4.8 四方理论的应用-interactive

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• 选择节四方理论的应用

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  四方理论的应用

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  M any properties of special quadrilaterals can be proved from the specific definitions of the quadrilaterals.
  ::从四边形的具体定义中可以证明特殊四边形的许多特性。

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  Below is a summary of quadrilateral definitions and properties .
  ::以下是四边定义和属性摘要。

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  Parallelogram
  ::平行图

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  A quadrilateral with two pairs of parallel sides.
  ::一个四边形,两对平行面

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  • Parallelogram Theorem #1 : Each diagonal of a divides the parallelogram into two congruent triangles.
    ::平行图定理# # 1: 每个对角将平行图分为两个相似的三角形。
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  • Parallelogram Theorem #1 Converse: If each of the diagonals of a quadrilateral divide the quadrilateral into two congruent triangles, then the quadrilateral is a parallelogram.
    ::平行图定理# 1 反面 : 如果四边形的对角将四边形分为两个相似的三角形, 那么四边形就是一个平行图。
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  • Parallelogram Theorem #2 : The opposite sides of a parallelogram are congruent.
    ::平行图定理 # 2 : 平行图的对面是相似的 。
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  • Parallelogram Theorem #2 Converse: If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
    ::平行图定理#2 反面:如果四边形的对面是相似的,那么四边形是平行图。
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  • Parallelogram Theorem #3 : The opposite angles of a parallelogram are congruent.
    ::3号平行图定理:平行图的相反角度是相同的。
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  • Parallelogram Theorem #3 Converse: If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
    ::3 反面:如果四边形的相反角度是相似的,那么四边形是平行的。
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  • Parallelogram Theorem #4 : The diagonals of a parallelogram bisect each other.
    ::平行图定理4: 平行图双形的对角。
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  • Parallelogram Theorem #4 Converse: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
    ::平行图定理 # 4 : 如果四边形的对角相对, 则四边形是平行图。

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  Rectangle
  ::矩形

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  A quadrilateral with four right angles.
  ::一个四边形,有四个右角度

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  • Rectangle Theorem #1 : A rectangle is a parallelogram.
    ::矩形定理 # 1 : 矩形是平行图。
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  • Rectangle Theorem #2 : A rectangle has congruent diagonals.
    ::矩形定理 #2: 矩形具有相似的对数法。
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  • Rectangle Theorem #2 Converse: If a parallelogram has congruent diagonals, then it is a rectangle.
    ::矩形定理 # 2 反面 : 如果平行图具有相似的对角线, 那么它是一个矩形 。

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  Rhombus
  ::滚轮

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  A quadrilateral with four congruent sides.
  ::一个四边形 四个对齐面

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  • Rhombus Theorem #1 : A rhombus is a parallelogram.
    ::Rhombus Theorem # 1 : 暴风车是一个平行图。
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  • Rhombus Theorem #2 : The diagonals of a rhombus are perpendicular .
    ::Rhombus Theorem # 2: 伦布鲁斯的对角是垂直的。
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  • Rhombus Theorem #3 : The diagonals of a rhombus bisect its angles.
    ::Rhombus Theorem # 3: 龙卷风的对角线,

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  Square : A quadrilateral with four right angles and four congruent sides.
  ::方形:四边形,有四个右角和四个相近的两面。

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  Kite
  ::Kite 键

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  A quadrilateral with two pairs of adjacent, congruent sides.
  ::一个四边形,有两对相邻、相近的两面。

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  • Kite Theorem #1 : One diagonal of a bisects the other diagonal.
    ::Kite Theorem #1:一个对角线,一个对角线,一个对角线,一个对角线,一个对角线,另一个对角线。
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  • Kite Theorem #2 : The diagonals of a kite are perpendicular.
    ::Kite Theorem # 2 : 风筝的对角是垂直的 。
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  • Kite Theorem #3 : One diagonal of a kite bisects its angles.
    ::Kite Theorem # 3: 风筝两角的一个对角,
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  • Kite Theorem #4 : A kite has one pair of opposite angles congruent.
    ::Kite Theorem # 4: 风筝有一对相反角度的对齐。

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  Investigate Kites and Rhombi
  ::调查Kites和Rhombi

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  Finding the Perimeter
  ::寻找周边

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  Find the perimeter of the quadrilateral below. 
  ::在下面找到四边形的周界

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  The markings indicate that the quadrilateral has four congruent sides . This means that the quadrilateral is a rhombus. One property of a rhombus is that its diagonals are perpendicular . This means that the four inner triangles are actually right triangles . You can use to find the hypotenuse of one of these triangles, which will be the length of each side of the rhombus.
  ::标记显示,四边形有四个相近面。 这意味着四边形是圆柱形。 圆柱形的一个属性是其对角形是垂直的。 这意味着四个内三角形实际上是右三角形。 您可以找到这些三角形中的一个, 也就是圆柱形两侧的长度。

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  $\begin{array}{rl}{3}^2+4^2& =c^2\\ 9+16& =c^2\\ 5& =c\end{array}$
  ::32+42=c29+16=c25=c

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  Since each side of the rhombus is 5 units long, the perimeter of the rhombus is $\begin{array}{r}P=5\cdot 4=20\end{array}$ units.
  ::由于暴风车的两侧各有5个长,因此暴风车的周边是P=54=20个。

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  Making and Proving Conjectures
  ::预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测 预测

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  The midpoints of each of the sides of parallelogram $\begin{array}{r}ABCD\end{array}$ have been connected, as shown below. Make a conjecture about the inner quadrilateral that has been formed. Then, prove the conjecture.
  ::平行图形 ABCD 的两侧的中点已经连接, 如下文所示。 对已经形成的内四边进行猜测。 然后, 证明这一猜测 。

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  The inner quadrilateral looks to be a parallelogram. To prove it is a parallelogram, prove that its opposite sides are congruent. (Parallelogram Theorem #2 Converse states that if the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.)
  ::内四边形看起来是一个平行图。 要证明它是一个平行图, 请证明它的对面是相似的 。 ( Parallelect Theorem # 2 相形之下说, 如果四边形的对面是相似的, 那么四边形是平行图 。 )

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  Given: Parallelogram $\begin{array}{r}ABCD\end{array}$ with midpoints $\begin{array}{r}E\end{array}$ , $\begin{array}{r}F\end{array}$ , $\begin{array}{r}H\end{array}$ , $\begin{array}{r}G\end{array}$ .
  ::中点为E、F、H、G的平行ABCD

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  Prove: $\begin{array}{r}EFHG\end{array}$ is a parallelogram
  ::证明:EFHG是一个平行图

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  Here is paragraph proof:
  ::此处为防排段落 :

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  Because $\begin{array}{r}ABCD\end{array}$ is a parallelogram , its opposite sides and angles are congruent. This means that $\begin{array}{r}\mathrm{\angle }C\cong \mathrm{\angle }A\end{array}$ , $\begin{array}{r}\mathrm{\angle }B\cong \mathrm{\angle }D\end{array}$ , $\begin{array}{r}\overline{AB}\cong \overline{DC}\end{array}$ , and $\begin{array}{r}\overline{AD}\cong \overline{BC}\end{array}$ . $\begin{array}{r}E\end{array}$ , $\begin{array}{r}F\end{array}$ , $\begin{array}{r}G\end{array}$ and $\begin{array}{r}H\end{array}$ are midpoints , so they divide each segment into two congruent segments. Because opposite sides of $\begin{array}{r}ABCD\end{array}$ are congruent, it must be true that $\begin{array}{r}\overline{AE}\cong \overline{CH}\end{array}$ , $\begin{array}{r}\overline{AF}\cong \overline{GC}\end{array}$ , $\begin{array}{r}\overline{FB}\cong \overline{GD}\end{array}$ , and $\begin{array}{r}\overline{ED}\cong \overline{BH}\end{array}$ . Therefore, $\begin{array}{r}\mathrm{△}AEF\cong \mathrm{△}CHG\end{array}$ and $\begin{array}{r}\mathrm{△}BFH\cong \mathrm{△}DGE\end{array}$ by $\begin{array}{r}SAS\cong \end{array}$ . This means that corresponding parts of these triangles are congruent and therefore $\begin{array}{r}\overline{EF}\cong \overline{GH}\end{array}$ and $\begin{array}{r}\overline{FH}\cong \overline{EG}\end{array}$ . Because opposite sides of $\begin{array}{r}EFHG\end{array}$ are congruent, $\begin{array}{r}EFHG\end{array}$ is a parallelogram .
  ::因为ABCD是一个平行图, 其对面和角度是相近的。 这意味着 C A, B D, AB DC, 和 AD B B 。 E, F, G 和 H 是中点 。 因此, 它们将每个段分为两个相近的段。 由于ABCD 的对面是相近的, AE CH , 和 ED B 。 因此, AEFCHG 和 BH DG 由 SAS 组成。 这意味着这些三角的对应部分是相近的, 因此 EFG 和 FH G。 因为 EFHG 的对面是相近的, EFHG 是一个平行的图表 。

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  Naming Shapes
  ::命名形状

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  Name the shape below based on its markings as precisely as you can. Don't assume that the shape is drawn to scale.
  ::尽可能精确地根据形状的标记命名下面的形状。 不要假设形状是被画成比例的 。

  

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  All that is marked is that the shape has four congruent sides. This is the definition of a rhombus so it must be a rhombus. Note that even though it might look like a square , if you don't know for sure that the shape has right angles, you cannot be sure it is a square.
  ::标记的只是形状有四个相容的面。 这是一个 rhombus 的定义, 所以它必须是一个 rhombus 。 请注意, 即使它看起来像一个正方形, 如果您不确定形状有正确的角度, 您也不能确定它是一个正方形 。

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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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  Consider parallelogram $\begin{array}{r}ABCD\end{array}$ below. $\begin{array}{r}F\end{array}$ is the midpoint of $\begin{array}{r}\overline{AB}\end{array}$ and $\begin{array}{r}G\end{array}$ is the midpoint of $\begin{array}{r}\overline{DC}\end{array}$ . Make at least one conjecture about how $\begin{array}{r}\overline{FG}\end{array}$ is related to $\begin{array}{r}\overline{AD}\end{array}$ .
  ::F是AB的中点,G是DC的中点。至少猜测一下FG与AD的关系。

  

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  Two possible conjectures are:
  ::两种可能的假设是:

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  1. $\begin{array}{r}\overline{FG}\Vert \overline{AD}\end{array}$
     ::FG 自动
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  2. $\begin{array}{r}\overline{FG}\cong \overline{AD}\end{array}$
     ::FG 发声

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  Example 2
  ::例2

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  In parallelogram $\begin{array}{r}ABCD\end{array}$ , $\begin{array}{r}F\end{array}$ is the midpoint of $\begin{array}{r}\overline{AB}\end{array}$ and $\begin{array}{r}G\end{array}$ is the midpoint of $\begin{array}{r}\overline{DC}\end{array}$ . Prove that $\begin{array}{r}\overline{FG}\cong \overline{AD}\end{array}$ .
  ::F是AB的中点,G是DC的中点。

  

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  Draw in $\begin{array}{r}\overline{AG}\end{array}$ . Prove that the two triangles formed are congruent, and therefore corresponding parts (the desired sides) are congruent.
  ::绘制在 AG 中。 证明形成的两个三角形是相近的, 因此相应的部分( 期望的侧面) 是相匹配的 。

  Statements	Reasons
  Parallelogram $\begin{array}{r}ABCD\end{array}$ with midpoints $\begin{array}{r}F\end{array}$ and $\begin{array}{r}G\end{array}$	Given
  $\begin{array}{r}\overline{AB}\cong \overline{DC}\end{array}$ , $\begin{array}{r}\overline{AB}\Vert \overline{DC}\end{array}$	Opposite sides of a parallelogram are congruent and parallel
  $\begin{array}{r}\overline{AF}\cong \overline{FB}\end{array}$ , $\begin{array}{r}\overline{DG}\cong \overline{GC}\end{array}$	Definition of midpoint
  $\begin{array}{r}\overline{AF}\cong \overline{DG}\end{array}$	Half of congruent segments are congruent
  $\begin{array}{r}\overline{AG}\cong \overline{AG}\end{array}$	Reflexive Property
  $\begin{array}{r}\mathrm{\angle }GFA\cong \mathrm{\angle }ADG\end{array}$	If lines are parallel then are congruent
  $\begin{array}{r}\mathrm{△}GFA\cong \mathrm{△}ADG\end{array}$	$\begin{array}{r}SAS\cong \end{array}$
  $\begin{array}{r}\overline{FG}\cong \overline{DA}\end{array}$

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  Example 3
  ::例3

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  Name the shape below based on its markings as precisely as you can. Don't assume that the shape is drawn to scale.
  ::尽可能精确地根据形状的标记命名下面的形状。 不要假设形状是被画成比例的 。

  

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  It is marked that the diagonals are perpendicular. Shapes with perpendicular diagonals are rhombuses, , and squares. Without additional information, you cannot say whether this is a rhombus, a kite, or a square. Therefore, all you can say for certain is that this shape is a quadrilateral.
  ::标记二角形是垂直的。 具有垂直对角形的形状是正方形, 和正方形。 没有其他信息, 您不能说它是圆柱形、 风筝还是正方形。 因此, 您只能说这个形状是四边形 。

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  Example 4
  ::例4

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  Use the markings on the shape to name the shape. Then, solve for $\begin{array}{r}x\end{array}$ .
  ::使用形状上的标记来命名形状。 然后, 解析 x 。

  

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  The shape has four congruent sides which makes it a rhombus. It also has one right angle . Opposite sides of a rhombus are parallel, so are supplementary. This means that $\begin{array}{r}x=90\end{array}$ . Because opposite angles of a rhombus are congruent, all four angles must be right angles. Therefore, a more precise name for this shape is a square.
  ::形状有四个相容的边, 使它成为圆柱形。 它也有一个右角。 圆柱形的对面面是平行的, 也是补充的。 这意味着 x=90。 因为圆柱形的对角是相近的, 所有四个角度必须是正角。 因此, 这个形状的更精确的名称是正方形 。

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

  Summary

  播放段落 A parallelogram is a quadrilateral with two pairs of parallel sides. 播放段落 Each diagonal of a parallelogram divides the parallelogram into two congruent triangles. ::每个平行方形的对角将平行方形分为两个相似的三角形。 播放段落 The opposite sides of a parallelogram are congruent. ::平行图的对面是相同的。 播放段落 The opposite angles of a parallelogram are congruent. ::平行图的相反角度是相同的。 播放段落 The diagonals of a parallelogram bisect each other. ::平行相形对齐的对角线 ::平行图是一个四边形, 有两对平行边。 平行图的对角将平行图分为两个相似的三角形。 平行图的对角是相似的。 平行图的对角是相似的。 平行图的对角是相似的。 平行图的对角是相似的。 平行图的对角是双形的对角。 播放段落 A rectangle is a quadrilateral with four right angles. 播放段落 A rectangle is a parallelogram. ::矩形是一个平行图。 播放段落 A rectangle has congruent diagonals. ::矩形具有相容的对角学。 ::矩形是四个右角度的四边形。矩形是平行图。矩形具有相同的对角形。 播放段落 A rhombus is a quadrilateral with four congruent sides. 播放段落 A rhombus is a parallelogram. ::暴风车是平行图。 播放段落 The diagonals of a rhombus are perpendicular. ::暴龙的对角是垂直的。 播放段落 The diagonals of a rhombus bisect its angles. ::龙卷风的对角线 将它的角度分成两部分 ::圆柱是一个四边形的四边形。 圆柱形是一个平行图。 圆柱形的对角是垂直的。 圆柱形的对角是两面的对角。 圆柱形的对角是其角度的对角。 播放段落 A kite is a quadrilateral with two pairs of adjacent, congruent sides. 播放段落 One diagonal of a kite bisects the other diagonal. ::一只风筝的对角线 另一只对角线的对角线 播放段落 The diagonals of a kite are perpendicular. ::风筝的对角是垂直的。 播放段落 One diagonal of a kite bisects its angles. ::风筝两面的对角 将它的角度分解成两面 播放段落 A kite has one pair of opposite angles congruent. ::风筝有一对相反角度的对立面 ::风筝是一种四边形, 配有两对相邻、 相近的两面。 一对风筝的对角分解了另一对对角。 一对风筝的对角是垂直的。 一对风筝的对角是垂直的。 一对风筝的对角是其角度的对角。 一对风筝的对角是相近的对角。

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  Review
  ::审查审查审查审查

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  1. True or false: A rectangle is always a square.
  ::1. 真实或虚假:矩形始终是方形。

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  2. True or false: A rhombus is always a parallelogram.
  ::2. 真实或虚假:暴风雪总是平行图。

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  3. True or false: A square is always a rectangle.
  ::3. 真实或虚假:方形总是矩形。

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  4. True or false: A kite is always a quadrilateral.
  ::4. 真实的或虚假的:风筝始终是四边形。

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  Name each shape below based on its markings as precisely as you can. Don't assume that the shape is drawn to scale.
  ::根据每个形状的标记, 请尽可能精确地在下面命名每个形状。 不要假设形状是被拉到比例的 。

  5.

  

  6.

  

  7.

  

  8.

  

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  For each quadrilateral below, name the shape. Then, solve for $\begin{array}{r}x\end{array}$ .
  ::对于以下的每个四边形, 命名形状。 然后, 解答 x 。

  9.

  

  10.

  

  11.

  

  12.

  

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  13. Earlier you proved that if points $\begin{array}{r}F\end{array}$  and $\begin{array}{r}G\end{array}$  are midpoints on the edges of parallelogram $\begin{array}{r}ABCD\end{array}$ then $\begin{array}{r}\mathrm{△}AGF\cong \mathrm{△}GAD.\end{array}$  Given this, prove that $\begin{array}{r}\overline{FG}\Vert \overline{AD}\end{array}$ .
  ::13. 早些时候,你证明如果F点和G点是ABCD平面边缘的中点,那么“AGF+GAD”。 有鉴于此,你证明FG'AD'。

  

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  14. The midpoints of each of the sides of rectangle $\begin{array}{r}ABCD\end{array}$ have been connected, as shown below. Make a conjecture about the inner quadrilateral that has been formed.
  ::14. 如下文所示,矩形ABCD两侧的中点相互连接,对所形成的内四边作出推测。

  

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  15. Prove your conjecture from #14.
  ::15. 从14号证明你的猜想。

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  16. Given: $\begin{array}{r}\overline{AH}\cong GC\end{array}$ ; $\begin{array}{r}\overline{DG}\cong \overline{HB}\end{array}$ ; $\begin{array}{r}\overline{DG}\parallel \overline{HB}\end{array}$
  ::16. 鉴于:AH ZGC;DG HB;DG HB

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  Prove: $\begin{array}{r}ABCD\end{array}$  is a parallelogram
  ::证明: ABCD 是平行图

  

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  17. Given: $\begin{array}{r}ABCD\end{array}$  is a parallelogram; $\begin{array}{r}\overline{AG}\cong \overline{CH}\end{array}$
  ::17. 参考:ABCD是一个平行图;AG CH

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  Prove: $\begin{array}{r}DGBH\end{array}$  is a parallelogram
  ::证明: DGBH 是一个平行图示

  

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  Review (Answers)
  ::审查(答复)

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