Course: 3.6 同一侧内地角

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同一侧内侧角度
Same Side Interior Angles
::同一侧内侧角度

Same side interior angles are two angles that are on the same side of the transversal and on the interior of (between) the two lines.
::同一侧内角是横贯同一侧和两条线(两线之间)内部的两个角度。

Same Side Interior Angles Theorem : If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.
::相同的侧侧内角定理 : 如果两条平行线被横切, 那么同一侧内角是辅助的。

If $\begin{aligned} l | | m \end{aligned}$ , then $\begin{aligned} m ∠ 1 + m ∠ 2 = 180^{\circ} \end{aligned}$ .
::如果lm,那么m1+m2=180。

Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the same side interior angles are supplementary, then the lines are parallel .
::反面的相同侧侧内侧角度理论 : 如果两条线被横向截断, 而同一侧内侧角度是补充的, 那么线是平行的。

If then $\begin{aligned} l | | m \end{aligned}$
::如果当时我

Suppose you were presented with two angles that are on the same side of a transversal and between the two parallel lines crossed by the transversal . How would you describe these angles and what could you conclude about their measures?
::假设有人向你们展示了两个角度,它们位于横跨横跨横跨的两条平行线之间。您如何描述这些角度,以及您可以得出什么结论来衡量这些角度呢?

Examples
::实例

Example 1
::例1

Is $\begin{aligned} l | | m \end{aligned}$ ? How do you know?
::你怎么会知道?

These angles are Same Side Interior Angles. So, if they add up to $\begin{aligned} 180^{\circ}, then l ∥ m . \end{aligned}$
::这些角度是同一侧内部角。所以,如果它们加起来等于180,那么我就会。

$\begin{aligned} 130^{\circ} + 67^{\circ} = 197^{\circ} \end{aligned}$ , therefore the lines are not parallel.
::13067197,因此线条不平行。

Example 2
::例2

Give two examples of same side interior angles in the diagram :
::在图表中举两个相同侧内角的例子:

There are MANY examples of same side interior angles in the diagram. Two are $\begin{aligned} ∠ 6 \end{aligned}$ and $\begin{aligned} ∠ 10 \end{aligned}$ , and $\begin{aligned} ∠ 8 \end{aligned}$ and $\begin{aligned} ∠ 12 \end{aligned}$ .
::图表中有许多同侧内部角度的例子。有两个是 6 和 10, 8 和 12 。

Example 3
::例3

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

The given angles are same side interior angles. Because the lines are parallel, the angles add up to $\begin{aligned} 180^{\circ} \end{aligned}$ .
::给定角度是相同的侧内侧角度。由于线条是平行的, 角度加起来等于 180 。

$\begin{aligned} (2 x + 43)^{\circ} + (2 x - 3)^{\circ} & = 180^{\circ} \\ (4 x + 40)^{\circ} & = 180^{\circ} \\ 4 x & = 140 \\ x & = 35 \end{aligned}$

:2x+43) (2x-3) (180) (4x+40) (180) (4x+40) ) (4x+40) (180) (4x=140x=35)

Example 4
::例4

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

$\begin{aligned} y \end{aligned}$ is a same side interior angle with the marked right angle . This means that $\begin{aligned} 90^{\circ} + y = 180 \end{aligned}$ so $\begin{aligned} y = 90 \end{aligned}$ .
::y 是带有标记右角的同一侧内侧角。这意味着 90\\ y= 180soy= 90 。

Example 5
::例5

Find the value of $\begin{aligned} x \end{aligned}$ if $\begin{aligned} m ∠ 3 = (3 x + 12)^{\circ} \end{aligned}$ and $\begin{aligned} m ∠ 5 = (5 x + 8)^{\circ} \end{aligned}$ .
::查找 x 的值, 如果 m3=( 3x+12) @ 和 m5=( 5x+8) @ 。

These are same side interior angles, so set up an equation and solve for $\begin{aligned} x \end{aligned}$ . Remember that same side interior angles add up to $\begin{aligned} 180^{\circ} \end{aligned}$ .
::这些是相同的侧内角, 所以设置 x 的方程式和解析。记住相同的侧内角共等于 180 。

$\begin{aligned} (3 x + 12)^{\circ} + (5 x + 8)^{\circ} & = 180^{\circ} \\ (8 x + 20)^{\circ} & = 180^{\circ} \\ 8 x & = 160 \\ x & = 20 \end{aligned}$

:3x+12) (5x+8) (18x+20) (8x+20) (180) (8x+20) (8x=160x=20)

Review
::回顾

For questions 1-2, use the diagram to determine if each angle pair is congruent, supplementary or neither.
::对于问题1-2,使用图表确定每个角度对是否一致、补充或两者兼而有之。

$\begin{aligned} ∠ 5 \end{aligned}$ and $\begin{aligned} ∠ 8 \end{aligned}$
::5和8

$\begin{aligned} ∠ 2 \end{aligned}$ and $\begin{aligned} ∠ 3 \end{aligned}$
::2和3

Are the lines parallel? Justify your answer.
::线条平行吗?

In 4-5, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.
::在4-5中,使用给定的信息来确定哪些线是平行的,如果没有,请写无,逐个考虑每个问题。

$\begin{aligned} ∠ A F D \end{aligned}$ and $\begin{aligned} ∠ B D F \end{aligned}$ are supplementary
::阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-民主-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-阿拉伯-

$\begin{aligned} ∠ D I J \end{aligned}$ and $\begin{aligned} ∠ F J I \end{aligned}$ are supplementary
::正义与正义倡议是补充性的

For 6-8, what does the value of $\begin{aligned} x \end{aligned}$ have to be to make the lines parallel?
::对于 6-8, x 的值是多少才能使线平行 ?

$\begin{aligned} m ∠ 3 = (3 x + 25)^{\circ} \end{aligned}$ and $\begin{aligned} m ∠ 5 = (4 x - 55)^{\circ} \end{aligned}$
::3=( 3x+25) 5=( 4x- 55)

$\begin{aligned} m ∠ 4 = (2 x + 15)^{\circ} \end{aligned}$ and $\begin{aligned} m ∠ 6 = (3 x - 5)^{\circ} \end{aligned}$
::m*4=( 2x+15) 和 m*6=( 3x- 5) * * *

$\begin{aligned} m ∠ 3 = (x + 17)^{\circ} \end{aligned}$ and $\begin{aligned} m ∠ 5 = (3 x - 5)^{\circ} \end{aligned}$
::3=(x+17) *和m*5=(3x-5) *

For 9-10, determine whether the statement is true or false.
::9-10时,确定声明是真实的还是虚假的。

Same side interior angles are on the same side of the transversal.
::同一侧的内部角度在横贯的同一侧。

Same side interior angles are congruent when lines are parallel.
::当线条平行时,同一侧内部角度是相似的。

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

同一侧内侧角度

Same Side Interior Angles ::同一侧内侧角度

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)

Same Side Interior Angles
::同一侧内侧角度

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)