6.6 了解三角的角度量

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• 选择节了解三角角度量

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  了解三角角度量

  

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  A roof truss is a pre-fabricated, triangular wooden structure used to support the roof of a home or building. The triangular shape is used because it is incredibly strong. A particular roof truss is an isosceles triangle such that the base angle has a measure of 50°. How can the worker figure out the measure of the other angles of the triangular roof truss to ensure that all trusses for this roof are identical?
  ::屋顶三角锥是一个预制的三角木制结构,用来支撑房屋或建筑物的屋顶。三角形的形状之所以被使用,是因为它非常坚固。一个特定的屋顶三角锥是一个等骨三角形,其基角的度量为50°。工人如何找到三角屋顶三角三角锥其他角度的度量,以确保这个屋顶的所有三角三角锥都完全相同?

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  In this concept, you will learn to understand the angle measures of triangles.
  ::在此概念中,您将学会理解三角形的角度度量。

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  Equiangular Triangles
  ::等角三角形

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  The sum of the interior angles of a triangle is 180°. This fact is true for all triangles regardless of the type of triangle. An equiangular triangle has three equal angles which each measure 60°. The following diagram is an equiangular triangle.
  ::三角形的内部角总和为 180 °。 对所有三角形来说, 不论三角形类型如何, 事实都是如此。 矩形三角形有三个相等的角, 每度度为 60 °。 下图是一个相等的三角形 。

  

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  Let’s look at another triangle to see how this information can be applied.
  ::让我们看看另一个三角形, 看看如何应用这些信息。

  

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  The above triangle has two acute angles that each measure 25° and one obtuse angle . The measure of the obtuse angle is not known.
  ::上述三角形有两个急性角,每个角测量25°,一个斜角。斜角的度量未知。

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  The sum of the three angles of the triangle equals 180°. Write an equation to represent this statement.
  ::三角形的三个角度之和等于 180 °。 写一个方程来代表此语句 。

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  $$\begin{array}{r}m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C={180}^{\circ }\end{array}$$

  
  ::mA+mB+mC=180\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\B\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\B\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

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  Next, substitute the known information into the equation.
  ::接下来,将已知信息替换为等式。

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  $$\begin{array}{r}m\mathrm{\angle }A+{25}^{\circ }+{25}^{\circ }={180}^{\circ }\end{array}$$

  
  ::@A+2525180

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }A+{25}^{\circ }+{25}^{\circ }& =& {180}^{\circ }\\ m\mathrm{\angle }A+{50}^{\circ }& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::@A+25 @ 25 @ @ 180 @ m @A+50 @ 180 @

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  Then, subtract 50° from both sides of the equation to solve for $\begin{array}{r}m\mathrm{\angle }A\end{array}$ .
  ::然后,从方程两侧减去50度,以解答 mA。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }A+{50}^{\circ }& =& {180}^{\circ }\\ m\mathrm{\angle }A+{50}^{\circ }-{50}^{\circ }& =& {180}^{\circ }-{50}^{\circ }\\ m\mathrm{\angle }A& =& {130}^{\circ }\end{array}\end{array}$$

  
  ::@A+50180A+5050501805050A=130

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  The answer is 130.
  ::答案是130。

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  The measure of the obtuse angle is 130°.
  ::斜角的度量为130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  the worker with the roof truss. He needs to figure out the measure of the interior angles in the roof truss. How can he do this?
  ::早些时候,有人给了你一个问题 工人用屋顶的铁丝网。他需要弄清楚 房顶铁丝网内角的大小。他怎么能这样做呢?

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  He can use the facts that the triangle is isosceles and the sum of the interior angles of the triangle equals 180°.
  ::他可以使用三角形是等离子体的事实,三角形内部角的总和等于180度。

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  First, draw and label an isosceles triangle to model the roof truss.
  ::首先,绘制和标注一个等分形三角形,以模拟屋顶三角形。

  

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  The triangular roof truss is an isosceles triangle. The angles opposite the equal sides are equal in measure.
  ::三角屋顶三角形三角形为等骨形三角形。对平面的角在量上是相等的。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }A& =& {50}^{\circ }\\ m\mathrm{\angle }A& =& m\mathrm{\angle }B\\ m\mathrm{\angle }B& =& {50}^{\circ }\end{array}\end{array}$$

  
  ::mA=50*m*m*m*m*m*m*m*m*m=50*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*m*b*B=50*

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  Next, write an equation to represent the sum of the interior angles of the triangle.
  ::下一步,写一个方程以表示三角形内部角度的总和。

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  $$\begin{array}{r}\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }\end{array}$$

  
  ::ABC=180

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  Next, fill into the equation, the measures of the equal angles.
  ::接下来,填入等式, 平等角度的度量。

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  $$\begin{array}{r}\begin{array}{rcl}\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C& =& {180}^{\circ }\\ {50}^{\circ }+{50}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::ABC=1805050C=180

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}\begin{array}{rcl}{50}^{\circ }+{50}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\\ {100}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::50C=180100C=180C

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  Next, subtract 100° from both sides of the equation to determine the measure of $\begin{array}{r}\mathrm{\angle }C\end{array}$ .
  ::下一步,从方程两侧减去100°,以确定 C 的度量。

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  $$\begin{array}{r}\begin{array}{rcl}{100}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\\ {100}^{\circ }-{100}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }-{100}^{\circ }\\ \mathrm{\angle }C& =& {80}^{\circ }\end{array}\end{array}$$

  
  ::100C=180100C=180100C=80C=80

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  The answer is 80.
  ::答案是80

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  The measures of the other two interior angles of the roof truss are 50° and 80°.
  ::屋顶三角形另外两个内部角的测量为50度和80度。

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  Example 2
  ::例2

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  If the measure of the vertex angle of an isosceles triangle is 50°, what is the measure of the base angles of the triangle?
  ::如果一等分三角形的顶角为50度,则三角形基角的测量值是多少?

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  First, draw and label a triangle to represent the given information.
  ::首先,绘制并标记三角形以代表给定的信息。

  

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  Next, write down what you know from the problem.
  ::接下来,写下你从问题中知道的东西

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }1+m\mathrm{\angle }2+m\mathrm{\angle }3& =& {180}^{\circ }\text{Sum of the angles of a triangle}\\ m\mathrm{\angle }3& =& {50}^{\circ }\text{Measure of the vertex angle}\\ m\mathrm{\angle }1& =& m\mathrm{\angle }2\text{Equal angles of an isosceles triangle}\end{array}\end{array}$$

  
  ::1+m2+m3=1803=1803=50三角形角度的m1+m2=m2赤道角的度量

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  Next, write down an equation to represent the information.
  ::下一步,写一个方程式来代表信息。

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  $$\begin{array}{r}m\mathrm{\angle }1+m\mathrm{\angle }2+{50}^{\circ }={180}^{\circ }\end{array}$$

  
  ::m1+m2+50180

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  Next, let ‘ $\begin{array}{r}x\end{array}$ ’ represent each of the equal angles of the isosceles triangle.
  ::接下来,让“x”代表等分形三角形的每一个等角。

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  $$\begin{array}{r}x+x+{50}^{\circ }={180}^{\circ }\end{array}$$

  
  ::x+x+50+________________________________________________________________________________________________

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}2x+{50}^{\circ }={180}^{\circ }\end{array}$$

  
  ::2+50180

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  Next, subtract 50° from both sides of the equation and simplify to isolate the variable.
  ::接下来,从方程两侧减去50度,并简化以孤立变量。

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  $$\begin{array}{r}\begin{array}{rcl}2x+{50}^{\circ }& =& 180{}^{\circ }\\ 2x+{50}^{\circ }-{50}^{\circ }& =& {180}^{\circ }-{50}^{\circ }\\ 2x& =& {130}^{\circ }\end{array}\end{array}$$

  
  ::2x50=180=2x50=50=50=50=50=50=50=50=50=2x130=130

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  Then, divide both sides of the equation by ‘2’ to solve for ‘ $\begin{array}{r}x\end{array}$ .’
  ::然后将方程式两边除以 '2 ' ,

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  $$\begin{array}{r}\begin{array}{rcl}2x& =& {130}^{\circ }\\ \frac{\stackrel{1}{\cancel{2}}x}{\cancel{2}}& =& \frac{{130}^{\circ }}{2}\\ x& =& {65}^{\circ }\end{array}\end{array}$$

  
  ::2x=130=21x2=130}2xx=65*

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  The answer is 65.
  ::答案是65岁

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  $\begin{array}{r}m\mathrm{\angle }1={65}^{\circ }\end{array}$  and $\begin{array}{r}m\mathrm{\angle }2={65}^{\circ }\end{array}$ .
  ::m1=65 和m2=65。

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  Example 3
  ::例3

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  For the following diagram, determine the measure of angles ‘ $\begin{array}{r}x\end{array}$ ’ and ‘ $\begin{array}{r}y\end{array}$ .’
  ::对于下图,确定角度 'x ' 和`y ' 的度量。

  

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  First, write down what you know from the diagram.
  ::首先,写下你从图表中知道的

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }x+{140}^{\circ }& =& {180}^{\circ }\text{Straight angle formed by adjacent angles}\\ m\mathrm{\angle }y+{125}^{\circ }& =& {180}^{\circ }\text{Straight angle formed by adjacent angles}\\ m\mathrm{\angle }x+m\mathrm{\angle }y+{85}^{\circ }& =& {180}^{\circ }\text{Sum of the angles of a triangle}\end{array}\end{array}$$

  
  ::三角形角的相邻角度======================================================================================================================================= =========== ======================================================================================================================== 方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方是方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方方

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  Next, use what you have written down to determine the measure of $\begin{array}{r}\mathrm{\angle }x\end{array}$ .
  ::接下来,使用您写下来的 确定 x 的度量 。

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  $$\begin{array}{r}m\mathrm{\angle }x+{140}^{\circ }={180}^{\circ }\end{array}$$

  
  ::mx+140

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  Next, subtract 140° from both sides of the equation.
  ::接下来,从方程两侧减去140度。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }x+{140}^{\circ }& =& {180}^{\circ }\\ m\mathrm{\angle }x+{140}^{\circ }-{140}^{\circ }& =& {180}^{\circ }-{140}^{\circ }\end{array}\end{array}$$

  
  ::@ mx+140 @ @ 180 @ m# x+140 @ 140 @ 180 @ 140 @ @

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  Next, simplify both sides of the equation.
  ::接下来,简化方程式的两边

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }x+{140}^{\circ }-{140}^{\circ }& =& {180}^{\circ }-{140}^{\circ }\\ m\mathrm{\angle }x& =& {40}^{\circ }\end{array}\end{array}$$

  
  ::@ mx+140 @ 140 @ @ 180 @ @ 140 @ mx=40 @

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  The answer is 40.
  ::答案是40岁

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  The measure of  $\begin{array}{r}\mathrm{\angle }x\end{array}$ is 40°.
  ::x的度量为40度。

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  Now, use what you know about the measures of the interior angles of the triangle to solve for the measure of $\begin{array}{r}\mathrm{\angle }y\end{array}$ .
  ::现在,使用您所知道的三角形内角的度量来解析 y 的度量。

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  $$\begin{array}{r}m\mathrm{\angle }x+m\mathrm{\angle }y+{85}^{\circ }={180}^{\circ }\end{array}$$

  
  ::mx+my+85180

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  Next, substitute the measure of  $\begin{array}{r}\mathrm{\angle }x\end{array}$ into the equation.
  ::接下来,将 x 的度量替换为等式 。

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  $$\begin{array}{r}\begin{array}{rcl}m\mathrm{\angle }x+m\mathrm{\angle }y+{85}^{\circ }& =& {180}^{\circ }\\ {40}^{\circ }+m\mathrm{\angle }y+{85}^{\circ }& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::@ mx+my+851840my+85180

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}\begin{array}{rcl}{40}^{\circ }+m\mathrm{\angle }y+{85}^{\circ }& =& {180}^{\circ }\\ {125}^{\circ }+m\mathrm{\angle }y& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}40y+8518125my=180

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  Next, subtract 125° from both sides of the equation.
  ::接下来,从方程两侧减去125度。

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  $$\begin{array}{r}\begin{array}{rcl}{125}^{\circ }+m\mathrm{\angle }y& =& {180}^{\circ }\\ {125}^{\circ }-{125}^{\circ }+m\mathrm{\angle }y& =& {180}^{\circ }-{125}^{\circ }\end{array}\end{array}$$

  
  ::125 @125 @my=180 @125 @125 @my=180 @125 @

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  Next, simplify both sides of the equation.
  ::接下来,简化方程式的两边

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  $$\begin{array}{r}\begin{array}{rcl}{125}^{\circ }-{125}^{\circ }+m\mathrm{\angle }y& =& {180}^{\circ }-{125}^{\circ }\\ m\mathrm{\angle }y& =& {55}^{\circ }\end{array}\end{array}$$

  
  ::125my=180125my=55

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  The answer is 55.
  ::答案是55岁

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  The measure of  $\begin{array}{r}\mathrm{\angle }y\end{array}$ is 55°.
  ::y的度量为55度。

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  Example 4
  ::例4

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  Given $\begin{array}{r}\mathrm{△}ABC\end{array}$ , an obtuse scalene triangle with  $\begin{array}{r}\mathrm{\angle }A={37}^{\circ }\end{array}$ and $\begin{array}{r}\mathrm{\angle }b={28}^{\circ }\end{array}$ , what is the measure of the obtuse angle?
  ::鉴于A=37和b=28的斜度三角形,斜度角的度量是多少?

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  First, draw and label a triangle to model the problem.
  ::首先,绘制并标记三角形来模拟问题。

  

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  Next, write an equation to represent the sum of the interior angles of the triangle.
  ::下一步,写一个方程以表示三角形内部角度的总和。

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  $$\begin{array}{r}\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C={180}^{\circ }\end{array}$$

  
  ::ABC=180

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  Next, fill into the equation, the measures of the angles given in the diagram.
  ::接下来,填入方程,测量图表中给出的角度。

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  $$\begin{array}{r}\begin{array}{rcl}\mathrm{\angle }A+\mathrm{\angle }B+\mathrm{\angle }C& =& {180}^{\circ }\\ {37}^{\circ }+{28}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::ABC=1803728C=180

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}\begin{array}{rcl}{37}^{\circ }+{28}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\\ {65}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::3728C=18065C=180C

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  Then, subtract 65° from both sides of the equation to solve for the measure of $\begin{array}{r}\mathrm{\angle }C\end{array}$ .
  ::然后从方程的两侧减去65度,以解答 QC 的量值。

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  $$\begin{array}{r}\begin{array}{rcl}{65}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }\\ {65}^{\circ }-{65}^{\circ }+\mathrm{\angle }C& =& {180}^{\circ }-{65}^{\circ }\\ \mathrm{\angle }C& =& {115}^{\circ }\end{array}\end{array}$$

  
  ::65C= 1806565C= 18065°C=115

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  The answer is 115.
  ::答案是115

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  The measure of the obtuse angle is 115° .
  ::斜角的度量为115度。

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  Example 5
  ::例5

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  Given $\begin{array}{r}\mathrm{△}DEF\end{array}$ , such that  $\begin{array}{r}\mathrm{\angle }D={42}^{\circ }\end{array}$ and $\begin{array}{r}\mathrm{\angle }E={123}^{\circ }\end{array}$ , what is the measure of $\begin{array}{r}\mathrm{\angle }F\end{array}$ ?
  ::鉴于"DEF",所以"D=42"和"E=123","F"的衡量标准是什么?

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  First, draw and label a triangle to model the problem.
  ::首先,绘制并标记三角形来模拟问题。

  

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  Next, write an equation to represent the sum of the measures of the interior angles of the triangle.
  ::下一步,写一个方程,以表示三角形内角的测量总和。

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  $$\begin{array}{r}\mathrm{\angle }D+\mathrm{\angle }E+\mathrm{\angle }F={180}^{\circ }\end{array}$$

  
  ::F

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  Next, fill into the equation, the measures of the angles given in the diagram.
  ::接下来,填入方程,测量图表中给出的角度。

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  $$\begin{array}{r}\begin{array}{rcl}\mathrm{\angle }D+\mathrm{\angle }E+\mathrm{\angle }F& =& {180}^{\circ }\\ {42}^{\circ }+{123}^{\circ }+\mathrm{\angle }F& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::

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  Next, simplify the left side of the equation.
  ::接下来,简化方程的左侧。

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  $$\begin{array}{r}\begin{array}{rcl}{42}^{\circ }+{123}^{\circ }+\mathrm{\angle }F& =& {180}^{\circ }\\ {165}^{\circ }+\mathrm{\angle }F& =& {180}^{\circ }\end{array}\end{array}$$

  
  ::42123F=180165F=180

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  Then, subtract 165° from both sides of the equation to solve for the measure of $\begin{array}{r}\mathrm{\angle }C\end{array}$ .
  ::然后从方程的两侧减去165度,以解答 C 的量度。

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  $$\begin{array}{r}\begin{array}{rcl}{165}^{\circ }+\mathrm{\angle }F& =& {180}^{\circ }\\ {165}^{\circ }-{165}^{\circ }+\mathrm{\angle }F& =& {180}^{\circ }-{165}^{\circ }\\ \mathrm{\angle }F& =& {15}^{\circ }\end{array}\end{array}$$

  
  ::F=1800 165165F=1800 165F=1500F=15

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  The answer is 15.
  ::答案是15岁

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  The measure of $\begin{array}{r}\mathrm{\angle }F={15}^{\circ }\end{array}$ .
  ::F=15的量度

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  Review
  ::回顾

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  Using what you have learned about the interior angles of a triangle, determine the missing angle in each triangle.
  ::使用您所了解的三角形内部角度, 确定每个三角形中缺失的角度 。

  1. 45°, 45°, ?

  2. 60°, 60°, ?

  3. 90°, 50°, ?

  4. 100°, 40°, ?

  5. 110°, 30°, ?

  6. 50°, 10°, ?

  7. 145°, 15°, ?

  8. 55°, 45°, ?

  9. 70°, 35°, ?

  10. 50°, 50°, ?

  11. 63°, 42°, ?

  12. 18°, 75°, ?

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  Identify three triangles in the room around you.
  ::确定你周围房间的三个三角形

  13.

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