4.3 共和三角

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  共成三角形

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  Congruent Triangles
  ::共成三角形

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  Two figures are congruent if they have exactly the same size and shape. If two triangles are congruent, they will have exactly the same three sides and exactly the same three angles . In other words, two triangles are congruent if you can turn, flip, and/or slide one so it fits exactly on the other.
  ::如果两个数字的大小和形状完全相同,两个数字是相同的。如果两个三角是相同的,它们将具有完全相同的三边和完全相同的三个角度。换句话说,如果您可以翻转、翻转和/或幻灯片1,两个三角是相同的,因此它完全适合另一边。

  

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  $\begin{align*}\triangle ABC\end{align*}$ and $\begin{align*}\triangle DEF\end{align*}$ are congruent because
  ::ABC和ZZDEF是相同的,因为

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  $$\begin{align*}\overline{AB} \cong \overline{DE} \qquad \angle{A} \cong \angle{D}\\ \overline{BC} \cong \overline{EF} \ \text{and} \ \angle{B} \cong \angle{E}\\ \overline{AC} \cong \overline{DF} \qquad \angle{C} \cong \angle{F}\end{align*}$$
  ::AB DE

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  Notice that when two triangles are congruent their three pairs of and their three pairs of corresponding sides are congruent.
  ::请注意,当两个三角对齐时,三对三角对齐时,三对对对齐时,三对对对齐时,三对对对齐时,三对对对齐。

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  When referring to corresponding congruent parts of , you can use the phrase C orresponding P arts of C ongruent T riangles are C ongruent, or its abbreviation .
  ::当指向对应的一致部分时,您可以使用Congruent三角的对应部分是Congruent, 或其缩略语。

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  What if you were given two triangles with all the angle measures and all the side lengths marked? How could you tell if the two triangles were congruent?
  ::如果给您两个三角形, 加上所有角度的度量和所有侧长标记呢? 您如何知道这两个三角形是否相容 ?

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  Examples
  ::实例

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  Example 1
  ::例1

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  Determine if the triangles are congruent using the definition of congruent triangles.
  ::使用对等三角形的定义,确定三角形是否一致。

  

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  We can see from the markings that $\begin{align*}\angle B \cong \angle C\end{align*}$ , $\begin{align*}\angle A \cong \angle D\end{align*}$ , and $\begin{align*}\angle AEB \cong \angle DEC\end{align*}$ because they are . Also, we know that $\begin{align*}\overline{BA} \cong \overline {CD}\end{align*}$ , $\begin{align*}\overline{EA} \cong \overline {ED}\end{align*}$ , and $\begin{align*}\overline{BE} \cong \overline {CE}\end{align*}$ . Because three pairs of sides and three pairs of angles are all congruent and they are corresponding parts, this means that the two triangles are congruent.
  ::我们从这些标记中可以看到,BC,AAD,和AEBDE, 因为它们是。此外,我们知道BACD,EAEEED,和BECE。因为三对两边和三对角度是相同的,它们都是相应的部分,这意味着两个三角是相同的。

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  Example 2
  ::例2

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  Determine if the triangles are congruent using the definition of congruent triangles.
  ::使用对等三角形的定义,确定三角形是否一致。

  

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  While there are congruent corresponding parts, there are only two pairs of congruent sides, the marked ones and the shared side. Without knowing whether or not the third pair of sides is congruent we cannot say if the triangles are congruent using the definition of congruent triangles. Note, this does not mean that the triangles are not congruent , it just means that we need more information in order to say they are congruent using the definition of congruent triangles (congruent triangles have three pairs of congruent angles and three pairs of congruent sides).
  ::虽然有一致的对应部分, 但只有两对一致的两边, 即标记的两边和共同的一面。 在不知道第三对是否一致的情况下, 我们无法用一致的三角的定义来判断三角是否一致。 注意, 这并不意味着三角是不一致的, 仅仅意味着我们需要更多的信息, 才能用一致的三角的定义来表示它们是一致的( 一致的三角有三对一致的角和三对一致的两边)。

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  Example 3
  ::例3

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  Are the two triangles below congruent?
  ::这两个三角形是否在一致之下?

  

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  To determine if the triangles are congruent, match up sides with the same number of tic marks: $\begin{align*}\overline{BC}\cong \overline{MN}, \ \overline{AB}\cong \overline{LM}, \ \overline{AC} \cong \overline{LN}\end{align*}$ .
  ::确定三角形是否吻合, 用相同数目的标记对齐两边: BC _NN, AB LM, AC LN 。

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  Next match up the angles with the same markings:
  ::下一个匹配角度时使用相同的标记 :

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  $\begin{align*}\angle{A} \cong \angle{L}, \ \angle{B} \cong \angle{M}\end{align*}$ , and $\begin{align*}\angle{C} \cong \angle{N}\end{align*}$ .
  ::A*L,B*M,和C*N。

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  Lastly, we need to make sure these are corresponding parts. To do this, check to see if the congruent angles are opposite congruent sides. Here, $\begin{align*}\angle{A}\end{align*}$ is opposite $\begin{align*}\overline{BC}\end{align*}$ and $\begin{align*}\angle{L}\end{align*}$ is opposite $\begin{align*}\overline{MN}\end{align*}$ . Because $\begin{align*}\angle{A} \cong \angle{L}\end{align*}$ and $\begin{align*}\overline{BC} \cong \overline{MN}\end{align*}$ , they are corresponding. Doing this check for the other sides and angles, we see that everything matches up and the two triangles are congruent.
  ::最后,我们需要确认这些是相应的部分。 要做到这一点, 请检查对应角度是否是对齐的一边。 这里, A是对BC 的对面, L是对齐的对齐的 。 因为 L 和 BC NN 是对应的 。 在对齐的一边和角度进行检查时, 我们可以看到所有的东西都是对齐的, 两个三角是对齐的 。

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  Example 4
  ::例4

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  If all three pairs of angles for two given triangles are congruent does that mean that the triangles are congruent?
  ::如果两个给定三角形的所有三对角度都一致,这是否意味着三角形是一致的?

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  Without knowing anything about the lengths of the sides you cannot tell whether or not two triangles are congruent. The two triangles described above might be congruent, but we would need more information to know for sure.
  ::在不知道两边长度的情况下, 你无法判断两个三角是否相容。 上述两个三角可能是相容的, 但我们需要更多信息才能确定。

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  Example 5
  ::例5

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  Determine if the triangles are congruent using the definition of congruent triangles.
  ::使用对等三角形的定义,确定三角形是否一致。

  

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  From the tic marks we can see that $\begin{align*}\overline{AB} \cong \overline {DE}\end{align*}$ . We also know that $\begin{align*}\angle ACB \cong \angle ECD \end{align*}$ because they are vertical angles. However, this is not enough information to know whether or not the triangles are congruent.
  ::我们还知道,“ACB”和“ECD”是垂直角度。然而,这还不足以说明三角形是否一致。

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  Review
  ::回顾

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  The following illustrations show two parallel lines cut by a transversal (You may assume that even though the lines are not marked parallel, the two lines that look parallel, actually are). Are the triangles formed by them definitively congruent?
  ::下面的插图显示了两条平行的横线被横切(你可以假设,即使两条线没有标出平行线,但两条看起来平行的线实际上是平行线 ) 。 三角形是否完全一致?

  1. 
  2. 
  3. 
  4. 
  5. 

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  Based on the following details, are the triangles definitively congruent?
  ::根据以下细节,三角形是否完全一致?

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  6. Both triangles are right triangles in which one angle measures $\begin{align*}55^{\circ}\end{align*}$ . All of their corresponding sides are congruent.
     ::两个三角形都是右三角形, 其中一个角度测量 55。 它们对应的两边都是相似的 。
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  7. Both triangles are equiangular triangles.
     ::两个三角形都是等角三角形。
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  8. Both triangles are equilateral triangles. All sides are 5 inches in length.
     ::两个三角形都是等边三角形。所有侧面长度为5英寸。
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  9. Both triangles are obtuse triangles in which one angle measures $\begin{align*}35^{\circ}\end{align*}$ . Two of their corresponding sides are congruent.
     ::两个三角形都是模糊的三角形,其中一个角度是35。两个对应的三角形是相同的。
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  10. Both triangles are obtuse triangles in which two of their angles measure $\begin{align*}40^{\circ}\end{align*}$ and $\begin{align*}20^{\circ}\end{align*}$ . All of their corresponding sides are congruent.
      ::两个三角形都是模糊的三角形,其中两个角度测量了40°和20°和20°。所有对应的两边都是相同的。
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  11. Both triangles are isosceles triangles in which one angle measures $\begin{align*}15^{\circ}\end{align*}$ .
      ::两个三角为等骨三角,其中一个角度测量 15 。
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  12. Both triangles are isosceles triangles with two equal angles of $\begin{align*}55^{\circ}\end{align*}$ . All corresponding sides are congruent.
      ::两个三角形都是等分三角形,两个等角为55。所有对应的两边都是相同的。
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  13. Both triangles are acute triangles in which two of their angles measure $\begin{align*}40^{\circ}\end{align*}$ and $\begin{align*}80^{\circ}\end{align*}$ . All of their corresponding sides are congruent.
      ::两个三角形都是急性三角形,其中两个角度测量了40°和80°。它们对应的两边都是相同的。
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  14. Both triangles are acute triangles in which one angle measures $\begin{align*}60^{\circ}\end{align*}$ . Two of their corresponding sides are congruent.
      ::两个三角形都是急性三角形,其中一个角度是60°Z。两个三角形对应的两边是相同的。
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  15. Both triangles are equilateral triangles.
      ::两个三角形都是等边三角形。

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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