3.4 SSS 三角和谐-interactive

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• 选择节SSS 三角和谐

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  SSS 三角和谐

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  If two triangles are congruent it means that all corresponding angle pairs and all corresponding sides are congruent. However, in order to be sure that two triangles are congruent, you do not necessarily need to know that all angle pairs and side pairs are congruent. Consider the triangles below.
  ::如果两个三角是相同的,这意味着所有相应的角对和所有对应的边是相同的。然而,为了确保两个三角是相同的,您不必知道所有角对和侧对都是相同的。请看看下面的三角。

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  • In these triangles, you can see that all three pairs of sides are congruent. This is commonly referred to as "side-side-side" or "SSS".
    ::在这些三角形中, 您可以看到所有三对对齐是相似的。 这通常被称为“ 边对边” 或“ SSS ” 。
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  • The criterion for triangle congruence states that if two triangles have three pairs of congruent sides, then the triangles are congruent.
    ::三角一致的标准是,如果两个三角有三对相近的两边,则三角是相近的。

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  SSS Triangle Congruence
  ::SSS 三角和谐

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  Click the blue arrow next to the image below and then drag the orange vertices to reshape the triangle $\begin{align*}ABC\end{align*}$ and observe the other triangle $\begin{align*}DEF\end{align*}$ change accordingly to remain congruent.
  ::单击下方图像旁边的蓝色箭头,然后拖动橙色的脊椎重塑三角形ABC,并观察其他三角形的DEF相应变化以保持一致。

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  In the example, you will use rigid transformations to show why the above SSS triangles must be congruent overall, even though you don't know the measures of any of the angles .
  ::例如,您会使用僵硬的变换来显示为什么上面的 SSS 三角形在整体上必须是一致的, 即使您不知道任何角度的度量 。

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  SSS by Basic Rigid Transformation
  ::基本硬变换 SSS

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  1. Perform a rigid transformation to bring point $\begin{align*}E\end{align*}$ to point $\begin{align*}B\end{align*}$ .
  ::1. 进行僵硬的转变,使E点达到B点。

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  Draw a vector from point $\begin{align*}E\end{align*}$ to point $\begin{align*}B\end{align*}$ . Translate $\begin{align*}\triangle DEF\end{align*}$ along the vector to create $\begin{align*}\triangle D^\prime E^\prime F^\prime\end{align*}$ .
  ::从 E 点到 B 点绘制矢量。 沿矢量绘制 TranslateDEF 以创建 {D_E_F} 。

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  2. Measure $\begin{align*}\angle{ABD^\prime}\end{align*}$ .
  ::2. 措施 ABD。

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  In this case, $\begin{align*}m\angle{ABD^\prime}=26^\circ\end{align*}$ .
  ::在这种情况下,MáABD#26__________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  3. Rotate $\begin{align*}\triangle D'E'F'\end{align*}$ clockwise that number of degrees to create $\begin{align*}\triangle D^{\prime\prime}E^{\prime\prime} F^{\prime\prime}\end{align*}$ .
  ::3. 旋转 {D}E}F} 顺时针该度数来创建 {D}E}F} 。

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  Note that because $\begin{align*}\overline{DE}\cong \overline{AB}\end{align*}$ and rigid transformations preserve distance , $\begin{align*}\overline{D^{\prime\prime}E^{\prime\prime}}\end{align*}$ matches up perfectly with $\begin{align*}\overline{AB}\end{align*}$ .
  ::注意,因为"AB"和"僵硬变形" 保持距离,"E"与"AB"完全吻合。

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  4. Reflect $\begin{align*}\triangle D^{\prime \prime} E^{\prime \prime} F^{\prime \prime}\end{align*}$ across $\begin{align*}\overline{D^{\prime\prime}E^{\prime\prime}}\end{align*}$ (which is the same as $\begin{align*}\overline{AB}\end{align*}$ ).
  ::4. 向全方位(与AB相同)反映“D”和“E”之间的情况。

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  In this case, it looks like the triangles match up exactly and are therefore congruent, but how can you always be confident that $\begin{align*}F^{\prime \prime}\end{align*}$ will map to $\begin{align*}C\end{align*}$ ? Consider the previous step, with the two triangles below:
  ::在这种情况下,三角形看起来完全吻合,因此是相近的,但是你怎么能总是相信F会映射到C?考虑前一步,下面两个三角形:

  

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  You know that wherever $\begin{align*}F^{\prime \prime}\end{align*}$ ends up after it is reflected, it has to stay 5 units away from $\begin{align*}E^{\prime \prime}\end{align*}$ and 9 units away from $\begin{align*}D^{\prime \prime}\!.\end{align*}$
  ::你知道,不管F在被反映后最后在哪里, 它必须留在5个单位 远离E,9个单位远离D。

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  Create a circle centered at $\begin{align*}E^{\prime \prime}\end{align*}$ with radius 5 units to find all the points besides $\begin{align*}F^{\prime \prime}\end{align*}$ that are 5 units away from $\begin{align*}E^{\prime \prime}\! .\end{align*}$
  ::创建以 E为中心、 半径为 5 单位的圆圈, 以查找除 F 以外、 离 E 5 个单位之外的所有点 。

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  C reate a circle centered at $\begin{align*}D^{\prime \prime}\end{align*}$ with radius 9 units to find all the points besides $\begin{align*}F^{\prime \prime}\end{align*}$ that are 9 units away from $\begin{align*}D^{\prime \prime}\!.\end{align*}$
  ::创建以 D为中心、 半径为 9 单位的圆圈, 以查找除 F 以外的所有点, 即离 D 以外的 9 个单位 。

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  Notice that there are only two points in the whole plane that are both 5 units away from $\begin{align*}E^{\prime \prime}\end{align*}$ and 9 units away from $\begin{align*}D^{\prime \prime}\!,\end{align*}$ specifically point $\begin{align*}F^{\prime \prime}\end{align*}$ and point $\begin{align*}C.\end{align*}$ Since reflections preserve distance, when $\begin{align*}F^{\prime \prime}\end{align*}$ is reflected, it must end up at point $\begin{align*}C.\end{align*}$ Therefore, a reflection will always map $\begin{align*}\triangle D^{\prime\prime}E^{\prime\prime} F^{\prime\prime}\end{align*}$ to $\begin{align*}\triangle ABC\end{align*}$ at this step.
  ::请注意,整个飞机只有两点,即离E5个单位和离D9个单位,具体指FC点。 由于反射保持距离,当反映F时,反射必须到C点。 因此,反射总是会在这个步骤上映到QEFABC。

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  • This means that even though you didn't know the angle measures, because you knew three pairs of sides were congruent, the triangles had to be congruent overall.
    ::这意味着,即使你不知道角度的测量, 因为你知道三对方是一致的, 三角形必须是一致的整体。
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  • Now you can use the SSS criterion for showing triangles are congruent without having to go through all of these transformations each time, and you can explain why SSS works in terms of the rigid transformations.
    ::现在,您可以使用 SSS 标准来显示三角形的一致性,而不必每次经历所有这些变换,您可以解释为什么SSS在硬质变换方面起作用。

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  Examples
  ::实例实例实例实例

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  Example 1
  ::例1

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  How can you use the SSS criterion for triangle congruence to show that the triangles below are congruent?
  ::您如何使用三角一致的 SSS 标准来显示下面的三角是相同的 ?

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  Because these are right triangles , you can use the Pythagorean Theorem to find the third side of each triangle. The third side of each triangle will be $\begin{align*}\sqrt{15^2-12^2}=9\end{align*}$ . Now you know that all three pairs of sides are congruent , so the triangles are congruent by SSS.
  ::因为这些是正确的三角形, 您可以使用 Pythagorean Theorem 来找到每个三角形的第三面。 每个三角形的第三面是 152 - 122=9。 现在你知道所有三边都是相同的, 所以三角形都是由 SSS 匹配的 。

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  In general, anytime you have the hypotenuses congruent and one pair of legs congruent for two right triangles, the triangles are congruent. This is often referred to as "HL" for "hypotenuse-leg". Remember, it only works for right triangles because you can only use the Pythagorean Theorem for right triangles.
  ::一般来说, 每当您对两个右三角形的下限相近, 一对双腿相匹配时, 三角形是相匹配的。 这通常被称为“ HL ” 。 记住, 它只对右三角形有效, 因为您只能对右三角形使用 Pythagoren 理论 。

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  Example 2
  ::例2

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  Are the following triangles congruent? Explain.
  ::以下三角形是否一致?解释。

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  Yes, the triangles are congruent by SSS.
  ::是的 三角形和SSS的吻合

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  Example 3
  ::例3

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  Are the following triangles congruent? Explain.
  ::以下三角形是否一致?解释。

  

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  There is not enough information to determine if the triangles are congruent. You need to know how the unmarked side compares to the other sides, or if there are right angles.
  ::没有足够的信息来确定三角形是否一致。 您需要知道无标记的侧面与其他侧面的对比如何, 或者是否有正确的角度 。

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  Example 4
  ::例4

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  Are the following triangles congruent. Explain.
  ::以下三角形是相同的。请解释 。

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  What additional information would you need in order to be able to state that the triangles below are congruent by ?
  ::您需要哪些补充信息才能声明以下三角形的吻合性 ?

  

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  You would need to know that the triangles are right triangles in order to use HL.
  ::您需要知道三角形是正确的三角形才能使用 HL 。

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  CK-12 PLIX Interactive
  ::CK-12 PLIX 交互式互动

  Summary

  播放段落 Side-side-side (SSS) criterion states that if two triangles have three pairs of congruent sides, then the triangles are congruent. ::侧边标准(SSS)规定,如果两个三角形有三对相近的两边,三角形是相近的。

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  Review
  ::审查审查审查审查

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  1. What does SSS stand for? How is it used?
  ::1. SSS代表什么?如何使用SSS?

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  2. What does HL stand for? How is it used?
  ::2. HL代表什么?HL是如何使用的?

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  3. Draw an example of two triangles that must be congruent due to SSS.
  ::3. 绘制两个三角形的示例,这些三角形因SSS而必须相容。

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  4. Draw an example of two triangles that must be congruent due to HL.
  ::4. 绘制两个三角形的示例,这些三角形必须因 HL 而相容。

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  For each pair of triangles below, state if they are congruent by SSS, congruent by HL, or if there is not enough information to determine whether or not they are congruent.
  ::对于以下每一对三角形,请说明它们是否与SSS一致,是否与HL一致,或者是否没有足够的信息来确定它们是否一致。

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  10. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by HL?
  ::10. 为了能够说明以下三角形与HL相同,您需要哪些最低限度的额外信息?

  

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  11. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by SSS?
  ::11. 为了能够说明以下三角形与SSS的吻合,你最起码需要哪些补充信息?

  

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  12. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by HL?
  ::12. 为了能够说明以下三角形与HL相同,您需要哪些最低限度的额外信息?

  

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  13. Point $\begin{align*}A\end{align*}$ is the center of the circle below. What is the minimum additional information you would need in order to be able to state that the triangles below are congruent by SSS?
  ::13. A点是以下圆圈的中心。为了能够说明以下三角形与SSS的三角形相匹配,您需要多少最低的额外信息?

  

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  14. If you can show that two triangles are congruent by HL, can you also show that they are congruent by SAS?
  ::14. 如果你能证明两个三角形与HL是相同的,你能否也证明它们与SAS是相同的?

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  15. Show how the SSS criterion for triangle congruence works: use rigid transformations to help explain why the triangles below are congruent.
  ::15. 显示三角一致的 SSS 标准是如何工作的:使用硬质变换来解释以下三角为何是一致的。

  

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  16. Below is another example which demonstrates why SSS can be used to establish triangle congruence. Three sides of $\begin{align*}\triangle ABC\end{align*}$ are given as congruent to three sides of $\begin{align*}\triangle DEF\end{align*}$ as shown. Rigid motion transformations were performed to map $\begin{align*}\overline{AB}\end{align*}$ to $\begin{align*}\overline{DE}\end{align*}$ . What kind of triangle is  $\begin{align*}\triangle ACF\end{align*}$ ? So what is known about $\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 2\end{align*}$ ? What about $\begin{align*}\angle 3\end{align*}$ and $\begin{align*}\angle 4\end{align*}$ ? Why? So what about $\begin{align*}\angle ACE\end{align*}$ and $\begin{align*}\angle DFE\end{align*}$ ? Now, is there enough information to conclude the triangles are congruent? Why? Explain the process just used in words to a peer.
  ::16. 下面是另一个例子,说明为什么SSS可以用来建立三角一致。如所示, ABC的三面与 ZDEF 的三面一致。 进行了硬运动变换以映射 AB 到 DE。 QACF 是什么三角形? 有关 & 1 和 _ 2 的已知信息是什么? 3 和 _ 4 是什么? 那么为什么? QACE 和 QDFE 如何呢? 现在, 是否有足够的信息来得出三角是否一致? 为什么? 解释刚刚用语言向同行解释的过程 。

  

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  17. The two triangles below have two pairs of congruent sides and a pair of non-included congruent angles. Are the triangles congruent? Why or why not? Under what conditions would the triangles be congruent?
  ::17. 下面的两个三角有两对相似的两边和两对未包含的相同角度,三角是否一致?为什么或为什么没有?三角在什么条件下是一致的?

  

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  18. The two triangles below have one pair of congruent legs and congruent hypotenuses. Remember, SSA is not a congruence theorem. But a similar method used in number 16 can be used to establish that these two triangles are congruent. Explain the process. Based on this, can one conclude that HL is a congruence theorem that can be used for any pair of right triangles that have a pair of congruent legs and a pair of congruent hypotenuses? Why or why not?
  ::18. 下面的两个三角有一对齐合的双腿和齐合的极低位。记住,特别服务协定不是一对齐合的定理。但是,第16号中的类似方法可以用来确定这两个三角是齐合的。解释一下这个过程。根据这一点,人们可以得出结论,HL是齐合的定理,可以用于任何对齐合的直立三角,它们都有一对齐合的双腿和一对齐合的低位?为什么不行?

  

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  Review (Answers)
  ::审查(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 。