4.3 线线和角定理的应用-interactive

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  线和角定理的应用

  There are four categories of theorems to remember that have to do with lines and angles :

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  1. When two lines intersect, two pairs of vertical angles are formed.
  ::1. 当两条线交叉时,形成两对垂直角度。

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  In the diagram above, $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }3\end{array}$ are . $\begin{array}{r}\mathrm{\angle }2\end{array}$ and $\begin{array}{r}\mathrm{\angle }4\end{array}$ are also vertical angles. Vertical angles are always congruent .
  ::在上图中,%1和%3是.%2和%4也是垂直角度。垂直角度总是一致的。

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  2. When two lines are cut by a transversal , many different angle pairs are formed. If the two lines are parallel , these angle pairs have special properties. 
  ::2. 当两条线被横切切割时,会形成许多不同的角对,如果两条线是平行的,这些角对具有特殊特性。

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  All Angles
  ::全部角

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  • $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }5\end{array}$ are corresponding angles because their locations are corresponding . If lines are parallel, then are congruent. Other examples of corresponding angles are $\begin{array}{r}\mathrm{\angle }2\end{array}$ and $\begin{array}{r}\mathrm{\angle }6\end{array}$ , $\begin{array}{r}\mathrm{\angle }3\end{array}$ and $\begin{array}{r}\mathrm{\angle }7\end{array}$ , and $\begin{array}{r}\mathrm{\angle }4\end{array}$ and $\begin{array}{r}\mathrm{\angle }8.\end{array}$
    ::\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\
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  • $\begin{array}{r}\mathrm{\angle }4\end{array}$ and $\begin{array}{r}\mathrm{\angle }5\end{array}$ are same side interior angles because they are inside the lines and on the same side of the transversal . If lines are parallel, then are supplementary. Another example of same side interior angles is $\begin{array}{r}\mathrm{\angle }3\end{array}$ and $\begin{array}{r}\mathrm{\angle }6.\end{array}$
    ::4 和 5 是相同的侧内侧角度, 因为它们位于线内, 并且位于横跨线的同一侧。 如果线是平行的, 那么是补充的 。 相同的侧内侧角度的另一个例子是 3 和 6 。
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  • $\begin{array}{r}\mathrm{\angle }3\end{array}$ and $\begin{array}{r}\mathrm{\angle }5\end{array}$ are alternate interior angles because they are inside the lines and on opposite sides of the transversal. If lines are parallel, then are congruent. Another example of alternate interior angles is $\begin{array}{r}\mathrm{\angle }4\end{array}$ and $\begin{array}{r}\mathrm{\angle }6.\end{array}$
    ::3 和 5 是其它内部角度, 因为它们在横贯线的内侧和对面。 如果线是平行的, 那么是相似的。 另一个替代内部角度的例子就是 4 和 6 。
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  • $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }7\end{array}$ are alternate exterior angles . If lines are parallel, then are congruent. Another example of alternate exterior angles is $\begin{array}{r}\mathrm{\angle }2\end{array}$ and $\begin{array}{r}\mathrm{\angle }8.\end{array}$  
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  • $\begin{array}{r}\mathrm{\angle }2\end{array}$ and $\begin{array}{r}\mathrm{\angle }7\end{array}$ are same side exterior angles . If lines are parallel, then same side exterior angles are supplementary. Another example of same side exterior angles is $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }8.\end{array}$  
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  3. The converses of all of the above theorems and postulates are also true and are ways to show that lines are parallel.
  ::3. 上述所有理论和假设的对立面也是真实的,是显示线条平行的方法。

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  For example, if corresponding angles are congruent then the lines that form them must be parallel. Similarly, if same side interior angles are supplementary then the lines that form them must be parallel.
  ::例如,如果相应的角度是相似的,则构成这些角度的线条必须是平行的。 同样,如果相同的侧侧内角是补充的,则构成这些角度的线条必须是平行的。

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  4. When a line segment is bisected by a perpendicular line ,  all points on the perpendicular bisector are exactly equidistant from the segment's endpoints (points $\begin{array}{r}A\end{array}$  and $\begin{array}{r}B\end{array}$  in the image below).
  ::4. 当一条线段被一条垂直线分两部分时,横直线两部分上的所有点与该段端点(下图中的A点和B点)完全相等。

  

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  For the figure above, as $\begin{array}{r}C\end{array}$ moves along the perpendicular bisector, it will always be true that $\begin{array}{r}\overline{AC}\cong \overline{CB}\end{array}$ .
  ::上图中,C在直角两侧区域移动时,AC ZCB永远是真实的。

  

  INTERACTIVE

  Theorems about Lines and Angles

  

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  Move the blue point to change the position of point $C$ . Move the red point into the open circle to draw  $\overline{AC}$  and $\overline{BC}$ .  
  ::移动蓝色点以改变 C点的位置。 将红点移到开放圆以绘制 AC 和 BC 。

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  If you remember all of the above postulates and theorems, you can use them to help solve problems.
  ::如果您记得上面所有的假设和定理, 您可以用它们来帮助解决问题 。

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  Measuring Angles
  ::测量角度

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  If $\begin{array}{r}m\mathrm{\angle }KIB={105}^{\circ }\end{array}$ , what is:
  ::如果MKIB=105,什么是:

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      a.  $\begin{array}{r}m\mathrm{\angle }JIE\end{array}$
  ::a. 公 约

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  • $\begin{array}{r}m\mathrm{\angle }JIE={105}^{\circ }\end{array}$  because it is a vertical angle with $\begin{array}{r}\mathrm{\angle }KIB\end{array}$ and vertical angles are congruent.
    ::mJIE=105,因为它是一个垂直角度,与 KIB 相近,而垂直角度是相近的。

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      b. $\begin{array}{r}m\mathrm{\angle }DEI\end{array}$
  ::b. m DEI

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  • $\begin{array}{r}m\mathrm{\angle }DEI={105}^{\circ }\end{array}$  because it is a corresponding angle with $\begin{array}{r}\mathrm{\angle }KIB\end{array}$ and corresponding angles are congruent when lines are parallel (note that the $\begin{array}{r}>\end{array}$ markings indicate that the lines are parallel).
    ::mDEI=105 因为它是一个与 KIB 相对应的角,而当线线平行时对应的角是相同的(注意 > 标记表示线线是平行的)。

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      c. $\begin{array}{r}m\mathrm{\angle }GEI\end{array}$
  ::c. m GEI

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  • $\begin{array}{r}m\mathrm{\angle }GEI={75}^{\circ }\end{array}$  because it forms a straight line with $\begin{array}{r}\mathrm{\angle }DEI\end{array}$ and so those angles are supplementary .
    ::mGEI=75 因为它与 DEI 形成直线, 所以这些角度是补充的 。

  

  INTERACTIVE

  Applications of Line and Angle Theorems

  

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  Move the red point into the red circle to bring the two blue parallel lines together.
  ::将红点移入红圆,将两条蓝色平行线连在一起。

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  Find $m\angle 2.$
  
  ::找m% 2 。

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  Measuring Line Segments
  ::测量线段

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  Find the length of $\begin{array}{r}\overline{CB}\end{array}$ .
  ::查找 CB 的长度 。

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  The markings in the picture indicate that $\begin{array}{r}D\end{array}$ is the midpoint of $\begin{array}{r}\overline{AB}\end{array}$ and $\begin{array}{r}\mathrm{\angle }CDB\end{array}$ is a right angle . This means that $\begin{array}{r}\stackrel{\leftrightarrow }{CD}\end{array}$ is the perpendicular bisector of $\begin{array}{r}\overline{AB}\end{array}$ . Therefore, $\begin{array}{r}C\end{array}$ must be equidistant from $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ , and $\begin{array}{r}CB=2\text{cm}\end{array}$ .
  ::图片中的标记显示 D 是 AB 的中点, 并且 {CDB 是一个正确角度。 这意味着 CD {} 是 AB 的垂直双形区块。 因此, C 必须与 A 和 B 等距, 和 CB = 2 cm 。

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  Writing a Proof
  ::a 写作证明书

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  $\begin{array}{r}ABCD\end{array}$ is shown below. Prove that $\begin{array}{r}\mathrm{△}ABC\cong \mathrm{△}CDA\end{array}$ .
  ::ABC+CDA 证明这一点。

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  Recall that the definition of a parallelogram is a quadrilateral with two pairs of parallel sides. Since this is a parallelogram, you know that  $\begin{array}{r}\overline{AD}\parallel \overline{BC}\end{array}$  and  $\begin{array}{r}\overline{AB}\parallel \overline{DC}\end{array}$  (Remember that  $\begin{array}{r}\parallel \end{array}$  means parallel). With parallel lines comes lots of congruent angles. These angles will help you to show that the triangles are congruent.  
  ::回顾平行图形的定义是一个具有两对平行边的四边形。 由于这是一个平行图, 你知道 AD {BC} 和 AB {DC } (记住 {BC 和 AB {DC 意指平行 ) 。 平行线有很多相似角度。 这些角度会帮助您显示三角形是相似的 。

  播放段落 Statements ::声明声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明 声明	播放段落 Reasons ::原因原因原因原因
  播放段落 $\begin{array}{r}\overline{AD}\Vert \overline{BC}\end{array}$ and $\begin{array}{r}\overline{AB}\Vert \overline{DC}\end{array}$ ::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}AB 和AB DC {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}AB	播放段落 Definition of a parallelogram ::平行图的定义
  播放段落 $\begin{array}{r}\mathrm{\angle }DAC\cong \mathrm{\angle }BCA\end{array}$ , $\begin{array}{r}\mathrm{\angle }ACD\cong \mathrm{\angle }CAB\end{array}$ ::* DACBCA, ACDCAB	播放段落 Alternate interior angles are congruent if lines are parallel ::如果线条平行, 替代的内侧角度是相似的
  $\begin{array}{r}\overline{AC}\cong \overline{AC}\end{array}$	播放段落 Reflexive Property ::弹性财产
  $\begin{array}{r}\mathrm{△}ABC\cong \mathrm{△}CDA\end{array}$	播放段落 $\begin{array}{r}ASA\cong \end{array}$ ::亚塞俄比亚州

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  If you have trouble seeing the alternate interior angles, try extending the lines that form the parallelogram and focusing on one pair of parallel lines at a time.
  ::如果您无法看到其它的内部角度, 请尝试扩展形成平行图的线条, 并一次关注一对平行线条 。

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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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  What can you say about the relationship between $\begin{array}{r}\overline{AB}\end{array}$ and $\begin{array}{r}\overline{CD}\end{array}$ ? What does this have to do with ?
  ::你能对AB和CD之间的关系说些什么?

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  From the markings in the picture, you can see that $\begin{array}{r}\overline{AC}\cong \overline{CB}\end{array}$ and $\begin{array}{r}\overline{AD}\cong \overline{DB}\end{array}$ . This means that both $\begin{array}{r}C\end{array}$ and $\begin{array}{r}D\end{array}$ are equidistant from $\begin{array}{r}A\end{array}$ and $\begin{array}{r}B\end{array}$ . Therefore, both $\begin{array}{r}C\end{array}$ and $\begin{array}{r}D\end{array}$ are on the perpendicular bisector of $\begin{array}{r}\overline{AB}\end{array}$ . Therefore, $\begin{array}{r}\overline{CD}\end{array}$ must BE the perpendicular bisector of $\begin{array}{r}\overline{AB}\end{array}$ .
  ::从照片中的标记中,你可以看到AC {CB}和AD {DB}。这意味着 C和D都与A和B相距相当。因此,C和D都位于AB的垂直两侧区块上。因此,CD必须是AB的垂直两侧区块。

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  Quadrilateral $\begin{array}{r}ACBD\end{array}$ is a because it has two pairs of adjacent congruent sides. The above reasoning shows that one of the diagonals of a kite is the perpendicular bisector of the other diagonal .
  ::“四边ACBD”是因为它有两对相邻相近的对等面。 以上推理表明风筝的对角是另一对对角的垂直对角。

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  Example 2
  ::例2

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  In the diagram below, $\begin{array}{r}m\mathrm{\angle }ABC={50}^{\circ }\end{array}$ and $\begin{array}{r}m\mathrm{\angle }KIJ={80}^{\circ }\end{array}$ .
  ::在下图中, mABC=50 和 mKIJ=80 。

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  1. Find $\begin{array}{r}m\mathrm{\angle }EBI\end{array}$ .
  ::1. 寻找MQEBI。

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  $\begin{array}{r}m\mathrm{\angle }EBI={50}^{\circ }\end{array}$ because it is a vertical angle with $\begin{array}{r}\mathrm{\angle }ABC\end{array}$ and vertical angles are congruent.
  ::mEBI=50 因为它是一个垂直角度 与 ABC 具有相似的垂直角度 。

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  2. Find $\begin{array}{r}m\mathrm{\angle }BIE\end{array}$ .
  ::2. 找到MBIE。

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  $\begin{array}{r}m\mathrm{\angle }BIE={80}^{\circ }\end{array}$ because it is a vertical angle with $\begin{array}{r}\mathrm{\angle }KIJ\end{array}$ and vertical angles are congruent.
  ::mBIE=80 因为它是一个垂直角度 与 KIJ 和垂直角度是相似的。

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  3. Find $\begin{array}{r}m\mathrm{\angle }BEI\end{array}$ .
  ::3. 找到MBEI。

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  $\begin{array}{r}m\mathrm{\angle }BEI=180-(80+50)={50}^{\circ }\end{array}$ because $\begin{array}{r}\mathrm{\angle }EBI\end{array}$ , $\begin{array}{r}\mathrm{\angle }BIE\end{array}$ , and $\begin{array}{r}\mathrm{\angle }BEI\end{array}$ form a triangle , and the sum of the measures of the interior angles of a triangle is $\begin{array}{r}{180}^{\circ }\end{array}$ .
  ::mBEI=180-(80+50)=50,因为 @EBI、ZBIE和ZBEI组成三角形,三角形内角的测量总和是180+。

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  4. Find $\begin{array}{r}m\mathrm{\angle }GEI\end{array}$ .
  ::4. 寻找MQGEI。

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  $\begin{array}{r}m\mathrm{\angle }GEI={80}^{\circ }\end{array}$ because it is a corresponding angle with $\begin{array}{r}\mathrm{\angle }KIJ\end{array}$ and corresponding angles are congruent when lines are parallel.
  ::mGEI=80 因为它是QKIJ的对应角度, 当线条平行时相应的角度是相同的 。

  

  INTERACTIVE

  Applications of Line and Angle Theorems

  

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  Move the red points to move the blue transversal .
  
  ::移动红点以移动蓝色横轴 。

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  INTERACTIVE

  Applications of Line and Angle Theorems

  

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  The red lines $l$  and $m$  are parallel. Move the red point to change the position of the blue transversal $t.$  Move the blue point into the blue circle to bring the two red lines together and compare the blue angles.
  ::红线 I 和 m 是平行的。 移动红点以改变蓝色横向 t 的位置。 将蓝点移动到蓝色圆上, 将两条红线连接起来, 比较蓝色角度 。

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

  Summary

  播放段落 Alternate exterior angles are two angles that are on the exterior of two parallel lines, but on the opposite sides of the transversal. ::不同的外部角度是两个角度 位于两条平行线的外侧, 而在横向的对面。 播放段落 Corresponding angles are called that because their locations correspond: they are formed on different lines but in the same position. ::相应的角度被称作, 因为它们的位置相对应: 它们由不同的线条组成, 但位置相同 。 播放段落 Same side interior angles (sometimes called consecutive interior angles) are on the interior of the figure (between the parallel lines) and on the same side of the transversal.  ::同侧内角(有时称为连续内角)位于图的内部(平行线之间)和横向的同一侧。 播放段落 Alternate interior angles are also on the interior of the figure, but on opposite sides of the transversal.  ::其他内部角度也位于图的内侧,但位于横面的对面。 播放段落 For same s ide exterior angles (sometimes consecutive exterior angles), if lines are parallel, then same side exterior angles are supplementary. ::对于相同的侧外部角度(有时是连续的外部角度),如果线条平行,则相同的侧外部角度是补充的。

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  Review
  ::审查审查审查审查

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  1. Draw an example of vertical angles.
  ::1. 举一个垂直角度的例子。

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  Use the diagram below for questions 2-4.
  ::问题2-4使用下图。

  

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  2. Give an example of same side interior angles. Name each angle with three letters.
  ::2. 举一个同侧内部角度的例子,用三个字母给每个角度命名。

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  3. Give an example of alternate interior angles. Name each angle with three letters.
  ::3. 举一个替代内部角度的例子,用三个字母给每个角度命名。

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  4. Give an example of corresponding angles. Name each angle with three letters.
  ::4. 举一个相应角度的例子,用三个字母给每个角度命名。

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  5. If lines are not parallel, are corresponding angles still congruent?
  ::5. 如果线条不平行,相应的角度是否仍然一致?

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  For 6-9, determine whether or not the lines are parallel based on the given angle measures. Explain your answer in each case.
  ::对于 6- 9, 根据给定角度计量, 确定线条是否平行。 请解释您在每种情况下的回答 。

  6.

  

  7.

  

  8.

  

  9.

  

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  10. In the diagram below, $\begin{array}{r}C\end{array}$ is the midpoint of $\begin{array}{r}\overline{BD}\end{array}$ . Prove that $\begin{array}{r}\mathrm{△}ABC\cong \mathrm{△}EDC\end{array}$ .
  ::10. 在下图中,C是BD的中点。

  

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  11. Extend your proof from #10 to prove that $\begin{array}{r}\overline{AC}\cong \overline{CE}\end{array}$ .
  ::11. 将证据从10号延伸至10号 以证明AC CE。

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  12. Which two line segments must be parallel in the picture below?
  ::12. 在下文中,哪些线段必须平行?

  

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  13. The measures of two angles are given below. Solve for $\begin{array}{r}x\end{array}$ .
  ::13. 以下列出两个角度的计量。

  

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  14. The measures of two angles are given below. Solve for $\begin{array}{r}x\end{array}$ .
  ::14. 以下列出两个角度的计量。

  

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  15. $\begin{array}{r}D\end{array}$ is the midpoint of $\begin{array}{r}\overline{AB}\end{array}$ and $\begin{array}{r}\overline{AC}\cong \overline{CB}\end{array}$ . Find the length of $\begin{array}{r}\overline{AB}\end{array}$ .
  ::15. D是AB和AC CB的中点。

  

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  16. Given: $\begin{array}{r}\stackrel{\leftrightarrow }{AB}\parallel \stackrel{\leftrightarrow }{DE}\end{array}$ ; $\begin{array}{r}\overline{AB}\cong \overline{DE}\end{array}$ ; $\begin{array}{r}\overline{CB}\cong \overline{EF}\end{array}$
  ::16. 鉴于:AB-DE_;AB_DE_;AB_DE_;CB_EF_

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  Prove: $\begin{array}{r}\mathrm{\angle }BCA\cong \mathrm{\angle }EFD\end{array}$
  ::证明:_BCA_EFD

  

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  17. Given: $\begin{array}{r}\mathrm{\angle }1\cong \mathrm{\angle }2\end{array}$ ; congruent segments as marked in diagram
  ::17. 参照:%1 @%2;图表中标注的相近区段

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  Prove: $\begin{array}{r}\mathrm{\angle }BCA\cong \mathrm{\angle }EFD\end{array}$
  ::证明:_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

  

    

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  18. Given $\begin{array}{r}h\parallel i\end{array}$ , f ind $\begin{array}{r}x\end{array}$ in the diagram shown.
  ::18. 鉴于hi,请在图表中找到x。

  

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  19. Given: $\begin{array}{r}f\parallel g\end{array}$
  ::19. 参照:f_____________________________________________________________________________________

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  Prove: $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }3\end{array}$ are supplementary
  ::证明:%1和%3是补充的

  

    

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  20. Given: $\begin{array}{r}f\parallel g\end{array}$ ; $\begin{array}{r}h\parallel i\end{array}$
  ::20. 参照:f___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Prove: $\begin{array}{r}\mathrm{\angle }1\end{array}$ and $\begin{array}{r}\mathrm{\angle }3\end{array}$ are congruent  
  ::证明:% 1 和% 3 相同

  

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  21. Given $\begin{array}{r}f\parallel g\end{array}$ , find $\begin{array}{r}x\end{array}$ and $\begin{array}{r}y\end{array}$ in the diagram below.
  ::21. 根据fg,在下图中找到x和y。

  

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  22. Given: $\begin{array}{r}\mathrm{\angle }1\cong \mathrm{\angle }2\end{array}$
  ::22. 鉴于:%1%2

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  Prove: $\begin{array}{r}\mathrm{\angle }3\cong \mathrm{\angle }4\end{array}$
  ::证明:

  

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  23. Given: $\begin{array}{r}g\parallel f\end{array}$ and $\begin{array}{r}M\end{array}$ is the midpoint of $\begin{array}{r}\overline{AC}\end{array}$ . Describe a rigid motion transformation that maps $\begin{array}{r}\mathrm{\angle }BAM\end{array}$ to $\begin{array}{r}\mathrm{\angle }DCM\end{array}$ .
  ::23. 鉴于:gf和M是AC的中点。描述一个刻板的动作转换,将 BAM 映射为 DCM 。

  

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  Review (Answers) 
  ::审查(答复)

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