6.4 SAS三角相似性-interactive

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  SAS 三角相似性

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  If two triangles are similar , then all corresponding angle pairs are congruent and all corresponding sides are proportional.
  ::如果两个三角形相似,则所有相应的角对均匀,所有对应边均成正比。

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  It is possible to verify that two triangles are similar without information about all sides and all angles .
  ::可以核实两个三角形是相似的,没有关于各方和所有角度的信息。

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  • The criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.
    ::三角形相似性的标准是,如果一个三角形的两面与另一个三角形的两面成正比,而且其中的角是相同的,那么三角形是相似的。

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  This statement can be expressed as follows:
  ::这一陈述可表述如下:

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  If two triangles $\begin{align*}ABC\end{align*}$ and $\begin{align*}DEF\end{align*}$ are such that $\begin{align*}\angle A = \angle D\end{align*}$ and $\begin{align*}\frac{AC}{DF}\end{align*}$ $\begin{align*}=\end{align*}$ $\begin{align*}\frac{AB}{DE}\end{align*}$ , then $\begin{align*}\triangle ABC \sim \triangle DEF\end{align*}$ .
  ::如果两个三角的ABC和DEF都达到AQD和ACDF=ABDE,那么ABC=DEF。

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  This implies that
  ::这意味着:

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  $\begin{align*}\frac{BC}{EF}\end{align*}$ $\begin{align*}=\end{align*}$ $\begin{align*}\frac{AC}{DF}\end{align*}$ $\begin{align*}=\end{align*}$ $\begin{align*}\frac{AB}{DE},\end{align*}$ and also that $\begin{align*}\angle B = \angle E\end{align*}$ , and $\begin{align*}\angle C = \angle F\end{align*}$ .
  ::BCEF=ACDF=ADEBDE,还有BE和CF。

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  SAS Triangle Similarity
  ::SAS 三角相似性

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  Click the small 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 similar.
  ::单击下方图像旁边的蓝色小箭头,然后拖动橙色顶部以重塑三角形ABC,并观察其他三角形的DEF相应变化以保持相似。

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  1. Consider the triangles below with $\begin{align*}\frac{AC}{DE}=\frac{BC}{FE}=k\end{align*}$ and $\begin{align*}\angle C \cong \angle E.\end{align*}$ Dilate $\begin{align*}\triangle DEF\end{align*}$ with a scale factor of $\begin{align*}k\end{align*}$ to create $\begin{align*}\triangle D^\prime E^\prime F^\prime.\end{align*}$
  ::1. 考虑下方带有ACDE=BCFE=k和QCQE的三角形。

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  What do you know about the sides and angles of $\begin{align*}\triangle D^\prime E^\prime F^\prime?\end{align*}$ How do they relate to the sides and angles of $\begin{align*}\triangle ABC?\end{align*}$
  ::你对ZDEF的侧面和角度了解多少?它们与ABC的侧面和角度有何关系?

  

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  Below, $\begin{align*}\triangle DEF\end{align*}$ is dilated about point $\begin{align*}P\end{align*}$ with a scale factor of $\begin{align*}k\end{align*}$ to create $\begin{align*}\triangle D^\prime E^\prime F^\prime.\end{align*}$
  ::DEF在P点上放大, 比例因子为 k , 以创建 D§E§F§。

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  are congruent after a dilation is performed, so $\begin{align*}\angle E\cong \angle E^\prime \!.\end{align*}$ Therefore, $\begin{align*}\angle E^\prime \cong \angle C\end{align*}$ as well. The scale factor was $\begin{align*}k\end{align*}$ , which is equal to $\begin{align*}\frac{AC}{DE}\end{align*}$ and $\begin{align*}\frac{BC}{FE}\end{align*}$ . This means:
  ::进行比照后是相同的, 所以 E E 。 因此, E C 也是。 比例系数是 k, 等于 ACDE 和 BCFE 。 这意味着 :

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  • $\begin{align*}D^\prime E^\prime=k\cdot DE=\frac{AC}{DE}\cdot DE=AC\end{align*}$ . Therefore, $\begin{align*}\overline{D^\prime E^\prime}\cong \overline {AC}\end{align*}$ .
    ::DEAAC,所以,是AC。
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  • $\begin{align*}F^\prime E^\prime=k\cdot FE=\frac{BC}{FE}\cdot FE=BC\end{align*}$ . Therefore, $\begin{align*}\overline{F^\prime E^\prime}\cong \overline {BC}\end{align*}$ .
    ::FEFE=BCFEFE,所以,是FEBE。

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  2. P rove that $\begin{align*}\triangle ABC\thicksim \triangle DFE.\end{align*}$
  ::2. 证明ABCDFE。

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  From #1, you know that $\begin{align*}\angle E^\prime \cong \angle C\end{align*}$ , $\begin{align*}\overline{D^\prime E^\prime}\cong \overline{AC}\end{align*}$ and $\begin{align*}\overline{F^\prime E^\prime} \cong \overline{BC}\end{align*}$ . This means $\begin{align*}\triangle ABC \cong \triangle D^\prime F^\prime E^\prime\end{align*}$ by $\begin{align*}SAS\cong.\end{align*}$
  ::从1,你知道C,EAC和F'E'BC,这意味着ASAS的ABC'D'F'E。

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  Therefore, there must exist a sequence of rigid transformations that will carry $\begin{align*}\triangle ABC\end{align*}$ to $\begin{align*}\triangle D^\prime F^\prime E^\prime.\end{align*}$
  ::因此,必须存在一系列的僵硬变换,将ABC带到ZD'F'E。

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  $\begin{align*}\triangle ABC\thicksim \triangle DFE\end{align*}$ because a series of rigid transformations will carry $\begin{align*}\triangle ABC\end{align*}$ to $\begin{align*}\triangle D^\prime F^\prime E^\prime\end{align*}$ , and then a dilation will carry to $\begin{align*}\triangle D^\prime F^\prime E^\prime\end{align*}$ to $\begin{align*}\triangle DFE\end{align*}$ .
  ::ABC-DFEE 因为一系列的硬质变换 将把ABC 带到ZDFEE, 然后将放大到DFEE。

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  The only given information regarding  the original two triangles  was that the  two pairs of proportional sides included congruent angles.
  ::唯一提供的关于最初两个三角的资料是,两对比例方包括相近角度。

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  You have proved that SAS is a criterion for triangle similarity.
  ::您已经证明SAS是三角相似性的标准。

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  Now, let's take a look at determining if two triangles are similar.
  ::现在,让我们来看看 确定两个三角形是否相似。

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  Are the two triangles below similar? Explain.
  ::下面两个三角形相似吗?

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  First look at what is marked. $\begin{align*}\angle C \cong \angle D\end{align*}$ . Also, $\begin{align*}\frac{AC}{DE}=2\end{align*}$ and $\begin{align*}\frac{AB}{FE}=2\end{align*}$ . Two sides are proportional but the congruent angle is not the included angle .
  ::首先看标记的是什么。 CD. 另外, ACDE=2 和 ABFE=2 。 双面是比例的, 但相容角度不是包含的角度 。

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  This is SSA which is not a way to prove that triangles are similar (just like it is not a way to prove that triangles are congruent).
  ::这是SSA, 无法证明三角形是相似的(就像它不能证明三角形是相似的一样)。

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  Look carefully at the two triangles. Notice that the longest side in $\begin{align*}\triangle ABC\end{align*}$ is $\begin{align*}\overline{BC}\end{align*}$ , which is unmarked. The longest side in $\begin{align*}\triangle EFD\end{align*}$ appears to be $\begin{align*}\overline {DE}\end{align*}$ , which is marked. $\begin{align*}\overline{BC}\end{align*}$ and $\begin{align*}\overline{DF}\end{align*}$ will not be proportional to the other pairs of sides.
  ::仔细看这两个三角形。 请注意, ABC 中最长的一面是 BC , 没有标记 。 $EFD 中最长的一面似乎是 DE , 有标记 。 $BC 和 $FD 将不会与其他两边成比例 。

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

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  SAS Similarity
  ::SAS SAS 相似性

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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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  SAS is a criterion for both triangle similarity and triangle congruence . What's the difference between the two criteria?
  ::SAS是三角相似性和三角一致性的标准。 两种标准之间有什么区别?

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  With $\begin{align*}SAS\cong,\end{align*}$ you must show that two pairs of sides are congruent and their included angles are congruent as well. With $\begin{align*}SAS\thicksim,\end{align*}$ you must show that two pairs of sides are proportional and their included angles are congruent. Two triangles that are similar by $\begin{align*}SAS\thicksim\end{align*}$ with a scale factor of 1 will be congruent.
  ::在 SAS 中, 您必须显示两对对齐, 它们包含的角度也是一致的。 在 SAS 中, 您必须显示两对对齐, 它们包含的角度是正比的。 两个与 SAS 相类似的三角, 其比例系数为 1 将会是正比的 。

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  Example 2
  ::例2

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  For the triangles to be similar, corresponding angles must be congruent and corresponding sides must be proportional. $\begin{align*}\frac{AC}{ED}\end{align*}$ $\begin{align*}=\end{align*}$ $\begin{align*}\frac{BC}{FD}\end{align*}$ $\begin{align*}=2\end{align*}$ . $\begin{align*}\angle C \cong \angle D\end{align*}$ so the included angles are congruent. Therefore, the triangles are similar by $\begin{align*}SAS \thicksim.\end{align*}$
  ::三角形要相似, 相应的角必须是相近的, 相应的边必须成正比 。 ACED= BCFD=2. CD, 所以包含的角是相近的 。 因此, 三角形与 SAS 相似 。

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  Example 3
  ::例3

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  Are the triangles similar? Explain.
  ::三角形相似吗?

  

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  While two pairs of sides are proportional and one pair of angles are congruent, the angles are not the included angles. This is SSA, which is not a similarity criterion. Therefore, you cannot say for sure that the triangles are similar.
  ::虽然两边是成比例的,一对角度是相近的, 但角度不是包含的角度。 这是 SSA, 这不是相似的标准 。 因此, 您不能肯定三角形是相似的 。

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  Example 4
  ::例4

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  What additional information would you need to be able to say that $\begin{align*}\triangle ABC\thicksim \triangle EBD?\end{align*}$
  ::您还需要什么补充信息才能说出@ABC_EBD?

  

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  Because $\begin{align*}\overline{BD}\cong \overline{DC}\end{align*}$ , $\begin{align*}\frac{BC}{BD}=2\end{align*}$ . The two triangles share $\begin{align*}\angle B\end{align*}$ , so that is a pair of congruent angles.
  ::因为BCB=2,两个三角形共享B, 所以这是一对相似的角度。

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  To prove that the triangles are similar, you would need to know that $\begin{align*}\frac{BA}{BE}=2\end{align*}$ as well. If you knew that $\begin{align*}\overline{BE}\cong \overline{EA}\end{align*}$ or $\begin{align*}E\end{align*}$ was the midpoint of $\begin{align*}\overline{AB}\end{align*}$ , then you could say that the triangles are similar.
  ::要证明三角形是相似的,你也需要知道BABE=2。如果你知道“BE”或“E”是AB的中点,那么你可以说这些三角形是相似的。

  Summary

  播放段落 The SAS criterion for triangle similarity states that if two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar. ::关于三角相似性的SAS标准规定,如果一个三角的两面与另一个三角的两面成正比,而且其中的角是相同的,那么三角就是相似的。 播放段落 SSA which is not a way to prove that triangles are similar (just like it is not a way to prove that triangles are congruent). ::SSA不是用来证明三角形相似的一种方法(就像它不是用来证明三角形相似的一种方法) 。

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  Review
  ::审查审查审查审查

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  1. What does SAS stand for? What does it have to do with similar triangles?
  ::1. SAS代表什么?它与类似的三角有什么关系?

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  2. What does SSA stand for? What does it have to do with similar triangles?
  ::2. 特别服务协定代表什么?它与类似的三角有什么关系?

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  3. Draw an example of two triangles that must be similar due to SAS.
  ::3. 举两个三角形的例子,这两个三角形必须类似于SAS。

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  4. Draw an example of two triangles that are not necessarily similar because all you know is SSA.
  ::4. 举两个三角形的例子,这两个三角形不一定相似,因为你们只知道特别服务协定。

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  For each pair of triangles below state if they are similar, congruent, or if there is not enough information to determine whether or not they are congruent. If they are similar or congruent, write a similarity or congruence statement.
  ::对于下方的每对三角形,如果它们相似、一致,或者如果没有足够的信息来确定它们是否一致。如果它们相似或一致,请写一个相似或一致的语句。

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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 similar by SAS?
  ::11. 为了能够说明以下三角与SAS相似,您需要哪些最低限度的额外信息?

  

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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 similar by SAS?
  ::12. 为了能够说明以下三角与SAS相似,您需要哪些最低限度的额外信息?

  

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

  

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  14. AAS and ASA are both criteria for triangle congruence. Are they also criteria for triangle similarity? Explain.
  ::14. AAS和ASA是三角一致的标准,它们也是三角相似性的标准吗?

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  15. Show how the SAS criterion for triangle similarity works: use transformations to help explain why the triangles below are similar given that $\begin{align*}\frac{AC}{FD}=\frac{CB}{DE}\end{align*}$ .
  ::15. 显示三角相近性SAS标准是如何运作的:利用变换来解释以下三角为何相似,因为ACFD=商用八溴二苯醚。

  

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  16. Given two right triangles whose legs are proportional to each other, are the triangles similar? Diagram and explain.
  ::16. 对于双腿对齐的两个右三角,三角形相似吗?图表和解释。

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  17. Given two isosceles triangles such that the vertex angles are congruent, are the triangles similar? Diagram and explain.
  ::17. 鉴于两个等分三角形,使顶端角相近,三角形相似吗?图表和解释。

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  18. Are all equilateral triangles similar? Why or why not?
  ::18. 所有等边三角是否都相似?为什么或为什么没有?

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  19. Given the diagram below, are the two triangles similar? Why or why not?
  ::19. 如下图所示,这两个三角是否相似?为什么或为什么没有?

  

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  20.  $\begin{align*}\triangle AEF\end{align*}$ is the image resulting from a dilation of $\begin{align*}\triangle ABC\end{align*}$ . Give the center of dilation and the approximate scale factor. Explain.
  ::20. QAEF是 QABC 放大后产生的图像。 给出放大中心和大致比例系数。 请解释 。

  

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  21. Given: Diagram as shown;  $\begin{align*}\frac{a}{a+b}=\frac{c}{c+d}\end{align*}$
  ::21. 参考:所示图表;aa+b=cc+d

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  Prove: $\begin{align*}\triangle AEF \sim \triangle ABC\end{align*}$
  ::证明:_AEF_ABC

  

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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 。