节: Convex 多边形中的内部角 | 6.1 Convex 多边形中的内角

章节大纲

Interior Angles in Convex Polygons
::Convex 多边形中的内部角

The interior angle of a polygon is one of the angles on the inside, as shown in the picture below. A polygon has the same number of interior angles as it does sides.
::多边形的内部角是内部角度之一,如下图所示。多边形的内部角与侧角的数目相同。

The sum of the interior angles in a polygon depends on the number of sides it has. The Polygon Sum Formula states that for any $\begin{aligned} n - \end{aligned}$ gon, the interior angles add up to $\begin{aligned} (n - 2) \times 180^{\circ} . \end{aligned}$
::多边形内角的总和取决于其边数。多边形总和表示,对于任何 n-gon,内角加起来等于(n-2)x180。

\begin{aligned} \to n = 8 \\ (8 - 2) & \times 180^{\circ} \\ 6 & \times 180^{\circ} \\ 1 & 080^{\circ} \end{aligned}

::=88(8- 2) ×180 ×6x180 1080

Once you know the sum of the interior angles in a polygon it is easy to find the measure of ONE interior angle if the polygon is regular : all sides are congruent and all angles are congruent. Just divide the sum of the angles by the number of sides.
::一旦您知道多边形内角的总和, 如果多边形是正态的, 很容易找到一个内部角的大小: 方形是正态的, 方形是正态的, 方形是正态的。只要将角度之和除以边数即可。

Regular Polygon Interior Angle Formula: For any equiangular $\begin{aligned} n - \end{aligned}$ gon, the measure of each angle is $\begin{aligned} \frac{(n - 2) \times 180^{\circ}}{n} \end{aligned}$ .
::常规多边形内部角公式:对于任何等角-角,每个角的度量为(n-2)x180n。

In the picture below, if all eight angles are congruent then each angle is $\begin{aligned} \frac{(8 - 2) \times 180^{\circ}}{8} = \frac{6 \times 180^{\circ}}{8} = \frac{1080^{\circ}}{8} = 135^{\circ} \end{aligned}$ .
::在下面的图片中,如果所有八个角度是相同的,那么每个角度是(8-2)x1808=6x1808=6x1808=10808=135}。

What if you were given an equiangular seven-sided convex polygon? How could you determine the measure of its interior angles?
::如果给您一个方形七面方形的方形多边形呢?您如何确定内角的大小 ?

Examples
::实例

Example 1
::例1

The interior angles of a pentagon are $\begin{aligned} x^{\circ}, x^{\circ}, 2 x^{\circ}, 2 x^{\circ}, \end{aligned}$ and $\begin{aligned} 2 x^{\circ} \end{aligned}$ . What is $\begin{aligned} x \end{aligned}$ ?
::五角形的内角是 x, x , 2x , 2x , 和 2x 。什么是 x?

From the Polygon Sum Formula we know that a pentagon has interior angles that sum to $\begin{aligned} (5 - 2) \times 180^{\circ} = 540^{\circ} \end{aligned}$ .
::从多边形和公式中,我们知道,一个五角形的内部角度等于(5-2)x180540。

Write an equation and solve for $\begin{aligned} x \end{aligned}$ .
::写入公式并解析 x。

\begin{aligned} x^{\circ} + x^{\circ} + 2 x^{\circ} + 2 x^{\circ} + 2 x^{\circ} & = 540^{\circ} \\ 8 x & = 540 \\ x & = 67.5 \end{aligned}

::{\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不... {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不 {\fn黑体\fs22\bord1\shad0\3aHBE\4aH00\fscx67\fscy66\2cHFFFFFF\3cH808080}不...

Example 2
::例2

What is the sum of the interior angles in a 100-gon?
::100公分内角的内角总和是多少?

Use the Polygon Sum Formula. $\begin{aligned} (100 - 2) \times 180^{\circ} = 17, 640^{\circ} \end{aligned}$ .
::使用多边形总和公式。 (100-2) ×18017, 640。

Example 3
::例3

The interior angles of a polygon add up to $\begin{aligned} 1980^{\circ} \end{aligned}$ . How many sides does it have?
::多边形的内角加到1980年。它有多少边?

Use the Polygon Sum Formula and solve for $\begin{aligned} n \end{aligned}$ .
::使用多边形总和公式并解决 n.

\begin{aligned} (n - 2) \times 180^{\circ} & = 1980^{\circ} \\ 180^{\circ} n - 360^{\circ} & = 1980^{\circ} \\ 180^{\circ} n & = 2340^{\circ} \\ n & = 13 \end{aligned}

::==============================================================================================

The polygon has 13 sides.
::多边形有13边。

Example 4
::例4

How many degrees does each angle in an equiangular nonagon have?
::等角非形中每个角有多少度?

First we need to find the sum of the interior angles; set $\begin{aligned} n = 9. \end{aligned}$
::首先我们需要找到内部角度的总和; 设定 n=9 。

\begin{aligned} (9 - 2) \times 180^{\circ} = 7 \times 180^{\circ} = 1260^{\circ} \end{aligned}

“Equiangular” tells us every angle is equal. So, each angle is $\begin{aligned} \frac{1260^{\circ}}{9} = 140^{\circ} \end{aligned}$ .
::“ 等角 ” 告诉我们每个角度都是相等的。所以, 每个角度是 1260\\\9=140\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\4\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Example 5
::例5

An interior angle in a regular polygon is $\begin{aligned} 135^{\circ} \end{aligned}$ . How many sides does this polygon have?
::普通多边形中的内角是 135 。这个多边形有多少边 ?

Here, we will set the Regular Polygon Interior Angle Formula equal to $\begin{aligned} 135^{\circ} \end{aligned}$ and solve for $\begin{aligned} n \end{aligned}$ .
::在此, 我们将设置常规多边形内部角公式, 等于 135 , 并解决 n 。

\begin{aligned} \frac{(n - 2) \times 180^{\circ}}{n} & = 135^{\circ} \\ 180^{\circ} n - 360^{\circ} & = 135^{\circ} n \\ - 360^{\circ} & = - 45^{\circ} n \\ n & = 8 The polygon is an octagon . \end{aligned}

n-2)x180n=13518n-360135n-36045nn=8)多边形是八边形。

Review
::回顾

Fill in the table. Round values to the nearest hundredths place if needed.
::填满表格。如果需要, 圆形值为最接近的百位数。

# of sides	Sum of the Interior Angles	Measure of Each Interior Angle in a Regular $\begin{aligned} n - \end{aligned}$ gon
3		$\begin{aligned} 60^{\circ} \end{aligned}$
4	$\begin{aligned} 360^{\circ} \end{aligned}$
5	$\begin{aligned} 540^{\circ} \end{aligned}$	$\begin{aligned} 108^{\circ} \end{aligned}$
6		$\begin{aligned} 120^{\circ} \end{aligned}$
7
8
9
10
11
12

What is the sum of the angles in a 15-gon?
::15角角角度的总和是多少?

What is the sum of the angles in a 23-gon?
::23角角角度的总和是多少?

The sum of the interior angles of a polygon is $\begin{aligned} 4320^{\circ} \end{aligned}$ . How many sides does the polygon have?
::多边形的内部角总和是4320。多边形有多少边?

The sum of the interior angles of a polygon is $\begin{aligned} 3240^{\circ} \end{aligned}$ . How many sides does the polygon have?
::多边形的内部角总和是 3240。多边形有多少边?

What is the measure of each angle in a regular 16-gon?
::在正常的 16 个角中,每个角度的量度是多少 ?

What is the measure of each angle in an equiangular 24-gon?
::方形 24 角方形 24 角每个角度的量度是多少 ?

Each interior angle in a regular polygon is $\begin{aligned} 156^{\circ} \end{aligned}$ . How many sides does it have?
::普通多边形中每个内部角为156。它有多少边?

Each interior angle in an equiangular polygon is $\begin{aligned} 90^{\circ} \end{aligned}$ . How many sides does it have?
::方形多边形内每个内角为 90 。它有多少边 ?

For questions 10-18, find the value of the missing variable(s).
::对于问题10-18,请找到缺失变量的价值。

The interior angles of a hexagon are $\begin{aligned} x^{\circ}, (x + 1)^{\circ}, (x + 2)^{\circ}, (x + 3)^{\circ}, (x + 4)^{\circ}, \end{aligned}$ and $\begin{aligned} (x + 5)^{\circ} . \end{aligned}$ What is $\begin{aligned} x \end{aligned}$ ?
::六边形的内角是 x 、 (x+1) 、 (x+2) 、 (x+3) 、 (x+4) 和 (x+5) 。什么是 x?

Review (Answers)
::回顾(答复)

To see the answer key for this book, go to the and click on the Answer Key under the ' ' option.
::要查看本书的答案键, 请在“ ” 选项下点击答案键。

章节大纲

Interior Angles in Convex Polygons ::Convex 多边形中的内部角

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