Course: 4.11 等边三角 | The Way To Learn

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等边三角形

Equilateral Triangle Theorem
::等边三角定理

Equilateral Triangle Theorem : All equilateral triangles are also equiangular. Furthermore, all equiangular triangles are also equilateral.
::等边三角理论:所有等边三角也都是等角。此外,所有等角三角也都是等边三角。

If $\begin{aligned} \bar{A B} ≅ \bar{B C} ≅ \bar{A C} \end{aligned}$ , then $\begin{aligned} ∠ A ≅ ∠ B ≅ ∠ C \end{aligned}$ . Conversely, if $\begin{aligned} ∠ A ≅ ∠ B ≅ ∠ C \end{aligned}$ , then $\begin{aligned} \bar{A B} ≅ \bar{B C} ≅ \bar{A C} \end{aligned}$ .
::如果AB BCAC,那么ABC。相反,如果ABC,那么ABBABAC。

What if you were presented with an equilateral triangle and told that its sides measure x , y , and 8? What could you conclude about x and y ?
::假若有人对你们提出一个对等三角形,并且对你们说,其两边的度量是X,y,和8,你们能得出什么结论呢?对于 x,y,你们能得出什么结论呢?

Examples
::实例

Example 1
::例1

Fill in the proof:
::填写证据:

Given : Equilateral $\begin{aligned} △ R S T \end{aligned}$ with
::给定: 等量 QRST

$\begin{aligned} \bar{R T} ≅ \bar{S T} ≅ \bar{R S} \end{aligned}$
::程程程程程

Prove : $\begin{aligned} △ R S T \end{aligned}$ is equiangular
::证明: RST 是等角

*Statement*	*Reason*
1.	1. Given
2.	2. Base Angles Theorem
3.	3. Base Angles Theorem
4.	4. Transitive PoC
5. $\begin{aligned} △ R S T \end{aligned}$ is equiangular	5.

*Statement*	*Reason*
1. $\begin{aligned} \bar{R T} ≅ \bar{S T} ≅ \bar{R S} \end{aligned}$	1. Given
2. $\begin{aligned} ∠ R ≅ ∠ S \end{aligned}$	2. Base Angles Theorem
3. $\begin{aligned} ∠ T ≅ ∠ R \end{aligned}$	3. Base Angles Theorem
4. $\begin{aligned} ∠ T ≅ ∠ S \end{aligned}$	4. Transitive PoC
5. $\begin{aligned} △ R S T \end{aligned}$ is equiangular	5. Definition of equiangular.

Example 2
::例2

True or false: All equilateral triangles are .
::真实的或虚假的: 所有等边三角形都是。

This statement is true. The definition of an isosceles triangle is a triangle with at least two congruent sides. Since all equilateral triangles have three congruent sides, they fit the definition of an isosceles triangle.
::此语句是真实的。等分三角形的定义是一个至少有两个相容面的三角形。由于所有等分三角形都有三个相近面, 它们符合等分三角形的定义。

Example 3
::例3

Find the value of $\begin{aligned} x \end{aligned}$ .
::查找 x 的值。

Because this is an equilateral triangle $\begin{aligned} 3 x - 1 = 11 \end{aligned}$ . Solve for $\begin{aligned} x \end{aligned}$ .
::因为这是一个等边三角形 3x-1=11。解决 x 。

\begin{aligned} 3 x - 1 & = 11 \\ 3 x & = 12 \\ x & = 4 \end{aligned}

::3x-1=113x=12x=4

Example 4
::例4

Find the values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::查找 x 和 y 的值。

The markings show that this is an equilateral triangle since all sides are congruent. This means all sides must equal $\begin{aligned} 10 \end{aligned}$ . We have $\begin{aligned} x = 10 \end{aligned}$ and $\begin{aligned} y + 3 = 10 \end{aligned}$ which means that $\begin{aligned} y = 7 \end{aligned}$ .
::标记显示,这是一个等边三角形, 因为所有边都是相似的。这意味着所有边必须等于 10 。我们有 x= 10 和 y+3= 10, 这意味着y= 7 。

Example 5
::例5

Two sides of an equilateral triangle are $\begin{aligned} 2 x + 5 \end{aligned}$ units and $\begin{aligned} x + 13 \end{aligned}$ units. How long is each side of this triangle?
::等边三角形的两面是 2x+5 单位和 x+13 单位。这个三角形的两面需要多久?

The two given sides must be equal because this is an equilateral triangle. Write and solve the equation for $\begin{aligned} x \end{aligned}$ .
::给定的两边必须是相等的, 因为这是一个等边三角形。写入并解析 x 的方程。

\begin{aligned} 2 x + 5 & = x + 13 \\ x & = 8 \end{aligned}

::2x+5=x+13x=8

To figure out how long each side is, plug in $\begin{aligned} 8 \end{aligned}$ for $\begin{aligned} x \end{aligned}$ in either of the original expressions. $\begin{aligned} 2 (8) + 5 = 21 \end{aligned}$ . Each side is $\begin{aligned} 21 \end{aligned}$ units.
::要找出每边的长度, 在原始表达式中的任意一种中, x 插入 8 。 2( 8)+5=21。每边为 21 个单位。

Review
::回顾

The following triangles are equilateral triangles. Solve for the unknown variables.
::下列三角形是等边三角形。解决未知变量。

Find the measures of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::查找 x 和 y 的量度。

Review (Answers)
::回顾(答复)

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。

Section outline

General

等边三角形

Equilateral Triangle Theorem ::等边三角定理

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源