6.10 承认和理解共形多边形

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  承认和理解共形多边形

  

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  A geodesic dome is a structure that looks like a half sphere and is made up of triangle supports. Geodesic domes are efficient and inexpensive to construct and therefore are used in home and commercial construction. Are the triangles in a geodesic dome congruent ?
  ::大地测量圆顶是一种结构,看上去像半个球体,由三角支持组成。 大地测量圆顶在建筑上效率高,成本低廉,因此用于家庭和商业建设。 三角形是否与大地测量圆顶相容?

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  In this concept, you will learn all about the importance of congruence and how to determine congruence.
  ::在这一概念中,你们将了解一致性的重要性以及如何确定一致性。

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  Congruent Polygons
  ::共形多边形

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  What does it mean when you say that two figures are congruent? The word congruent means exactly the same. When you have two figures of any kind that have the same size, shape and measure , you can say that these two figures are congruent.
  ::当你说两个数字是相同的时,它意味着什么?两个数字是相同的。一个词是相同的。如果两个数字具有相同的大小、形状和量度,你可以说这两个数字是相同的。

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  Let’s look at an example.
  ::让我们举个例子。

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  Are these two angles congruent?
  ::这两个角度一致吗?

  

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  If you look at these two angles, you will see that they are both 55°. The angles are labeled that way, so you can see that they are equal. This means that they are congruent. You can say that angle $\begin{array}{r}A\end{array}$  is congruent to angle $\begin{array}{r}B\end{array}$ .
  ::如果您看这两个角度, 你会看到它们都是55°。 角度被贴上这样的标签, 这样您就可以看到它们是相等的。 这意味着它们是相同的。 您可以说角度 A 和 角度 B 是相同的 。

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  The answer is $\begin{array}{r}\mathrm{\angle }A\cong \mathrm{\angle }B\end{array}$ .
  ::答案是"A"B

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  Just as you said that any two figures can be congruent, you can use this when you look at different types of polygons too. Two polygons can be considered congruent or not congruent.
  ::正如你说任何两个数字都可能一致,您也可以在查看不同类型的多边形时使用这两个数字。两个多边形可以被视为一致或不一致。

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  Let’s look at an example involving congruence.
  ::让我们看看一个涉及一致性的例子。

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  Are these two regular octagons congruent?
  ::这两个普通的八边形相似吗?

  

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  Look at these two octagons. They are exactly the same in every way. You can see that if you put one octagon on top of the other octagon that they would match up perfectly. The side lengths are also congruent and the angle measures are congruent. If two polygons are congruent, then it is a given that the side lengths and the angle measures are also congruent.
  ::查看这两个八边形。 它们在所有方面都是完全相同的。 您可以看到, 如果您将一个八边形放在其他八边形的顶部, 它们就会完全匹配。 侧边长度也是一样的, 角度测量也是一样的。 如果两个多边形是相同的, 那么则给定的边长和角测量也是一样的 。

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  Now that you know how to identify whether or not two figures are congruent, you can look at figuring out congruent parts and angles. 
  ::既然你知道如何确定两个数字是否一致, 你可以看一看如何找出相匹配的部件和角度。

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  First, let’s think again about the main characteristics of congruent polygons:
  ::首先,让我们再想想相容多边形的主要特点:

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  1. Same number of sides
     ::相同数边数
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  2. Corresponding sides are the same length
     ::对应方的长度相同
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  3. Corresponding interior angles are the same measure
     ::对应的内部角度是相同的量度

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  The last two characteristics can be a bit tricky. Sometimes, you will have two congruent figures, but all of the measures of the won’t be the same. For example, if you had two irregular congruent hexagons, this means that there are different angle measures in the two hexagons; however, they are congruent so there are “matching” angles between the two figures.
  ::最后两个特性可能有点棘手。 有时,你会有两个相近的数字,但所有对Won的测量都不一样。 比如,如果你有两个不规则的相匹配的六边形,这意味着在两个六边形中存在不同的角度量;然而,它们是相近的,因此两个数字之间有“对齐”的角度。

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  Take a look at these figures.
  ::看看这些数字。

  

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  Here you have two hexagons. They are irregular - which means that all of the side lengths and angles are not the same. However, they are congruent. You can see that one matches the other. Because of this, you have corresponding angles that are the same from the first hexagon to the second hexagon.
  ::这里您有两个六边形。 它们不规则, 这意味着所有的侧边长度和角度都不一样。 但是, 它们是相似的。 您可以看到一个和另一个吻合。 因此, 您的对应角度从第一个六边形到第二个六边形都是相同的。

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  You can identify . Corresponding parts can include side lengths and angle measures.
  ::您可以识别。对应的部件可以包括侧边长度和角度量度。

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  Let’s look at an example.
  ::让我们举个例子。

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  Name each pair of corresponding side lengths for these congruent figures.
  ::列出这些相容数字的每对对应的侧边长度。

  

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  First, look at these two congruent pentagons. To name the corresponding sides, you name the sides that match from one pentagon to another pentagon.
  ::首先,看看这两个相近的五边形。要给对应的边命名, 您要给对应的两边命名, 从一个五边形到另一个五边形。

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  Next, write the pairs of congruent sides. Here are the corresponding sides and how you can write them using mathematical notation.
  ::接下来, 写一对一致的两边。 这是对应的两边, 以及您如何用数学符号来写 。

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  $$\begin{array}{r}\begin{array}{rcl}\overline{BA}& \cong & \overline{GF}\\ \overline{BC}& \cong & \overline{GH}\\ \overline{CD}& \cong & \overline{HK}\\ \overline{AE}& \cong & \overline{FL}\\ \overline{ED}& \cong & \overline{LK}\end{array}\end{array}$$

  
  ::家 家 家 家 家 家 家 家

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  The answer is $\begin{array}{r}\overline{BA}\cong \overline{GF},\overline{BC}\cong \overline{GH},\overline{CD}\cong \overline{HK},\overline{AE}\cong \overline{FL},\overline{ED}\cong \overline{LK}\end{array}$ .
  ::答案是BA=GA=BA=BA=BA=BA=BA=CAE=AE=AE=AE=LK=LK=BA=BA=BA=BA=GA=BA=BA=BA=AE=AEE=AEE=LK=LK=LAY=LAY=LAY=LAY=YA=YAY=YA=YA=YA=YL=YA=YAY=YA=YAYYY=YA=YA=YA=YYYYYYYY=YY=YY=YYY=YY=YYY=YYY=YY=YY=YY=Y=YYYYY=YY=Y=Y===================================================================================== ===================================================================================================================================================================================================================================

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  Then, look at the angles. You can also look at the corresponding angles for two congruent figures. When two figures are congruent, then the matching angles will also be congruent.
  ::然后,请查看角度。您也可以查看两个相匹配的数字的相应角度。当两个数字相匹配时,匹配的角度也会相匹配。

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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 a  geodesic dome.
  ::早些时候,有人给了你一个关于大地测量圆顶的问题。

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  First, let’s think about what the word congruent means. Congruent means exactly the same. For an object to be congruent, the side lengths have to be the same. The triangles in the geodesic dome have to be congruent for it to stand up because the triangle is a structure that is well balanced to help with structure and security. Triangles are used in all kinds of construction like roofs and bridges.
  ::首先,让我们考虑一下“一致”一词的含义。 共性的意思是完全相同的。 一个目标要一致,侧长必须相同。 大地测量圆顶的三角必须一致才能站起来,因为三角形的结构非常平衡,有助于结构和安全。 三角形在屋顶和桥梁等各种建筑中都被使用。

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  The test for congruence of the triangles would be to see that the side lengths are the same.
  ::三角形一致性的检验标准是看侧边长度是否相同。

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  Example 2
  ::例2

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  Are these two hexagons congruent?
  ::这两个六边形相似吗?

  

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  First, look at these two shapes and compare them.
  ::首先,看看这两个形状,比较一下它们。

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  One shape is a regular hexagon and the other is an irregular hexagon.
  ::一个形状是普通的六边形,另一个形状是非常规的六边形。

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  Next, use the congruent polygon properties to see if the two hexagons are congruent.
  ::下一步,使用相容多边形属性来查看两个六边形是否相容。

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  In order to be congruent, corresponding sides must be the same length. The regular hexagon has all side lengths with the same measure. The irregular hexagon does not have sides that are the same length.
  ::为了做到一致,对应方必须是相同的长度。普通六边形具有所有侧长,且具有相同的度量。非常规六边形没有相同的边长。

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  The answer is no, the two hexagons are not congruent as their corresponding sides are not all the same measure.
  ::答案是否定的,两个六边形并不一致,因为它们对应的两边并非完全相同。

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  Use the following figures to answer each question.
  ::使用下列数字回答每个问题。

  

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  Example 3
  ::例3

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  What is the angle measure of angle $\begin{array}{r}F\end{array}$ ?
  ::F角度的角度量是什么?

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  First, find the corresponding angle in quadrilateral $\begin{array}{r}ABCD\end{array}$ .
  ::首先,在四边ABCD中找到相应的角度。

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  $$\begin{array}{r}\mathrm{\angle }F\cong \mathrm{\angle }B\end{array}$$

  
  ::财宝

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  Next, find the measure of the corresponding angle. This will be the measure of $\begin{array}{r}\mathrm{\angle }F\end{array}$ .
  ::下一步, 找到相应的角度的度量。 这将是 QF 的度量 。

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  The answer is 100°.
  ::答案是100摄氏度

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  Example 4
  ::例4

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  Angle $\begin{array}{r}D\end{array}$  is congruent to which other two angles?
  ::角度 D 是否与其他两个角度一致 ?

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  First, find the corresponding angle in quadrilateral $\begin{array}{r}EFGH\end{array}$ .
  
  

  $$\begin{array}{r}\mathrm{\angle }D\cong \mathrm{\angle }H\end{array}$$

  
  ::首先,在四边EFGH中找到相应的角度。

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  Next, look at quadrilateral $\begin{array}{r}ABCD\end{array}$ . What are the measures of all of angles?
  ::接下来,看看四边形ABCD。所有角度的量度是多少?

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  $$\begin{array}{r}\begin{array}{rcl}\mathrm{\angle }A& =& {80}^{\circ }\\ \mathrm{\angle }B& =& {100}^{\circ }\\ \mathrm{\angle }C& =& {80}^{\circ }\\ \mathrm{\angle }D& =& {100}^{\circ }\end{array}\end{array}$$

  
  ::A=80B=100C=80D=100

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  Then, if $\begin{array}{r}\mathrm{\angle }B=\mathrm{\angle }D\text{ and }\mathrm{\angle }B\cong \mathrm{\angle }F,\text{ then }\mathrm{\angle }D\cong \mathrm{\angle }H\cong \mathrm{\angle }F\end{array}$ .
  ::然后,如果"B"和"B", 然后"D"H"F。

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  The answer is $\begin{array}{r}\mathrm{\angle }D\cong \mathrm{\angle }H\cong \mathrm{\angle }F\end{array}$ .
  ::答案是'D'H'F.

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  Example 5
  ::例5

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  What is the measure of angle $\begin{array}{r}G\end{array}$ ?
  ::角度G的量度是多少?

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  First, find the corresponding angle in quadrilateral  $\begin{array}{r}ABCD\end{array}$ .
  ::首先,在四边ABCD中找到相应的角度。

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  $$\begin{array}{r}\mathrm{\angle }G\cong \mathrm{\angle }C\end{array}$$

  
  ::GC C C C C C C C C C C C C C C C C C C C C C

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  Next, find the measure of the corresponding angle. This will be the measure of $\begin{array}{r}\mathrm{\angle }G\end{array}$ .
  ::下一步, 找到相应角度的度量。 这将是% G 的度量 。

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  The answer is 80°.
  ::答案是80度

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  Review
  ::回顾

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  Answer each question true or false.
  ::回答每个问题是否真实或虚假。

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  1. Congruent means that a figure has the same side lengths but not the same angle measures.
  ::1. 共性是指数字的侧边长度相同,但角度尺度不同。

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  2. Congruent means exactly the same in every measure.
  ::2. 共性在每一项措施中都指完全相同的。

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  3. Similar means having the same shape, but not the same size.
  ::3. 类似的意思是形状相同,但大小不同。

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  4. Two congruent figures would have the same size and shape.
  ::4. 两个一致的数字大小和形状相同。

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  5. Corresponding parts are parts that are in the same figure.
  ::5. 相应的部分是同一数字中的部分。

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  6. You need to understand corresponding parts before you can determine if two figures are congruent.
  ::6. 在确定两个数字是否一致之前,需要了解相应的部分。

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  7. You can determine if two figures are congruent without knowing any of their measurements.
  ::7. 可以确定两个数字是否一致,而不知道它们的任何测量。

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  8. Similar figures are also congruent.
  ::8. 类似数字也是一致的。

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  9. If two triangles are equilateral triangles, then they are automatically congruent.
  ::9. 如果两个三角为等边三角,则它们自动相容。

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  10. If two quadrilaterals have measures of 360°, then they are congruent.
  ::10. 如果两个四边形的测量度为360°,那么它们就是相同的。

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  The two figures shown are congruent. Use the illustration to answer each question.
  ::显示的这两个数字是相同的。 使用插图回答每个问题 。

  

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  11. If angle $\begin{array}{r}B\end{array}$  has a measure of 75°, which other angle has the same measure?
  ::11. 如果角度B的测量度为75°,则其他哪个角度的测量度为75°?

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  12. If angle $\begin{array}{r}F\end{array}$  is 120°, which other angle has the same measure?
  ::12. 如果角度F为120度,其他哪个角度的测量尺度相同?

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  13. True or false. Angle $\begin{array}{r}E\end{array}$  and angle $\begin{array}{r}K\end{array}$  have the same measure.
  ::13. 真实或假的,角E和角K的量度相同。

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  14. True or false. Angle $\begin{array}{r}C\end{array}$  and angle $\begin{array}{r}H\end{array}$  have the same measure.
  ::14. 目标C和角度H具有相同的度量。

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  15. Name this figure.
  ::15. 请说明这一数字。

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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