6.5 分类三角形

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  分类三角形

  

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  Marjorie has just returned from her friend Shandra’s cottage, where Shandra has hung some beautiful homemade bunting from the ceiling. Marjorie has decided to brighten up her own space by making some bunting too. She wants the point of each triangle to come down at least 16 inches from the string. What type of triangle will Marjorie use to make her bunting?
  ::Marjorie刚从她的朋友Shandra的小屋返回,那里Shandra把一些漂亮的自制的包子吊在天花板上。 Marjorie决定通过做一些包子来亮亮自己的空间。 她希望每个三角点至少从弦上降下16英寸。 Marjorie用哪种三角形来做她的包子?

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  In this concept, you will learn to .
  ::在这个概念中,你会学会。

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  Classifying Triangles
  ::分类三角形

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  A triangle is a plane figure with three straight sides and three interior angles . A triangle can be classified by its angles or by its sides. There are seven types of triangles.
  ::三角形是一个平面图,有三个直角和三个内角。三角形可以按其角度或侧面进行分类。三角形有七种三角形。

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  Let’s look at triangles classified by their interior angles.
  ::让我们看看三角形内部角度分类的三角形。

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  The sum of the three interior angles of a triangle equals 180°. If one of the angles measures exactly 90° the tringle is called a right triangle . The other two angles of the triangle share the remaining ninety degrees. These angles are called acute angles since the measure of each of the angles is less than 90°. The following diagram is a right triangle. The right angle is indicated by the box in the corner of the triangle at the vertex labeled ‘ $\begin{array}{r}C\end{array}$ ’.
  ::三角形的三个内部角的总和等于 180 °。 如果一个角度精确度为 90 °, 矩形将被称作右三角形。 三角形的另外两个角度将共享其余的 90 度。 这些角度被称作急性角, 因为每个角度的测量量小于 90 °。 下图是一个右三角形。 右角由三角形角角的框标出, 在标有“ C ” 的顶端。

  

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  The angle at vertex ‘ $\begin{array}{r}A\end{array}$ ’ and the angle at vertex ‘ $\begin{array}{r}B\end{array}$ ’ are the acute angles of the right triangle.
  ::顶端 A 角和 顶端 B 角是右三角的急性角。

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  Another type of triangle is an acute triangle . As its name indicates, each of the three angles of an acute triangle measure less than 90°. The following diagram is an acute triangle.
  ::另一种三角形是急性三角形。如其名称所示,急性三角形的三个角度中每个角的测量度小于90度。下图为急性三角形。

  

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  . The other two angles of an obtuse triangle are acute angles. The following diagram is an obtuse triangle. obtuse triangle . A triangle that has an obtuse angle as one of its interior angles is called an obtuse angle An angle with a measure greater than 90° and less than 180° is called an
  ::。斜三角形的其他两个角度是急性角度。下图是一个隐性三角形.obtuse三角形。一个三角形,其内角中有一个隐性角度,称为“隐性角度”,其度量大于90度,小于180度。

  

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  An equiangular triangle is one in which the measure of each of the interior angles is 60°. If the sum of the angles of a triangle equals 180° then $\begin{array}{r}{180}^{\circ }÷3={60}^{\circ }\end{array}$ . The following diagram is an equiangular triangle. The following diagram is an equiangular triangle.
  ::矩形三角形是每个内角的度量为60°的三角形。如果三角形角的和等于180°,则等于180°3=60°。下图是角三角形的三角形。下图是角三角形的三角形。下图是角三角形的三角形。

  

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  In addition to these triangles named with respect to the interior angles, there are three types of triangles that are named with respect to the sides of the triangles. A triangle that has three angles equal in measure also has three sides that are equal in length. A triangle with three sides equal in length is called an . The following diagram is an equilateral triangle.
  ::除了以内角命名的三角形外,还有三种三角形,以三角形的两侧命名。一个三角形,三角度等同,三角度等同,三边长度等同。一个三角形,三边等同,称为“。下图是等边三角形”。

  

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  The same markings on the sides of the triangle indicate the sides are equal in length.
  ::三角形两侧同样的标记表明两侧长度相等。

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  Another triangle named with respect to its sides is one that has only two sides equal in length. A triangle with two sides equal in length is called an isosceles triangle . If a triangle with three sides equal in length has three angles equal in measure, then a triangle with two sides equal in length has the angles opposite each of the equal sides equal in measure. The following diagram is an isosceles triangle.
  ::另一边命名为另一三角, 其侧面只有两边相等。 两边相等的三角被称为一等分三角形。 如果一个三边相等的三角形的长度相等于三个角度, 则两边相等的三角形的长度相等于两边相等的角。 下图为一等分三角形。

  

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  The third type of triangle named with respect to its sides is called a scalene triangle and it has no sides equal in length and no angles equal in measure. The following diagram is a scalene triangle.
  ::第三种三角形以其侧面命名,称为缩放三角形,其边长不相等,角度不相等。下图是缩放三角形。

  

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  The following triangles have been grouped with respect to their interior angles. Let’s name the triangles.
  ::以下三角形按内部角度分组。让我们命名三角形。

  

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  Triangle one is a triangle which shows an angle greater than 90°. This triangle is an obtuse triangle.
  ::三角形一是一个三角形,其角大于90度。三角形是一个隐形三角形。

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  Triangle two is a triangle which shows an angle of 90°. The triangle is a right triangle.
  ::三角形二是三角形,角为90°。三角形是右三角形。

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  Triangle three is a triangle which shows three angles such that each angle is less than 90°. This triangle is an acute triangle.
  ::三角三角三是一个三角形,它显示三个角,每个角小于90度。这个三角形是一个急性三角形。

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  Triangle four is a triangle which shows three angles of equal measure. This triangle is an equiangular triangle.
  ::三角形 4 是三角形, 显示三个角度的等量度。 此三角形是一个矩形三角形 。

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  The following triangles have been grouped with respect to the measures of their sides. Name the triangles.
  ::下列三角形已按其侧边的度量分组。 请列出三角形名称 。

  

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  Triangle one shows a triangle with two sides equal in length – 7 m and 7 m. This triangle is an isosceles triangle.
  ::三角形一显示三角形,两边长度等于7米和7米。三角形是一个等骨三角形。

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  Triangle two shows a triangle with three sides equal in length – 4.3 in, 4.3 in and 4.3 in. This triangle is an equilateral triangle.
  ::三角形 2 显示三角形, 三角形三边长度相等 — — 4.3 英寸, 4.3 英寸和 4.3 英寸。 这个三角形是一个等边三角形 。

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  Triangle three shows a triangle with no sides equal in length. This triangle is a scalene triangle.
  ::三角三角三显示三角形,其边长度不相等。此三角形是一个缩放三角形。

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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  Marjorie and her new bunting.
  ::之前,你得到一个问题 关于玛乔莉 和她的新捆绑。

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  She needs to figure out what type of triangle to cut out for the curtains.
  ::她需要弄清楚 什么样的三角形 剪出窗帘。

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  First, she must figure out what she wants the valance to look like.
  ::首先,她必须弄清楚 她想让价钱长成什么样

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  She wants the triangles to be tall (16 inches from the base to the point) such that the two sides are equal in length.
  ::她希望三角形高(从基点到点点16英寸), 使两边长度相等。

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  Next, Marjorie should sketch the triangle so that she has a pattern to cut out.
  ::接下来 Marjorie应该画出三角形 这样她就可以切断一个模式了

  

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  $$\begin{array}{r}\text{One block}=2\text{ inches}\end{array}$$

  
  ::1个区块=2英寸

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  Marjorie should use an acute, isosceles triangle to make the valance for the curtains.
  ::Marjorie应该用一个尖锐的等骨三角形 来给窗帘起伏

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  Example 2
  ::例2

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  Identify the type of triangle below.
  ::确定以下三角形的类型 。

  

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  First, look at the triangle to determine if it has any same markings to indicate either sides equal in length or angles equal in measure.
  ::首先,看看三角形,以确定它是否有相同的标记来表示长度相等的两边或量角相等的两边。

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  There are no markings on any of the angles but the three sides of the triangle have a single tic mark on them.
  ::角上没有任何标记,但三角形的三边在角上有一个单点标记。

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  Next, decide the name of the triangle based on whether it has been named according to the sides or the angles.
  ::下一步,决定三角形的名称,根据三角形是按两边还是按角度命名的。

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  The triangle has been named according to the length of the sides.
  ::三角形是根据两边的长度命名的。

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  Then, name the type of triangle.
  ::然后,命名三角形的类型。

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  The triangle which has three sides equal in length is called an equilateral triangle.
  ::三边长度相等的三角形称为等边三角形。

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  Example 3
  ::例3

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  What measures can be used to classify triangles?
  ::可采取哪些措施对三角形进行分类?

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  Triangles can be classified by using the lengths of the sides or the measures of the interior angles.
  ::三角形可以使用边的长度或内角的量度进行分类。

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  Example 4
  ::例4

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  What type of triangle is shown in the diagram? Justify your answer.
  ::图表中显示的三角形类型是什么? 请说明答案的理由 。

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  First, look at the triangle and write down what you know from the diagram.
  ::首先,看看三角形 记下图中你所知道的

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  $\begin{array}{r}\overline{DE}=\overline{EF}\text{and}m\mathrm{\angle }D=m\mathrm{\angle }F\end{array}$ .
  ::和M'D'F。 和M'D'F。

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  Next, based on what you have written down, name the type of triangle.
  ::接下来,根据您所写的内容, 命名三角形的类型 。

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  A triangle that has two sides equal in length and the two angles, opposite the equal sides, equal in measure is called an isosceles triangle.
  ::一个三角形,其两边长度相等,而两个角度,对等角,等量相等,称为等分三角形。

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  Example 5
  ::例5

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  Name each of the following triangles with reference to both the side lengths and the interior angles.
  ::以侧边长度和内角为参考点,列出下列三角形的每一个名称。

  

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  Triangle one (left):
  ::三角形 1 (左):

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  First, look at the interior angles of the triangle.
  ::首先,看看三角形的内部角。

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  There are two acute angles equal in measure and one obtuse angle.
  ::有两个等量的急性角度和一个隐形角度。

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  Next, look at the side lengths of the triangle.
  ::接下来,看看三角形的侧长。

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  There are two sides equal in length.
  ::两边长度相等。

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  Then, name the triangle with reference to both the angles and the side lengths.
  ::然后,用角度和侧边长度的参考点命名三角形。

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  This triangle is an obtuse, isosceles triangle.
  ::此三角形是一个隐形, 等分形三角形 。

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  Triangle two (middle):
  ::三角形 2 (中) :

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  First, look at the interior angles of the triangle.
  ::首先,看看三角形的内部角。

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  There are two acute angles and one 90° angle.
  ::有两个急性角和一个90度角

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  Next, look at the side lengths of the triangle.
  ::接下来,看看三角形的侧长。

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  There are two sides equal in length.
  ::两边长度相等。

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  Then, name the triangle with reference to both the angles and the side lengths.
  ::然后,用角度和侧边长度的参考点命名三角形。

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  This triangle is a right, isosceles triangle.
  ::这个三角形是右侧的, 等分形三角形 。

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  Triangle three (right):
  ::三角三角三( 右) :

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  First, look at the interior angles of the triangle.
  ::首先,看看三角形的内部角。

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  There are two acute angles and one obtuse angle.
  ::有两个急性角度和一个隐形角度

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  Next, look at the side lengths of the triangle.
  ::接下来,看看三角形的侧长。

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  There are no sides equal in length.
  ::没有任何两边的长度相等。

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  Then, name the triangle with reference to both the angles and the side lengths.
  ::然后,用角度和侧边长度的参考点命名三角形。

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  This triangle is an obtuse, scalene triangle.
  ::此三角形是一个隐性缩放三角形。 NAME OF TRANSLATORS

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  Review
  ::回顾

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  Classify each triangle by the given angle measures as acute, obtuse or right.
  ::将每个三角形按给定角测量尺度分类为急性、隐形或右角。

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  1. A triangle with three 60° angles.
  ::1. 三角形,有3个60度角。

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  2. A triangle with one 110° angle.
  ::2. 三角形,角110°。

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  3. A triangle with one right angle and two acute angles.
  ::3. 一个右角和两个急性角的三角形。

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  4. A triangle with one 130° angle.
  ::4. 角为130°的三角形。

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  5. A triangle with three acute angles.
  ::5. 三角形,有三个急性角度。

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  6. A triangle with a 90° angle.
  ::6. 角为90度的三角形。

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  7. A triangle with three angles that are less than 90°.
  ::7. 三角形,三个角度小于90度。

  8. 

  

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  Identify each triangle by the side lengths described. Identify them as equilateral, isosceles or scalene.
  ::以描述的侧边长度识别每个三角形。 将其标记为等边、 等距或缩放 。

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  9. A triangle with side lengths of 6 in, 6 in and 4 inches.
  ::9. 三角形,侧长6英寸、6英寸和4英寸。

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  10. A triangle with side lengths of 3 ft., 4 ft., and 5 ft.
  ::10. 侧长3英尺、4英尺和5英尺的三角形。

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  11. A triangle with side lengths of 8 inches.
  ::11. 侧长8英寸的三角形。

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  12. A triangle with side lengths of 7 inches, 8 inches and 8 inches.
  ::12. 侧长7英寸、8英寸和8英寸的三角形。

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  13. A triangle with side lengths of 6 meters, 8 meters and 10 meters.
  ::13. 侧长6米、8米和10米的三角形。

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  14. A triangle with side lengths of 10 mm.
  ::14. 侧长10毫米的三角形。

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  15. A triangle with side lengths of 12 cm.
  ::15. 侧长12厘米的三角形。

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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 。