Course: 6.6 草类 | The Way To Learn

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细胞类
Trapezoids
::细胞类

A trapezoid is a quadrilateral with exactly one pair of parallel sides.
::甲状腺类是一种四边形, 有着一对齐的平行面。

An isosceles trapezoid is a trapezoid where the non-parallel sides are congruent .
::等离子体类是非平行面相似的类类。

The base angles of an isosceles trapezoid are congruent. If $\begin{align*}ABCD\end{align*}$ is an isosceles trapezoid, then $\begin{align*}\angle A \cong \angle B\end{align*}$ and $\begin{align*}\angle C \cong \angle D\end{align*}$ .
::asosceles capezoid 的底角是相似的。如果 ABCD 是 asosceles capezoid, 则 {AB 和 {CD } 。

The converse is also true. If a trapezoid has congruent base angles, then it is an isosceles trapezoid. The diagonals of an isosceles trapezoid are also congruent. The midsegment (of a trapezoid) is a line segment that connects the midpoints of the non-parallel sides:
::反之, 反之亦然。如果一个陷阱类具有相容的底角, 那么它就是一个等骨类底角。等骨类底座的对角也是相似的。中间部分( 类) 是连接非平行边中点的线条段 :

There is only one midsegment in a trapezoid. It will be parallel to the bases because it is located halfway between them.
::捕鲸类中只有一个中间部分。它会与基地平行, 因为它位于它们中间的半个位置。

Midsegment Theorem : The length of the midsegment of a trapezoid is the average of the lengths of the bases.
::密段定理 : 网格类的中间部分长度是基底长度的平均值。

If $\begin{align*}\overline{EF}\end{align*}$ is the midsegment, then $\begin{align*}EF = \frac{AB + CD}{2}\end{align*}$ .
::如果EF是中间部分,那么EF=AB+CD2。

What if you were told that the polygon $\begin{align*}ABCD\end{align*}$ is an isosceles trapezoid and that one of its base angles measures $\begin{align*}38^\circ\end{align*}$ ? What can you conclude about its other base angle ?
::如果有人告诉您多边形ABCDIS 等离子体细胞类和其基角之一测量38+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Examples
::实例

For Examples 1 and 2, use the following information:
::关于例1和例2,请使用以下信息:

$\begin{align*}TRAP\end{align*}$ is an isosceles trapezoid.
::TRAP是一个等骨细胞类。

Example 1
::例1

Find $\begin{align*}m \angle TPA\end{align*}$ .
::找到MTPA。

$\begin{align*}\angle TPZ \cong \angle RAZ\end{align*}$ so $\begin{align*}m\angle TPA=20^\circ + 35^\circ = 55^\circ\end{align*}$ .
::~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Example 2
::例2

Find $\begin{align*}m \angle ZRA\end{align*}$ .
::去找玛拉

Since $\begin{align*}m\angle PZA = 110^\circ\end{align*}$ , $\begin{align*}m\angle RZA=70^\circ\end{align*}$ because they form a linear pair . By the Triangle Sum Theorem, $\begin{align*}m\angle ZRA=90^\circ\end{align*}$ .
::自 mPZA=110, mRZA=70 因为它们形成直线对。依三角形sumTheorem, mRA=90。

Example 3
::例3

Look at trapezoid $\begin{align*}TRAP\end{align*}$ below. What is $\begin{align*}m \angle A\end{align*}$ ?
::看看下面的陷阱图是什么?

$\begin{align*}TRAP\end{align*}$ is an isosceles trapezoid. $\begin{align*}m \angle R = 115^\circ\end{align*}$ also.
::TRAP 是一个等离子细胞类。 mR=115。

To find $\begin{align*}m \angle A\end{align*}$ , set up an equation.
::为了找到maA, 设置一个方程。

$\begin{aligned} 115^{\circ} + 115^{\circ} + m ∠ A + m ∠ P & = 360^{\circ} \\ 230^{\circ} + 2 m ∠ A & = 360^{\circ} \to m ∠ A = m ∠ P \\ 2 m ∠ A & = 130^{\circ} \\ m ∠ A & = 65^{\circ} \end{aligned}$
$\begin{align*}115^\circ + 115^\circ + m \angle A + m \angle P & = 360^\circ\\ 230^\circ + 2m \angle A & = 360^\circ \qquad \rightarrow m \angle A = m \angle P\\ 2m \angle A & = 130^\circ\\ m \angle A & = 65^\circ\end{align*}$
::$115mA+mP=3602302A=360mA=mP2A=130mA=65

Notice that $\begin{align*}m \angle R + m \angle A = 115^\circ + 65^\circ = 180^\circ\end{align*}$ . These angles will always be supplementary because of the Consecutive Interior Angles Theorem.
::注意 mR+mA=11565 180。这些角度将永远是补充的, 因为有连续的内脏角度定理。

Example 4
::例4

Is $\begin{align*}ZOID\end{align*}$ an isosceles trapezoid? How do you know?
::ZOID是不是一个异骨骼类动物?你怎么知道的?

$\begin{align*}40^\circ \neq 35^\circ\end{align*}$ , $\begin{align*}ZOID\end{align*}$ is not an isosceles trapezoid.
::4035, ZOID 不是一个等离子体。

Example 5
::例5

Find $\begin{align*}x\end{align*}$ . All figures are trapezoids with the midsegment marked as indicated.
::查找 x. 所有数字都是插件,中间部分标记如上所示。

$\begin{align*}x\end{align*}$ is the average of 12 and 26. $\begin{align*}\frac{12 + 26}{2} = \frac{38}{2} = 19\end{align*}$
::x是12和26.12+262=382=19的平均值

24 is the average of $\begin{align*}x\end{align*}$ and 35.
::24是x和35的平均数。

$\begin{aligned} \frac{x + 35}{2} & = 24 \\ x + 35 & = 48 \\ x & = 13 \end{aligned}$
$\begin{align*}\frac{x + 35}{2} & = 24\\ x + 35 & = 48\\ x & = 13\end{align*}$
::x+352=24x+35=48x=13

20 is the average of $\begin{align*}5x - 15\end{align*}$ and $\begin{align*}2x - 8\end{align*}$ .
::20是5x-15和2x-8的平均值。

$\begin{aligned} \frac{5 x - 15 + 2 x - 8}{2} & = 20 \\ 7 x - 23 & = 40 \\ 7 x & = 63 \\ x & = 9 \end{aligned}$
$\begin{align*}\frac{5x - 15 + 2x - 8}{2} & = 20\\ 7x - 23 & = 40\\ 7x & = 63\\ x & = 9\end{align*}$
::5-15+2x-82=207x-23=407x=63x=9

Review
::回顾

1. Can the parallel sides of a trapezoid be congruent? Why or why not?
::1. 围网类的平行面能否一致?为什么或为什么不能?

For questions 2-8, find the length of the midsegment or missing side.
::对于问题2-8,请找到中间部分的长度或缺失的侧面。

Find the value of the missing variable(s).
::查找缺失变量的值。

Find the lengths of the diagonals of the trapezoids below to determine if it is isosceles.
::查找下面的捕捉类星体的对角长度,以确定它是否为等分形。

$\begin{align*}A(-3, 2), B(1, 3), C(3, -1), D(-4, -2)\end{align*}$
::A(3,3,2),B(1,3,3),C(3,-1),D(-4,4)-2

$\begin{align*}A(-3, 3), B(2, -2), C(-6, -6), D(-7, 1)\end{align*}$
::A(-3,3),B(2,-2),C(-6,-6),D(-7,1)

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

细胞类

Trapezoids ::细胞类

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)

Trapezoids
::细胞类

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)