4.1 三角定理

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  三角三角间苏门神话

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  Triangle Sum Theorem
  ::三角三角间苏门神话

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  The Triangle Sum Theorem says that the three interior angles of any triangle add up to $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::三角形Sumorem表示,任何三角形的三个内角加起来等于180。

  

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

  .
  ::m1+m%2+m%3=180%。

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  Here is one proof of the Triangle Sum Theorem.
  ::这是三角定理的证明

  

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  Given : $\begin{array}{r}\mathrm{△}ABC\end{array}$ with $\begin{array}{r}\stackrel{\leftrightarrow }{AD}||\overline{BC}\end{array}$
  ::以 : @ ABC 与 ADBC 的

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  Prove : $\begin{array}{r}m\mathrm{\angle }1+m\mathrm{\angle }2+m\mathrm{\angle }3={180}^{\circ }\end{array}$
  ::证明: m1+m2+m3=180

  Statement	Reason
  1. $\begin{array}{r}\mathrm{△}ABC\end{array}$ with $\begin{array}{r}\stackrel{\leftrightarrow }{AD}||\overline{BC}\end{array}$	Given
  2. $\begin{array}{r}\mathrm{\angle }1\cong \mathrm{\angle }4,\mathrm{\angle }2\cong \mathrm{\angle }5\end{array}$	Theorem
  3. $\begin{array}{r}m\mathrm{\angle }1=m\mathrm{\angle }4,m\mathrm{\angle }2=m\mathrm{\angle }5\end{array}$	$\begin{array}{r}\cong \end{array}$ angles have = measures
  4. $\begin{array}{r}m\mathrm{\angle }4+m\mathrm{\angle }CAD={180}^{\circ }\end{array}$	Linear Pair Postulate
  5. $\begin{array}{r}m\mathrm{\angle }3+m\mathrm{\angle }5=m\mathrm{\angle }CAD\end{array}$	Angle Addition Postulate
  6. $\begin{array}{r}m\mathrm{\angle }4+m\mathrm{\angle }3+m\mathrm{\angle }5={180}^{\circ }\end{array}$	Substitution PoE
  7. $\begin{array}{r}m\mathrm{\angle }1+m\mathrm{\angle }3+m\mathrm{\angle }2={180}^{\circ }\end{array}$	Substitution PoE

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  You can use the Triangle Sum Theorem to find missing angles in triangles.
  ::您可以使用三角 Sum 理论在三角形中找到缺失的角度 。

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  What if you knew that two of the angles in a triangle measured $\begin{array}{r}{55}^{\circ }\end{array}$ ? How could you find the measure of the third angle?
  ::如果你知道一个三角形中两个角度是55°C的呢?你如何找到第三个角度的度量?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Two interior angles of a triangle measure $\begin{array}{r}{50}^{\circ }\end{array}$ and $\begin{array}{r}{70}^{\circ }\end{array}$ . What is the third interior angle of the triangle?
  ::三角度为 50 和 70 的两个内角。 三角度的第三个内角是什么 ?

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  $\begin{array}{r}{50}^{\circ }+{70}^{\circ }+x={180}^{\circ }\end{array}$ .
  ::50 70x=180 。

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  Solve this equation and you find that the third angle is $\begin{array}{r}{60}^{\circ }\end{array}$ .
  ::解决这个方程式,你就会发现第三个角度是60

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  Example 2
  ::例2

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  Find the value of $\begin{array}{r}x\end{array}$ and the measure of each angle.
  ::查找 x 的值和每个角度的度量。

  

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  All the angles add up to $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::所有角度加起来等于180

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  $$\begin{array}{rl}(8x-1{)}^{\circ }+(3x+9{)}^{\circ }+(3x+4{)}^{\circ }& ={180}^{\circ }\\ (14x+12{)}^{\circ }& ={180}^{\circ }\\ 14x=168\\ x=12\end{array}$$

  
  :8x- 1) ( 3x+9) ( 3x+4) ( 3x+4) ( 14x+12) ( 14x+12) ( 180) ( 14x) ( 14x) ( 14x) = 168x=12)

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  Substitute in 12 for $\begin{array}{r}x\end{array}$ to find each angle.
  ::以 12 代替 x 查找每个角度 。

  $$\begin{array}{rlrlr}[3(12)+9{]}^{\circ }={45}^{\circ }& & [3(12)+4{]}^{\circ }={40}^{\circ }& & [8(12)-1{]}^{\circ }={95}^{\circ }\end{array}$$

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  Example 3
  ::例3

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  What is $\begin{array}{r}m\mathrm{\angle }T\end{array}$ ?
  ::什么是MT?

  

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  We know that the three angles in the triangle must add up to $\begin{array}{r}{180}^{\circ }\end{array}$ . To solve this problem, set up an equation and substitute in the information you know.
  ::我们知道三角形中的三个角度必须相加到 180 。 要解决这个问题, 请设置一个方程, 并替换您所知道的信息 。

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  $$\begin{array}{rl}m\mathrm{\angle }M+m\mathrm{\angle }A+m\mathrm{\angle }T& ={180}^{\circ }\\ {82}^{\circ }+{27}^{\circ }+m\mathrm{\angle }T& ={180}^{\circ }\\ {109}^{\circ }+m\mathrm{\angle }T& ={180}^{\circ }\\ m\mathrm{\angle }T& ={71}^{\circ }\end{array}$$

  
  ::M+mA+mT=1808227MT=180109MT=180MT=71

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  Example 4
  ::例4

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  What is the measure of each angle in an equiangular triangle ?
  ::方形三角形中每个角度的度量是多少?

  

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  To solve, remember that $\begin{array}{r}\mathrm{△}ABC\end{array}$ is an equiangular triangle, so all three angles are equal. Write an equation.
  ::要解析, 请记住 { ABC 是一个等角三角形, 所以所有三个角度都是相等的。 写入一个公式 。

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  $$\begin{array}{rl}m\mathrm{\angle }A+m\mathrm{\angle }B+m\mathrm{\angle }C& ={180}^{\circ }\\ m\mathrm{\angle }A+m\mathrm{\angle }A+m\mathrm{\angle }A& ={180}^{\circ }Substitute,allanglesareequal.\\ 3m\mathrm{\angle }A& ={180}^{\circ }Combineliketerms.\\ m\mathrm{\angle }A& ={60}^{\circ }\end{array}$$

  
  ::mA+mB+mC=180mA+mA+mA=180Substitutive, 所有角度都等于3mA=180Combine like terms.mA=60

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  If $\begin{array}{r}m\mathrm{\angle }A={60}^{\circ }\end{array}$ , then $\begin{array}{r}m\mathrm{\angle }B={60}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }C={60}^{\circ }\end{array}$ .
  ::如果mA=60,那么mB=60和mC=60。

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  Each angle in an equiangular triangle is $\begin{array}{r}{60}^{\circ }\end{array}$ .
  ::角角三角形的每个角是60。

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  Example 5
  ::例5

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  Find the measure of the missing angle.
  ::查找缺失角度的度量 。

  

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  We know that $\begin{array}{r}m\mathrm{\angle }O={41}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }G={90}^{\circ }\end{array}$ because it is a right angle . Set up an equation like in Example 3.
  ::我们知道 mO=41和 mG=90, 因为它是一个正确的角度。 设置一个像例3那样的方程 。

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  $$\begin{array}{rl}m\mathrm{\angle }D+m\mathrm{\angle }O+m\mathrm{\angle }G& ={180}^{\circ }\\ m\mathrm{\angle }D+{41}^{\circ }+{90}^{\circ }& ={180}^{\circ }\\ m\mathrm{\angle }D+{41}^{\circ }& ={90}^{\circ }\\ m\mathrm{\angle }D& ={49}^{\circ }\end{array}$$

  
  ::mD+mO+mG=180mD+4190180MD+4190MD+4190MD=49

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  Review
  ::回顾

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  Determine $\begin{array}{r}m\mathrm{\angle }1\end{array}$ in each triangle.
  ::在每个三角形中确定 m% 1 。

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  8. Two interior angles of a triangle measure $\begin{array}{r}{32}^{\circ }\end{array}$ and $\begin{array}{r}{64}^{\circ }\end{array}$ . What is the third interior angle of the triangle?
  ::8. 三角度为32°和64°的两个内角。三角度的第三个内角是什么?

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  9. Two interior angles of a triangle measure $\begin{array}{r}{111}^{\circ }\end{array}$ and $\begin{array}{r}{12}^{\circ }\end{array}$ . What is the third interior angle of the triangle?
  ::9. 三角形措施111和12的两个内角。三角形第三个内角是什么?

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  10. Two interior angles of a triangle measure $\begin{array}{r}{2}^{\circ }\end{array}$ and $\begin{array}{r}{157}^{\circ }\end{array}$ . What is the third interior angle of the triangle?
  ::10. 三角度量 2 和 157 的两个内角。三角度量 2 和 157 的第三个内角是什么?

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  Find the value of $\begin{array}{r}x\end{array}$ and the measure of each angle.
  ::查找 x 的值和每个角度的度量。

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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