $\begin{array}{r}m\mathrm{\angle }B\end{array}$  by subtracting ‘ $\begin{array}{r}x\end{array}$ ’ from both sides of the equation.
  ::接下来,通过从方程式两侧减去 'x ' 来解析 mB 的方程式 。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B+x& =& 180\\ m\mathrm{\angle }B+x-x& =& 180-x\\ m\mathrm{\angle }B& =& 180-x\end{array}\end{array}$$

  
  ::mB+x=180mB+x-x=180-xmB=180-x

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  Next, write an equation to model the difference between the measures of the complementary angles.
  ::接下来,写一个方程来模拟 互补角度的测量之间的差别。

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  $$\begin{array}{r}m\mathrm{\angle }B-m\mathrm{\angle }A=32\end{array}$$

  
  ::mB-mA=32

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  Next, substitute in the expression for $\begin{array}{r}m\mathrm{\angle }B\end{array}$  and the variable for $\begin{array}{r}m\mathrm{\angle }A\end{array}$ .
  ::接下来,在 mB 和 mA 变量的表达式中替换 。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B-m\mathrm{\angle }A& =& 32\\ (180-x)-x& =& 32\end{array}\end{array}$$

  
  ::mB-mA=32(180-x)-x=32

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  Next, multiply $\begin{array}{r}1(180-x)\end{array}$  to clear the parenthesis.
  ::下一步,乘1(180-x)来清除括号。

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  $$\begin{array}{r}\begin{array}{rcl}1(180-x)-x& =& 32\\ 180-x-x& =& 32\end{array}\end{array}$$

  
  ::1(180)-x=32180-x-x=32

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  Next, simplify the equation.
  ::接下来,简化方程

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  $$\begin{array}{r}\begin{array}{rcl}180-x-x& =& 32\\ 180-2x& =& 32\end{array}\end{array}$$

  
  ::180 - 2x=32

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  Next, isolate the variable by subtracting 90 from both sides of the equation. Simplify the equation.
  ::下一步, 从方程两侧减去 90 , 分离变量。 简化方程 。

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  $$\begin{array}{r}\begin{array}{rcl}180-2x& =& 32\\ 180-180-2x& =& 32-180\\ -2x& =& -148\end{array}\end{array}$$

  
  ::180-180-2x=32-180-2x=32-180-2x_2x_____________________________________________________________________________________________________________________________________________________________________________________

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  Then, solve the equation for ‘ $\begin{array}{r}x\end{array}$ ’ by dividing both sides of the equation by -2.
  ::然后,通过将方程式两侧除以-2, 解决`x ' 方程式的方程式。

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  $$\begin{array}{r}\begin{array}{rcl}-2x& =& -148\\ \frac{\stackrel{1}{\cancel{-2}}x}{\cancel{-2}}& =& \frac{-148}{-2}\\ x& =& 74\end{array}\end{array}$$

  The answer is $\begin{array}{r}x=74\end{array}$ .
  ::-2x 148 -21x-2 148 -2x=74 答案是 x=74 。

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  The $\begin{array}{r}m\mathrm{\angle }A={74}^{\circ }\end{array}$ .
  ::摩阿=74。

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  Then, use the value to determine $\begin{array}{r}m\mathrm{\angle }B\end{array}$ .
  ::然后,用这个值来确定 mB。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }B& =& 180-x\\ m\mathrm{\angle }B& =& 180-74\\ m\mathrm{\angle }B& =& {106}^{\circ }\end{array}\end{array}$$

  
  ::mB=180-xm_B=180-74m_B=106__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Review
  ::回顾

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  If the following angle pairs are complementary, then what is the measure of the missing angle?
  ::如果以下角度对是互补的,那么缺失角度的度量是多少?

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  1.  $\begin{array}{r}\text{If }\mathrm{\angle }A={55}^{\circ }\text{ then }\mathrm{\angle }B=?\end{array}$
  ::1. 如果A=55,那么B=?

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  2.  $\begin{array}{r}\text{If }\mathrm{\angle }C={33}^{\circ }\text{ then }\mathrm{\angle }D=?\end{array}$
  ::2. 如果C=33,那么D=?

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  3.  $\begin{array}{r}\text{If }\mathrm{\angle }E={83}^{\circ }\text{ then }\mathrm{\angle }F=?\end{array}$
  ::3. 如果E=83,那么F=?

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  4.  $\begin{array}{r}\text{If }\mathrm{\angle }G={73}^{\circ }\text{ then }\mathrm{\angle }H=?\end{array}$
  ::4. 如果G=73,那么H=4?

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  If the following angle pairs are supplementary, then what is the measure of the missing angle?
  ::如果以下角度对是补充的,那么缺失角度的量度是多少?

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  5.  $\begin{array}{r}\text{If }\mathrm{\angle }A={10}^{\circ }\text{ then }\mathrm{\angle }B=?\end{array}$
  ::5. 如果A=10,那么B=?

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  6.  $\begin{array}{r}\text{If }\mathrm{\angle }A={80}^{\circ }\text{ then }\mathrm{\angle }B=?\end{array}$
  ::6. 如果A=80,那么B=?

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  7.  $\begin{array}{r}\text{If }\mathrm{\angle }C={30}^{\circ }\text{ then }\mathrm{\angle }F=?\end{array}$
  ::7. 如果C=30,那么F=7?

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  8.  $\begin{array}{r}\text{If }\mathrm{\angle }D={15}^{\circ }\text{ then }\mathrm{\angle }E=?\end{array}$
  ::8. 如果"D=15",那么"E=8"?

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  9.  $\begin{array}{r}\text{If }\mathrm{\angle }M={112}^{\circ }\text{ then }\mathrm{\angle }N=?\end{array}$
  ::9. 如果M=112,那么N=?

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  10.  $\begin{array}{r}\text{If }\mathrm{\angle }O=2^{\circ }\text{ then }\mathrm{\angle }P=?\end{array}$
  ::10. 如果"O=2",那么"P"=?

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  State whether the following statements are True or False.
  ::说明以下声明是真实的还是虚假的。

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  11. Complementary angles are equal to 180°.
  ::11. 补充角度等于180°。

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  12. Complementary angles are equal to 90°.
  ::12. 补充角度等于90度。

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  13. Supplementary angles are equal to 90°.
  ::13. 补充角度等于90度。

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  14. Supplementary angles are equal to 180°.
  ::14. 补充角度等于180°。

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  15. Angle pairs less than 90 degrees are neither supplementary nor complementary.
  ::15. 90度以下的角对子既不是补充,也不是补充。

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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 。