6.2 识别相邻和垂直角

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  识别相邻和垂直角

  

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  Susan has taken a keen interest in Geometry and wants to expand her knowledge of angles . While looking through a magazine she saw the following picture:
  ::Susan对几何学非常感兴趣,想扩大对角度的了解。

  

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  Susan knew the angles had a relationship but she couldn’t remember what they were called or how she could use the information to figure out the size of the angles.
  ::Susan知道这些角度有某种关系, 但她不记得这些角度叫什么,

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  How can Susan use the given measures of the angles to find the measure of each angle in degrees?
  ::Susan如何使用给定角度的量度来找到每个角度的度量?

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  In this concept, you will learn to identify adjacent and .
  ::在这一概念中,你将学会识别相邻和.。

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  Adjacent and Vertical Angles
  ::相邻和垂直角

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  When two straight lines intersect each other, four angles are created such that the point of intersection is the vertex for each angle. If two of the angles have a common vertex and share a common side they are called adjacent angles . The adjacent angles formed by two intersecting lines are supplementary which means the sum of their measures is $\begin{align*}180^\circ\end{align*}$ .
  ::当两条直线相互交错时,将创建四个角度,使交叉点为每个角度的顶点。如果两个角度有一个共同的顶点并有一个共同的侧面,它们就被称为相邻角度。两个交叉线形成的相邻角度是补充性的,这意味着其测量量的总和是180\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

  

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  $\begin{align*}\angle 2 \end{align*}$  and $\begin{align*}\angle 4\end{align*}$  are adjacent angles. The angles are next to each other, have common vertex and share the common side.  $\begin{align*}m \angle 2 + m \angle 4 = 180^\circ\end{align*}$
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  These are not the only adjacent angles formed by the intersection of the lines. The remaining pairs of adjacent angles are shown below:
  ::这些角度并非线条交叉点形成的唯一相邻角度。 其余的对相邻角度如下所示:

  

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  When two lines intersect, vertical angles , which are non-adjacent angles are also formed. There are two pairs of vertical angles. These angles also have a common vertex but never share a common side. The vertical angles are opposite each other and are equal in measure.
  ::当两个线条交叉时, 垂直角度( 即非相邻角度) 也会形成。 有两对垂直角度。 这些角度也有一个共同的顶点, 但从不共享共同的侧面。 垂直角度是相互对立的, 在量上是相等的 。

  

   

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  There are two pair of vertical angles:
  ::有两对垂直角度:

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  $\begin{align*}\angle 1\end{align*}$  and  $\begin{align*}\angle 2\end{align*}$ ;  $\begin{align*}\angle 3 \end{align*}$ and  $\begin{align*}\angle 4\end{align*}$ . The  $\begin{align*}m \angle 1 = m \angle 2 \end{align*}$  and $\begin{align*}\ m \angle 3 = m \angle 4\end{align*}$ .
  ::1和2;3和4。 m1=m2和m3=m4。

  

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  Let’s apply all this information about angles to a problem.
  ::让我们将所有这些关于角度的信息应用到问题上。

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  Using the following diagram , calculate the measures of the remaining three angles.
  ::使用下图计算其余三个角度的度量。

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  First, state the relationship between $\begin{align*}\angle JKL\end{align*}$  and one other angle.
  
  $\begin{align*}\angle JKL\end{align*}$  and $\begin{align*}\angle LKN\end{align*}$  are adjacent angles.
  ::首先,请说明jKL与其他角度之间的关系。jKL和LKN是相邻角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle LKN\end{align*}$ .
  ::下一步,使用此关系来计算 LKN 的度量 。

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  $$\begin{align*}m \angle JKL + m \angle LKN = 180^\circ\end{align*}$$
  ::@ mJKL+mLKN=180

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  Next, substitute $\begin{align*}43^\circ\end{align*}$ , for the measure of $\begin{align*}\angle JKL\end{align*}$  in the equation.
  ::下面换43 以测量方程式中的jKL

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  $$\begin{align*}\begin{array}{rcl} m \angle JKL + m \angle LKN &=& 180^\circ \\ 43^\circ + m \angle LKN &=& 180^\circ \end{array}\end{align*}$$
  ::@ mJKL+mLKN=18043LKN=180

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  Next, subtract $\begin{align*}43^\circ\end{align*}$  from both sides of the equation to solve for $\begin{align*}m \angle MKN\end{align*}$ .
  ::下一步,从方程的两侧减去43 以解答 mMKN 。

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  $$\begin{align*}\begin{array}{rcl} 43^\circ + m \angle LKN &=& 180^\circ \\ 43^\circ - 43^\circ + m \angle LKN &=& 180^\circ - 43^\circ \\ {\color{red} m \angle LKN} &{\color{red}=}& {\color{red}137^\circ} \end{array}\end{align*}$$
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  First, state another relationship between $\begin{align*}\angle JKL\end{align*}$  and another angle.
  ::首先,说明ZQKL和另一个角度之间的另一关系。

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  $\begin{align*}\angle JKL\end{align*}$  and $\begin{align*}\angle MKN\end{align*}$  are vertical angles.
  ::@JKL和QMKN是垂直角度。

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  Next, use this relationship to calculate the measure of  $\begin{align*} \angle MKN\end{align*}$ .
  ::下一步,使用此关系来计算 QMKN 的度量 。

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  $$\begin{align*}m \angle JKL = m \angle MKN \end{align*}$$
  ::@ mjKL=mMKN

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  Next, substitute $\begin{align*}43^\circ\end{align*}$ , for the measure of  $\begin{align*} \angle JKL\end{align*}$  in the equation.
  ::下面换43 以测量方程式中的jKL

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  $$\begin{align*}\begin{array}{rcl} m \angle JKL &=& m \angle MKN \\ {\color{red}43^\circ} & {\color{red}=}& {\color{red} m \angle MKN} \end{array}\end{align*}$$
  ::@ mjKL=m @MKN43 @mKN @MKN @MKN @MKN

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  First, state the relationship between the remaining angle and one other angle.
  ::首先,说明剩余角与其他角之间的关系。

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  $\begin{align*}\angle LKN\end{align*}$  and $\begin{align*} \angle JKM\end{align*}$  are vertical angles.
  ::@LKN和QJKM是垂直角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle JKM\end{align*}$ .
  ::下一步,使用此关系来计算 JKM 的度量 。

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  $$\begin{align*}m \angle LKN = m \angle JKM\end{align*}$$
  ::@ mLKN=mJKM

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  Next, substitute $\begin{align*}137^\circ\end{align*}$ , for the measure of $\begin{align*}\angle LKN\end{align*}$  in the equation.
  ::接下来, 替代137, 在方程式中测量 LKN 。

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  $$\begin{align*}\begin{array}{rcl} m \angle LKN &=& m \angle JKM \\ {\color{red}137^\circ} &{\color{red}=}& {\color{red}m \angle JKM} \end{array}\end{align*}$$
  ::@ mLKN =mJKM137MQKM

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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 Susan and her interest in Geometry.
  ::早些时候,你得到一个问题 关于苏珊和她对几何的兴趣。

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  She needs to figure out the measure of two equal angles.
  ::她需要找出两个平等角度的度量

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  The two given angles are opposite each other. These angles are vertical angles.
  ::给定的两个角度对立。 这些角度是垂直角度 。

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  Susan can use the fact that vertical angles are equal in measure to calculate the measure of these angles.
  ::Susan可以使用垂直角度等量来计算这些角度的度量。

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  First, write the relation between the two vertical angles such that $\begin{align*} \angle 1 = (2x - 9)^\circ\end{align*}$  and  $\begin{align*}\angle 2 = (x+39)^\circ \end{align*}$ .
  ::首先,写下两个垂直角度之间的关系, 以便% 1 =( 2x- 9) 和% 2 =( x+39) * 。

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  $$\begin{align*}m \angle 1 = m \angle 2\end{align*}$$  
  ::m1=m%2

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  Next, substitute the values for each angle into the equation.
  ::接下来,替换方程式中每个角度的值。

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  $$\begin{align*}(2x - 9)^\circ = (x+39)^\circ\end{align*}$$
  :2x-9) (x+39)

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  Next, clear the parenthesis by multiplying both sides of the equation by one.
  ::接下来,通过将方程的两侧乘以一来清除括号。

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  $$\begin{align*}2x - 9^\circ = x+39^\circ\end{align*}$$
  ::2 - 9x+39

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  Next, add 9 to both sides of the equation to group the constants on one side of the equation.
  ::接下来,在等式的两侧加9,将常数组合在等式的一边。

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  $$\begin{align*}\begin{array}{rcl} 2x - 9^\circ &=& x+39^\circ \\ 2x - 9^\circ + 9^\circ &=& x + 39^\circ + 9^\circ \\ 2x &=& x+ 48^\circ \end{array}\end{align*}$$
  ::2 -9x+392x999999992x=x+48

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  Next, subtract ‘ $\begin{align*}x\end{align*}$ ’  from both sides of the equation to group the variables on one side of the equation.
  ::下一步,从方程的两侧减去 'x ' ,将变量组合在方程的一边。

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  $$\begin{align*}\begin{array}{rcl} 2x &=& x+ 48^\circ \\ 2x - x &=& x - x + 48^\circ \\ x &=& 48^\circ \end{array}\end{align*}$$
  ::2x=x+482x-x=x-x+48x=48

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  Use the value of the variable to calculate the measure of $\begin{align*}\angle 1\end{align*}$  and $\begin{align*}\angle 2\end{align*}$ .
  ::使用变量的值来计算 #% 1 和 # 2 的度量。

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  $$\begin{align*}\begin{array}{rcl} \angle 1 &=& (2x - 9)^\circ \ \text{and} \ \angle 2 = (x + 39)^\circ \\ \angle 1 &=& (2(48) - 9)^\circ \ \text{and} \ \angle 2 = (48 + 39)^\circ \\ \angle 1 &=& (96 - 9)^\circ \ \text{and} \ \angle 2 = (48 + 39)^\circ \\ \angle 1 &=& 87^\circ \ \text{and} \ \angle 2 = 87^\circ \end{array}\end{align*}$$
  ::1=(2x-9) 2=(x+39) 1=(2(48-9) 2=(48+39) 1=(96-9) 2=(48+39) 1=(96-9) 2=(48+39) 1=(87) 2=(87) 2=(87)

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  The measures of the two vertical angles are equal.
  ::两个垂直角度的测量是相等的。

   

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  Using the following diagram, calculate the measures of the remaining three angles.
  ::使用下图计算其余三个角度的度量。

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  First, state the relationship between  $\begin{align*} \angle 4\end{align*}$  and one other angle.
  ::首先,说明4和另一个角度之间的关系。

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  $\begin{align*}\angle 4\end{align*}$  and $\begin{align*}\angle 2\end{align*}$  are vertical angles.
  ::4和2是垂直角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle 2\end{align*}$ .
  ::接下来,使用此关系来计算 2 的度量。

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  $$\begin{align*}m \angle 4 = m \angle 2\end{align*}$$
  ::m4=m2

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  Next, substitute $\begin{align*}118^\circ \end{align*}$ , for the measure of $\begin{align*}\angle 4\end{align*}$  in the equation.
  ::接下来,替换118,以测量方程式中的 4。

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  $$\begin{align*}\begin{array}{rcl} m \angle 4 &=& m \angle 2 \\ {\color{red}118^\circ} &{\color{red}=}& {\color{red}m \angle 2} \end{array}\end{align*}$$
  ::===================================================================================================================================================================================

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  First, state another relationship between  $\begin{align*}\angle 4 \end{align*}$   and another angle.
  ::首先,说明4和另一个角度之间的另一关系。

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  $\begin{align*}\angle 4\end{align*}$  and  $\begin{align*}\angle 1 \end{align*}$   are adjacent angles.
  ::4和1是相邻角度

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  Next, use this relationship to calculate the measure of $\begin{align*} \angle 1 \end{align*}$ .
  ::下一步,使用此关系来计算 #% 1 的度量。

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  $$\begin{align*}m \angle 4 + m \angle 1 = 180^\circ\end{align*}$$
  
  ::m4+m1=180

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  Next, substitute $\begin{align*}118^\circ\end{align*}$ , for the measure of $\begin{align*}\angle 4\end{align*}$  in the equation.
  ::接下来,替换118,以测量方程式中的 4。

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  $$\begin{align*}\begin{array}{rcl} m \angle 4 + m \angle 1 &=& 180^\circ \\ 118^\circ + m \angle 1 &=& 180^\circ \end{array}\end{align*}$$
  ::m4+m1=1801181=180

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  Next, subtract $\begin{align*}118^\circ\end{align*}$  from both sides of the equation to solve for  $\begin{align*}m \angle 1 \end{align*}$ .
  ::接下来,从方程的两侧减去118 以解答 m1 。

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  $$\begin{align*}\begin{array}{rcl} 118^\circ + m \angle 1 &=& 180^\circ \\ 118^\circ - 118^\circ + m \angle 1 &=& 180^\circ - 118^\circ \\ {\color{red}m \angle 1} & {\color{red}=} & {\color{red} 62^\circ} \end{array}\end{align*}$$
  ::118 118 1 = 180 118 1 = 118 1 = 62

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  First, state the relationship between the remaining angle and one other angle.
  ::首先,说明剩余角与其他角之间的关系。

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  $\begin{align*}\angle 1\end{align*}$  and $\begin{align*}\angle 3\end{align*}$  are vertical angles.
  ::1和3是垂直角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle 3\end{align*}$ .
  ::接下来,使用此关系来计算 3 的度量 。

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  $$\begin{align*}m \angle 1 = m \angle 3\end{align*}$$
  ::m1=m3

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  Next, substitute $\begin{align*}62^\circ\end{align*}$ , for the measure of  $\begin{align*}\angle 1 \end{align*}$  in the equation.
  ::接下来,替换 62 , 以测量方程式中的 1 。

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  $$\begin{align*}\begin{array}{rcl} m \angle 1 &=& m \angle 3 \\ {\color{red}62^\circ} &{\color{red}=}& {\color{red}m \angle 3} \end{array}\end{align*}$$
  ::==========================================================================================================================================================================================

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  Example 2
  ::例2

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  Using the following diagram, calculate the measures of the remaining three angles.
  
  

  

  
  
  First, state the relationship between $\begin{align*} \angle 1 \end{align*}$  and one other angle.
  ::使用下图,计算其余三个角度的度量。首先,说明 1 和另一个角度之间的关系。

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  $\begin{align*}\angle 1\end{align*}$  and $\begin{align*}\angle 3\end{align*}$  are vertical angles.
  ::1和3是垂直角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle 3\end{align*}$ .
  ::接下来,使用此关系来计算 3 的度量 。

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  $$\begin{align*}m \angle 1 = m \angle 3\end{align*}$$
  ::m1=m3

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  Next, substitute $\begin{align*}40^\circ\end{align*}$ , for the measure of $\begin{align*}\angle 1\end{align*}$  in the equation.
  ::接下来,换成40 以测量方程式中的11。

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  $$\begin{align*}\begin{array}{rcl} m \angle 1 &=& m \angle 3 \\ {\color{red}40^\circ} &{\color{red}=}& {\color{red}m \angle 3} \end{array}\end{align*}$$
  ::===================================================================================================================================================================================

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  First, state another relationship between $\begin{align*}\angle 1\end{align*}$  and another angle.
  ::首先,说明1和另一个角度之间的另一关系。

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  $\begin{align*}\angle 1\end{align*}$  and $\begin{align*}\angle 4\end{align*}$  are adjacent angles.
  ::1和4是相邻角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle 4\end{align*}$ .
  ::接下来,使用此关系来计算 4 的度量 。

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  $$\begin{align*}m \angle 1 + m \angle 4 = 180^\circ\end{align*}$$
  ::m1+m4=180

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  Next, substitute $\begin{align*}40^\circ\end{align*}$ , for the measure of $\begin{align*}\angle 1\end{align*}$  in the equation.
  ::接下来,换成40 以测量方程式中的11。

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  $$\begin{align*}\begin{array}{rcl} m \angle 1 + m \angle 4 &=& 180^\circ \\ 40^\circ + m \angle 4 &=& 180^\circ \end{array}\end{align*}$$
  ::m1+m4=180404=180

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  Next, subtract $\begin{align*}40^\circ\end{align*}$  from both sides of the equation to solve for $\begin{align*}m \angle 4\end{align*}$ .
  ::接下来,从方程的两侧减去40 以解答 m 4 。

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  $$\begin{align*}\begin{array}{rcl} 40^\circ + m \angle 4 &=& 180^\circ \\ 40^\circ - 40^\circ + m \angle 4 &=& 180^\circ - 40^\circ \\ {\color{red}m \angle 4} &{\color{red}=}&{\color{red}140^\circ} \end{array}\end{align*}$$
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}4... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}

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  First, state the relationship between the remaining angle and one other angle.
  ::首先,说明剩余角与其他角之间的关系。

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  $\begin{align*}\angle 4\end{align*}$  and $\begin{align*}\angle 2\end{align*}$  are vertical angles.
  ::4和2是垂直角度。

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  Next, use this relationship to calculate the measure of $\begin{align*}\angle 3\end{align*}$ .
  ::接下来,使用此关系来计算 3 的度量 。

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  $$\begin{align*}m \angle 4 = m \angle 2\end{align*}$$
  ::m4=m2

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  Next, substitute $\begin{align*}140^\circ\end{align*}$ , for the measure of $\begin{align*}\angle 4\end{align*}$  in the equation.
  
  

  $$\begin{align*}\begin{array}{rcl} m \angle 4 &=& m \angle 2 \\ {\color{red}140^\circ} &{\color{red}=}& {\color{red}m \angle 2} \end{array}\end{align*}$$  
  ::下一位, 替代 140 , 以测量方程式中的 4 。 m4 = m2140 2 。

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  Review
  ::回顾

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  Identify whether each angle pair can be classified as adjacent angles or vertical angles or neither.
  ::确定每个角度对是否可划为相邻角度或垂直角度,或两者兼而有之。

  

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  1. $\begin{align*}\angle INK\end{align*}$  and  $\begin{align*} \angle MNL\end{align*}$
  ::1. 新加坡和荷兰

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  2.  $\begin{align*}\angle INJ \end{align*}$  and  $\begin{align*}\angle JNK\end{align*}$
  ::2. INJ和JNK

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  3.  $\begin{align*} \angle MNL\end{align*}$  and  $\begin{align*} \angle LNK\end{align*}$
  ::3. 扎扎扎纳尼和扎扎纳克

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  4. $\begin{align*}\angle JNL\end{align*}$  and  $\begin{align*}\angle INM\end{align*}$  
  ::4. JNL和INM

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  5.  $\begin{align*}\angle INM \end{align*}$ and  $\begin{align*}\angle KNL\end{align*}$
  ::5. 曼曼德·卡尼勒

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  6. If $\begin{align*}m \angle INJ = 63^\circ\end{align*}$  then $\begin{align*}m \angle MNL = \underline{\;\;\;\;\;\;\;\;\;\;\;\; ^\circ}\end{align*}$ .
  ::6. 如果MINJ=63,那么MMNL。

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  Use this diagram to answer the following questions.
  ::使用此图表回答下列问题 。

  

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  7. True or False. $\begin{align*}\angle 1\end{align*}$  and $\begin{align*}\angle 2\end{align*}$  are adjacent angles.
  ::7. 真实或虚假。% 1 和% 2 是相邻角度 。

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  8. What is the measure of $\begin{align*} \angle 1\end{align*}$  ?
  ::8. 衡量1的尺度是什么?

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  9. What is the measure of $\begin{align*}\angle 2\end{align*}$  ?
  ::9.2的衡量标准是什么?

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  10. What is the relationship between $\begin{align*}\angle 2\end{align*}$  and the angle opposite it?
  ::10. 2与与之相反的角度之间的关系是什么?

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  11. True or False. Adjacent angles 1 and 2 form a straight line with a value of  $\begin{align*}180^\circ\end{align*}$
  ::11. 真实或假。相邻角度1和2形成一条直线,值180____________________________________________________________________________________________________

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  Answer true or false for each question.
  ::回答每个问题是否真实或虚假。

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  12. Supplementary angles are also vertical angles.
  ::12. 补充角度也是垂直角度。

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  13. Vertical angles have the same measure.
  ::13. 垂直角度的测量尺度相同。

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  14. Adjacent angles always have a sum of $\begin{align*}180^\circ\end{align*}$ .
  ::14. 近邻角度总有180个。

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  15. Adjacent angles are also vertical angles.
  ::15. 近角也是垂直角。

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  16. Vertical angles are formed when lines intersect.
  ::16. 当线条交叉时形成垂直角度。

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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 。