Course: 6.4 角对角测量 | The Way To Learn

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角对角测量
The foul lines of a baseball diamond intersect at home plate to form a right angle . A baseball is hit from home plate and forms an angle of 36° with the third base foul line . What is the measure of the angle between the first base foul line and the bath of the baseball?
::棒球金刚石的底线在主盘上交叉,以形成一个右角。棒球从主盘上打,形成一个36°的角,与三基底底底底底底底底底底底底线相交。第一底底底底底底底底线与棒球浴缸之间的角的度量是多少?

How can you use your knowledge of angles to figure out the measure of the angle?
::您如何使用您对角度的知识来计算角度的度量 ?

In this concept, you will learn measure of angle pairs.
::在这个概念中,你会学习量度角度对。

Measuring Angle Pairs
::测量角对角

There are different types of angle pairs. are an angle pair formed by intersecting lines such that they are never adjacent. They have a common vertex and never share a common side. Vertical angles are equal in measure. The following diagram shows vertical angle pairs.
::角度对有不同类型。是由交叉线组成的角对, 因而它们从不相邻。它们有一个共同的顶点, 并且从不共享共同的侧面。垂直角度在量值上是相等的。下图显示垂直角度对。

$\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 2\end{align*}$ are vertical angles. $\begin{align*}m \angle 1 = m \angle 2\end{align*}$
::1和2是垂直角度。 m1=m2

$\begin{align*}\angle 3\end{align*}$ and $\begin{align*}\angle 4\end{align*}$ are vertical angles. $\begin{align*}m \angle 3 = m \angle 4\end{align*}$
::3 和 4 是垂直角。 m 3 = m 4

Adjacent angles are an angle pair also formed by two intersecting lines. Adjacent angles are side by side, have a common vertex and share a common side. The following diagram shows pairs of adjacent angles.
::相邻角度是一个角度对, 由两条交叉线组成。相邻角度相邻, 有着共同的顶点, 并有着共同的一面。下图显示相邻角度的一对。

Each pair of adjacent angles forms a straight angle . Therefore the sum of any two adjacent angles equals 180 ° .
::每对相邻角形成直角。因此,任何两个相邻角的总和等于180度。

$\begin{aligned} \begin{array}{rcl} m ∠ 1 + m ∠ 3 & = & 180^{\circ} \\ m ∠ 2 + m ∠ 4 & = & 180^{\circ} \\ m ∠ 2 + m ∠ 3 & = & 180^{\circ} \\ m ∠ 1 + m ∠ 4 & = & 180^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} m \angle 1 + m \angle 3 &=& 180^\circ\\ m \angle 2 + m \angle 4 &=& 180^\circ\\ m \angle 2 + m \angle 3 &=& 180^\circ\\ m \angle 1 + m \angle 4 &=& 180^\circ \end{array}\end{align*}$
::m1+m3=1802+m4=1802+m3=1801+m4=1801+m4=180

If the sum two angles is 180° then the angles are called . The following diagram shows two supplementary angles.
::如果两个角度的和值为180°,则这些角度被称为。下图显示两个补充角度。

In both diagrams , $\begin{align*}m \angle 1 + m \angle 2 = 180^\circ\end{align*}$ .
::在两个图表中, m1+m2=180。

If the sum of two angles equals 90° then the angles are called . The following diagram shows two complementary angles.
::如果两个角度的和等于 90 °,则这些角度被称为。下图显示两个互补角度。

$\begin{align*}m \angle 1 + m \angle 2 = 90^\circ\end{align*}$
::m1+m2=90

Let’s apply all this information about angles and their measure to determine the measure of $\begin{align*}\angle a\end{align*}$ , $\begin{align*}\angle b\end{align*}$ , $\begin{align*}\angle c\end{align*}$ in the following diagram.
::让我们应用所有这些关于角度及其测量尺度的信息来确定下图中的 a, b, c的度量。

There are four angles formed by intersecting lines. The measure of one of the angles is 70°.
::有四个角度由交叉线组成。其中一个角度的测量值为 70 ° 。

First, state the relationship between the angle of 70 ° and $\begin{align*}\angle b\end{align*}$ .
::首先,说明70°角和b角之间的关系。

The angle of 70 ° is adjacent to $\begin{align*}\angle b\end{align*}$ and the two angles form a straight angle.
::70°角与 b 相邻,两个角度形成直角。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} ∠ b + 70^{\circ} = 180^{\circ} \end{aligned}$
$\begin{align*}\angle b + 70^\circ = 180^\circ\end{align*}$
::*b+70180*

Next, subtract 70 ° from both sides of the equation.
::接下来,从方程两侧减去70度。

$\begin{aligned} \begin{array}{rcl} ∠ b + 70^{\circ} & = & 180^{\circ} \\ ∠ b + 70^{\circ} - 70^{\circ} & = & 180^{\circ} - 70^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle b + 70^\circ &=& 180^\circ\\ \angle b + 70^\circ - 70^\circ &=& 180^\circ-70^\circ \end{array}\end{align*}$
::++70 +180 +70 +70 +180 +70 +70

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} ∠ b + 70^{\circ} - 70^{\circ} & = & 180^{\circ} - 70^{\circ} \\ ∠ b & = & 110^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle b + 70^\circ - 70^\circ &=& 180^\circ - 70^\circ\\ \angle b &=& 110^\circ \end{array}\end{align*}$
::@b+707018070b=110#

The answer is 110°.
::答案是110度

$\begin{aligned} m ∠ b = 110^{\circ} \end{aligned}$
$\begin{align*}m \angle b = 110^\circ\end{align*}$
::mb=110

First, state the relationship between the angle of 70° and $\begin{align*}\angle a\end{align*}$ .
::首先,说明70°角与a角之间的关系。

The angle of 70° and $\begin{align*}\angle a\end{align*}$ are vertical angles and are equal in measure.
::70°和a角是垂直角度,在量度上相等。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} m ∠ a = 70^{\circ} \end{aligned}$
$\begin{align*}m \angle a = 70^\circ\end{align*}$
::

The answer is 70°.
::答案是70度

$\begin{aligned} m ∠ a = 70^{\circ} \end{aligned}$
$\begin{align*}m \angle a = 70^\circ\end{align*}$
::

First state the relationship between the angle of 70° and $\begin{align*}\angle c\end{align*}$ .
::首先说明70°角与°c角之间的关系。

The angle of 70 ° is adjacent to $\begin{align*}\angle c\end{align*}$ and the two angles form a straight angle.
::70°角与c相邻,两个角度形成直角。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} ∠ c + 70^{\circ} = 180^{\circ} \end{aligned}$
$\begin{align*}\angle c + 70^\circ = 180^\circ\end{align*}$
::*c+70180 *c+70180*

Next, subtract 70° from both sides of the equation.
::接下来,从方程两侧减去70度。

$\begin{aligned} \begin{array}{rcl} ∠ c + 70^{\circ} & = & 180^{\circ} \\ ∠ c + 70^{\circ} - 70^{\circ} & = & 180^{\circ} - 70^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle c + 70^\circ &=& 180^\circ\\ \angle c + 70^\circ - 70^\circ &=& 180^\circ - 70^\circ \end{array}\end{align*}$
::

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} ∠ c + 70^{\circ} - 70^{\circ} & = & 180^{\circ} - 70^{\circ} \\ ∠ c & = & 110^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} \angle c + 70^\circ - 70^\circ &=& 180^\circ - 70^\circ\\ \angle c &=& 110^\circ \end{array}\end{align*}$
::@c+707018070c=110#

The answer is 110°.
::答案是110度

$\begin{aligned} m ∠ c = 110^{\circ} \end{aligned}$
$\begin{align*}m \angle c = 110^\circ\end{align*}$
::mc=110_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the baseball field and the foul lines.
::早些时候,你得到一个问题关于棒球场和下流线。

The angle between the path of the ball and the first base foul line needs to be figured out. This can be done using complementary angles.
::需要找出球和第一底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底线之间的角。这可以用互补角度来完成。

First, draw a diagram to model the problem.
::首先,绘制一个图来模拟问题。

Next, state the relationship between 36° and $\begin{align*}\angle x\end{align*}$ .
::下一步,说明36°和xx之间的关系。

36° and $\begin{align*}\angle x\end{align*}$ are complementary angles. The sum of the angles is 90°.
::36° 和 x 是互补角度。角度之和为 90 ° 。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} 36^{\circ} + ∠ x = 90^{\circ} \end{aligned}$
$\begin{align*}36^\circ+ \angle x = 90^\circ\end{align*}$
::36x=90{_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, subtract 36° from both sides of the equation.
::接下来,从方程两侧减去36度。

$\begin{aligned} \begin{array}{rcl} 36^{\circ} + ∠ x & = & 90^{\circ} \\ 36^{\circ} - 36^{\circ} + ∠ x & = & 90^{\circ} - 36^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 36^\circ + \angle x &=& 90^\circ\\ 36^\circ - 36^\circ + \angle x &=& 90^\circ - 36^\circ \end{array}\end{align*}$

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} 36^{\circ} - 36^{\circ} + ∠ x & = & 90^{\circ} - 36^{\circ} \\ ∠ x & = & 54^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 36^\circ - 36^\circ + \angle x &=& 90^\circ - 36^\circ\\ \angle x &=& 54^\circ \end{array}\end{align*}$

The answer is 54°.
::答案是54度

An angle of 54° is made between the first base foul line and the path of the baseball.
::第一底底底底底线和棒球路之间的角为54度。

Example 2
::例2

If the following angles are complementary, find the measure of the missing angle.
::如果以下角度是互补的,请找到缺失角度的度量。

$\begin{align*}\angle A = 37^\circ \text{ then } \angle B = ?\end{align*}$

First, draw a diagram to model the problem.
::首先,绘制一个图来模拟问题。

Next, state the relationship between $\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ .
::下一步,请说明A和B之间的关系。

$\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ are complementary angles. The sum of the angles is 90°.
::A和B是互补角度。角度之和为90°。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} ∠ A + ∠ B = 90^{\circ} \end{aligned}$
$\begin{align*}\angle A + \angle B = 90^\circ\end{align*}$
::AB=90 B=90 B=90 B=90 B=90 B=90 B=90 B=90 B=90 B=90 B=90

Next, substitute the measure of $\begin{align*}\angle A\end{align*}$ into the equation.
::接下来,将 A 的度量替换为等式。

$\begin{aligned} 37^{\circ} + ∠ B = 90^{\circ} \end{aligned}$
$\begin{align*}37^\circ + \angle B = 90^\circ\end{align*}$
::37B=90

Next, subtract 37° from both sides of the equation.

$\begin{aligned} \begin{array}{rcl} 37^{\circ} + ∠ B & = & 90^{\circ} \\ 37^{\circ} - 37^{\circ} + ∠ B & = & 90^{\circ} - 37^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 37^\circ + \angle B &=& 90^\circ\\ 37^\circ-37^\circ + \angle B &=& 90^\circ-37^\circ \end{array}\end{align*}$
::接下来,从方程两侧减去37°。37°B=90°B=37°37°B=90°37°B=90°37°B=37°B=90°37°B=37°B=37°B=37°B=37°B

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} 37^{\circ} - 37^{\circ} + ∠ B & = & 90^{\circ} - 37^{\circ} \\ ∠ B & = & 53^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 37^\circ-37^\circ + \angle B &=& 90^\circ - 37^\circ\\ \angle B &=& 53^\circ \end{array}\end{align*}$
::3737B=9037B=53

The answer is 53°.
::答案是53度

$\begin{aligned} m ∠ B = 53^{\circ} \end{aligned}$
$\begin{align*}m \angle B = 53^\circ\end{align*}$
::mB=53

Example 3
::例3

If the following angles are supplementary, find the measure of the missing angle.
::如果以下角度是补充的,请找到缺失角度的度量。

$\begin{align*}\angle A = 102^\circ \text{ then } \angle B = ?\end{align*}$
::A=102,然后是B=?

First, draw a diagram to model the problem.
::首先,绘制一个图来模拟问题。

Next, state the relationship between $\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ .
::下一步,请说明A和B之间的关系。

$\begin{align*}\angle A\end{align*}$ and $\begin{align*}\angle B\end{align*}$ are supplementary angles. The sum of the angles is 180°.
::A和B是补充角度。角度之和为180°。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} ∠ A + ∠ B = 180^{\circ} \end{aligned}$
$\begin{align*}\angle A + \angle B = 180^\circ\end{align*}$
::AB=180

Next, substitute the measure of $\begin{align*}\angle A\end{align*}$ into the equation.
::接下来,将 A 的度量替换为等式。

$\begin{aligned} 102^{\circ} + ∠ B = 180^{\circ} \end{aligned}$
$\begin{align*}102^\circ + \angle B = 180^\circ\end{align*}$
::102B=180{_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Next, subtract 102° from both sides of the equation.
::接下来,从方程两侧减去102度。

$\begin{aligned} \begin{array}{rcl} 102^{\circ} + ∠ B & = & 180^{\circ} \\ 102^{\circ} - 102^{\circ} + ∠ B & = & 180^{\circ} - 102^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 102^\circ + \angle B &=& 180^\circ\\ 102^\circ - 102^\circ + \angle B &=& 180^\circ - 102^\circ \end{array}\end{align*}$
::102B=180102102B=180102

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} 102^{\circ} - 102^{\circ} + ∠ B & = & 180^{\circ} - 102^{\circ} \\ ∠ B & = & 78^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 102^\circ - 102^\circ + \angle B &=& 180^\circ - 102^\circ\\ \angle B &=& 78^\circ \end{array}\end{align*}$
::102B=180102B=78

The answer is 78°.
::答案是78度

$\begin{aligned} m ∠ B = 78^{\circ} \end{aligned}$
$\begin{align*}m \angle B = 78^\circ\end{align*}$
::mB=78

Example 4
::例4

Using the following diagram, determine the measures of the missing angles.
::使用下图,确定缺失角度的度量。

First, state the relationship between the angle of $\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 3\end{align*}$ .
::首先,说明1和3角之间的关系。

$\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 3\end{align*}$ are vertical angles and are equal in measure.
::1和3是垂直角度,在量度上相等。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} m ∠ 1 = m ∠ 3 \end{aligned}$
$\begin{align*}m \angle 1 = m \angle 3\end{align*}$
::m1=m3

Next, substitute the measure of $\begin{align*}\angle 1\end{align*}$ into the equation.
::接下来,将 1 的度量替换为等式。

$\begin{aligned} \begin{array}{rcl} m ∠ 1 & = & m ∠ 3 \\ 137^{\circ} & = & m ∠ 3 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} m \angle 1 &=& m \angle 3\\ 137^\circ &=& m \angle 3 \end{array}\end{align*}$
::====================================================================================================================================================================================

The answer is 137°.
::答案是137度

$\begin{aligned} m ∠ 3 = 137^{\circ} \end{aligned}$
$\begin{align*}m \angle 3 = 137^\circ\end{align*}$
::m3=137

First, state the relationship between the angle of $\begin{align*}\angle 1\end{align*}$ and $\begin{align*}\angle 2\end{align*}$ .
::首先,说明1和2角之间的关系。

$\begin{align*}\angle 1\end{align*}$ is adjacent to $\begin{align*}\angle 2\end{align*}$ and the two angles form a straight angle.
::1 与 2 相邻,两个角度形成直角。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} ∠ 1 + ∠ 2 = 180^{\circ} \end{aligned}$
$\begin{align*}\angle 1 + \angle 2 = 180^\circ\end{align*}$

Next, substitute the measure of $\begin{align*}\angle 1\end{align*}$ into the equation.
::接下来,将 1 的度量替换为等式。

$\begin{aligned} 137^{\circ} + ∠ 2 = 180^{\circ} \end{aligned}$
$\begin{align*}137^\circ + \angle 2 = 180^\circ\end{align*}$

Next, subtract 137° from both sides of the equation.
::接下来,从方程两侧减去137度。

$\begin{aligned} \begin{array}{rcl} 137^{\circ} + ∠ 2 & = & 180^{\circ} \\ 137^{\circ} - 137^{\circ} + ∠ 2 & = & 180^{\circ} - 137^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 137^\circ + \angle 2 &=& 180^\circ\\ 137^\circ - 137^\circ + \angle 2 &=& 180^\circ - 137^\circ \end{array}\end{align*}$

Then, simplify both sides of the equation.
::然后简化方程式的两边

$\begin{aligned} \begin{array}{rcl} 137^{\circ} - 137^{\circ} + ∠ 2 & = & 180^{\circ} - 137^{\circ} \\ ∠ 2 & = & 43^{\circ} \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} 137^\circ - 137^\circ + \angle 2 &=& 180^\circ - 137^\circ\\ \angle 2 &=& 43^\circ \end{array}\end{align*}$

The answer is 43°.
::答案是43度

$\begin{aligned} m ∠ 2 = 43^{\circ} \end{aligned}$
$\begin{align*}m \angle 2 = 43^\circ\end{align*}$
::m2=43

First, state the relationship between the angle of $\begin{align*}\angle 2\end{align*}$ and $\begin{align*}\angle 4\end{align*}$ .
::首先,说明2和4角度之间的关系。

$\begin{align*}\angle 2\end{align*}$ and $\begin{align*}\angle 4\end{align*}$ are vertical angles and are equal in measure.
::2和4是垂直角度,在量度上相等。

Next, express the relationship using symbols.
::接下来,用符号来表达关系。

$\begin{aligned} m ∠ 2 = m ∠ 4 \end{aligned}$
$\begin{align*}m \angle 2 = m \angle 4\end{align*}$
::m2=m4

Next, substitute the measure of $\begin{align*}\angle 2\end{align*}$ into the equation.
::接下来,将 2 的度量替换为等式。

$\begin{aligned} \begin{array}{rcl} m ∠ 2 & = & m ∠ 4 \\ 43^{\circ} & = & m ∠ 4 \end{array} \end{aligned}$
$\begin{align*}\begin{array}{rcl} m \angle 2 &=& m \angle 4\\ 43^\circ &=& m \angle 4 \end{array}\end{align*}$

The answer is 43°.
::答案是43度

$\begin{aligned} m ∠ 4 = 43^{\circ} \end{aligned}$
$\begin{align*}m \angle 4 = 43^\circ\end{align*}$
::m4=43

Review
::回顾

If the following angle pairs are complementary, then what is the measure of the missing angle?
::如果以下角度对是互补的,那么缺失角度的度量是多少?

1. $\begin{align*}\text{If } \angle A = 45^\circ \text{ then } \angle B = ?\end{align*}$
::1. 如果A=45,那么B=?

2. $\begin{align*}\text{If } \angle C = 83^\circ \text{ then } \angle D = ?\end{align*}$
::2. 如果C=83,那么D=?

3. $\begin{align*}\text{If } \angle E = 33^\circ \text{ then } \angle F = ?\end{align*}$
::3. 如果E=33,那么F=3?

4. $\begin{align*}\text{If } \angle G = 53^\circ \text{ then } \angle H = ?\end{align*}$
::4. 如果G=53,那么H=4?

If the following angle pairs are supplementary, then what is the measure of the missing angle?
::如果以下角度对是补充的,那么缺失角度的量度是多少?

5. $\begin{align*}\text{If } \angle A = 40^\circ \text{ then } \angle B = ?\end{align*}$
::5. 如果A=40,那么B=?

6. $\begin{align*}\text{If } \angle A = 75^\circ \text{ then } \angle B = ?\end{align*}$
::6. 如果A=75,那么B=?

7. $\begin{align*}\text{If } \angle C = 110^\circ \text{ then } \angle F = ?\end{align*}$
::7. 如果C=110,那么F=?

8. $\begin{align*}\text{If } \angle D = 125^\circ \text{ then } \angle E = ?\end{align*}$
::8. 如果"D=125",那么"E"=?

9. $\begin{align*}\text{If } \angle M = 10^\circ \text{ then } \angle N = ?\end{align*}$
::9. 如果M=10,那么N=9?

10. $\begin{align*}\text{If } \angle O = 157^\circ \text{ then } \angle P = ?\end{align*}$
::10. 如果"O=157",那么"P"=?

Define the following types of angle pairs.
::定义下列角度对的种类。

11. Vertical angles
::11. 垂直角度

12. Adjacent angles
::12. 近角

13. Complementary angles
::13. 补充角度

14. Supplementary angles
::14. 补充角度

15. Interior angles
::15. 内部角度

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。

Resources
::资源

Section outline

General

角对角测量

Measuring Angle Pairs ::测量角对角

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Measuring Angle Pairs
::测量角对角

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源