Section: 线性对等 | 1.9 线性对等

Section outline

Linear Pairs
::线性对等

Two angles are adjacent if they have the same vertex , share a side, and do not overlap. $\begin{aligned} ∠ P S Q \end{aligned}$ and $\begin{aligned} ∠ Q S R \end{aligned}$ are adjacent.
::两个角度相邻,如果两个角度的顶点相同,共用一面,不重叠。 @PSQ 和 @SR 是相邻的。

A linear pair is two angles that are adjacent and whose non-common sides form a straight line . If two angles are a linear pair, then they are supplementary (add up to $\begin{aligned} 180^{\circ} \end{aligned}$ ). $\begin{aligned} ∠ P S Q \end{aligned}$ and $\begin{aligned} ∠ Q S R \end{aligned}$ are a linear pair.
::线性对面是相邻的两个角度,其非共同面构成一条直线线。如果两个角度是线性对,那么它们是补充的(加到180)。 PSQ 和SR 是线性对。

What if you were given two angles of unknown size and were told they form a linear pair? How would you determine their angle measures?
::假若你们获得两个不明的角,并且被告知它们为线性对子,你们如何确定它们的角度呢?

Examples
::实例

For Examples 1 and 2, use the diagram below. Note that $\begin{aligned} \bar{N K} ⊥ \overset{\leftrightarrow}{I L} \end{aligned}$ .
::例1和例2使用下图。请注意NK L。

Example 1
::例1

Name one linear pair of angles.
::列出一个直线角度。

$\begin{aligned} ∠ M N L \end{aligned}$ and $\begin{aligned} ∠ L N J \end{aligned}$
::MNLANDLNJ(缅甸)

Example 2
::例2

What is $\begin{aligned} m ∠ I N L \end{aligned}$ ?
::什么是MINL?

$\begin{aligned} 180^{\circ} \end{aligned}$

Example 3
::例3

What is the measure of each angle?
::每个角度的量度是多少?

These two angles are a linear pair, so they add up to $\begin{aligned} 180^{\circ} \end{aligned}$ .
::这两个角度是线性对角, 所以它们加起来等于180。

$\begin{aligned} (7 q - 46)^{\circ} + (3 q + 6)^{\circ} & = 180^{\circ} \\ 10 q - 40^{\circ} & = 180^{\circ} \\ 10 q & = 220^{\circ} \\ q & = 22^{\circ} \end{aligned}$

:7q-46) (3q+6) (180) (10q)-40(180) (10q)=(220) (q)=(22)

Plug in $\begin{aligned} q \end{aligned}$ to get the measure of each angle. $\begin{aligned} m ∠ A B D = 7 (22^{\circ}) - 46^{\circ} = 108^{\circ} m ∠ D B C = 180^{\circ} - 108^{\circ} = 72^{\circ} \end{aligned}$
::Q 中的插件以获得每个角度的量度。 mABD=7( 22) - 46108 mDBC=18010872

Example 4
::例4

Are $\begin{aligned} ∠ C D A \end{aligned}$ and $\begin{aligned} ∠ D A B \end{aligned}$ a linear pair? Are they supplementary?
::CDA和DAB是线性对吗?

The two angles are not a linear pair because they do not have the same vertex. They are supplementary because they add up to $\begin{aligned} 180^{\circ} \end{aligned}$ : $\begin{aligned} 120^{\circ} + 60^{\circ} = 180^{\circ} \end{aligned}$ .
::这两个角度不是线性对, 因为它们没有相同的顶点。它们具有补充性, 因为它们加起来等于 180 : 120 60 180 。

Example 5
::例5

Find the measure of an angle that forms a linear pair with $\begin{aligned} ∠ M R S \end{aligned}$ if $\begin{aligned} m ∠ M R S \end{aligned}$ is $\begin{aligned} 150^{\circ} \end{aligned}$ .
::如果 m MRS 是 150 , 则会找到一个角度的量度, 以 QMRS 构成线性对和 QMRS 的线性对。

Because linear pairs have to add up to $\begin{aligned} 180^{\circ} \end{aligned}$ , the other angle must be $\begin{aligned} 180^{\circ} - 150^{\circ} = 30^{\circ} \end{aligned}$ .
::因为线性配对必须加到180, 另一个角度必须是18015030。

Review
::回顾

For 1-5, determine if the statement is true or false.
::1-5,确定声明是真实的还是虚假的。

Linear pairs are congruent.
::线性对子是相同的。

Adjacent angles share a vertex.
::相邻角度有一个共同的顶点。

Adjacent angles overlap.
::近角重叠。

Linear pairs are supplementary.
::线性配对是补充性的。

Supplementary angles form linear pairs.
::补充角度形成线性对子。

For exercise 6, find the value of $\begin{aligned} x \end{aligned}$ .
::对于练习6, 找到 x 的值。

Find the measure of an angle that forms a linear pair with $\begin{aligned} ∠ M R S \end{aligned}$ if $\begin{aligned} m ∠ M R S \end{aligned}$ is:
::如果 mMRS 是 , 则查找以 {MRS 构成 MRS 成线性对的角的量度 :

$\begin{aligned} 61^{\circ} \end{aligned}$

$\begin{aligned} 23^{\circ} \end{aligned}$

$\begin{aligned} 114^{\circ} \end{aligned}$

$\begin{aligned} 7^{\circ} \end{aligned}$

$\begin{aligned} 179^{\circ} \end{aligned}$

$\begin{aligned} z^{\circ} \end{aligned}$
::

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

Linear Pairs ::线性对等

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)

Linear Pairs
::线性对等

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)