Course: 6.3 平行图 | The Way To Learn

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平行图

Parallelograms
::平行图

A parallelogram is a quadrilateral with two pairs of parallel sides.
::平行图是四边形,两对平行面。

Notice that each pair of sides is marked parallel (for the last two shapes, remember that when two lines are perpendicular to the same line then they are parallel). Parallelograms have a lot of interesting properties.
::注意每个侧面的标记是平行的( 对于最后两个形状, 请记住当两条线与同一线垂直时, 两条线是平行的 ) 。平行图有许多有趣的属性。

Facts about Parallelograms
::平行图事实

Opposite Sides Theorem : If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent .
::对面理论:如果四边形是一个平行图,那么两对对对立面是相同的。

If then
::如果当时

2. Opposite Angles Theorem: If a quadrilateral is a parallelogram, then both pairs of opposite angles are congruent.
::2. 对角角理论:如果四边形是一个平行图,那么两对对相反角度是相同的。

If then
::如果当时

3. Consecutive Angles Theorem: If a quadrilateral is a parallelogram, then all pairs of consecutive angles are supplementary.
::3. 连续角度理论:如果四边形是一个平行图,那么所有连续角度的对子都是补充的。

If then
::如果当时

\begin{aligned} m ∠ A + m ∠ D = 180^{\circ} \\ m ∠ A + m ∠ B = 180^{\circ} \\ m ∠ B + m ∠ C = 180^{\circ} \\ m ∠ C + m ∠ D = 180^{\circ} \end{aligned}

::@A+mD=180mA+mB=180mB+mC=180mC+mD=180

4. Parallelogram Diagonals Theorem: If a quadrilateral is a parallelogram, then the diagonals bisect each other.
::4. 对角线理论:如果四边形是平行图,则对角线相互交叉。

If then
::如果当时

What if you were told that $\begin{aligned} F G H I \end{aligned}$ is a parallelogram and you are given the length of $\begin{aligned} F G \end{aligned}$ and the measure of $\begin{aligned} ∠ F \end{aligned}$ ? What can you determine about $\begin{aligned} H I \end{aligned}$ , $\begin{aligned} ∠ H \end{aligned}$ , $\begin{aligned} ∠ G \end{aligned}$ , and $\begin{aligned} ∠ I \end{aligned}$ ?
::如果有人告诉你,FGHI是一个平行的图象,而且你得到FG和FF的长度呢?你能确定HI、H、G和I是什么?

Examples
::实例

Example 1
::例1

Show that the diagonals of $\begin{aligned} F G H J \end{aligned}$ bisect each other.
::显示FGHJ的对角线相互对立。

Find the midpoint of each diagonal .
::查找每个对角的中点。

\begin{aligned} Midpoint of \bar{F H} : (\frac{- 4 + 6}{2}, \frac{5 - 4}{2}) = (1, 0.5) \\ Midpoint of \bar{G J} : (\frac{3 - 1}{2}, \frac{3 - 2}{2}) = (1, 0.5) \end{aligned}

::FH的中点

-4+62,5-42)=(1,0.5)GJ的中点

-3-12,3-22)=(1,0.5)

Because they are the same point , the diagonals intersect at each other’s midpoint. This means they bisect each other.
::因为它们是同一点,对角线相互交叉,相互交叉,这意味着它们相互交叉。

Example 2
::例2

Find the measures of $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ in the parallelogram below:
::在以下平行图中查找a和b的测量值:

Consecutive angles are supplementary so $\begin{aligned} 127^{\circ} + m ∠ b = 180^{\circ} \end{aligned}$ which means that $\begin{aligned} m ∠ b = 53^{\circ} \end{aligned}$ . $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ are and since the lines are parallel (since its a parallelogram), that means that $\begin{aligned} m ∠ a = m ∠ b = 53^{\circ} \end{aligned}$ .
::连续角度是补充性的, 因此是127mb=180, 这意味着 mb=53. a和b是平行的( 因为它是一个平行图) , 这意味着 ma=mb=53。

Example 3
::例3

$\begin{aligned} A B C D \end{aligned}$ is a parallelogram. If $\begin{aligned} m ∠ A = 56^{\circ} \end{aligned}$ , find the measure of the other angles.
::ABCD 是一个平行图。如果 mA=56, 请找到其他角度的测量值。

First draw a picture. When labeling the vertices , the letters are listed, in order.
::首先绘制图片。在标注顶点时,字母按顺序列出。

If $\begin{aligned} m ∠ A = 56^{\circ} \end{aligned}$ , then $\begin{aligned} m ∠ C = 56^{\circ} \end{aligned}$ by the Opposite Angles Theorem.
::如果 mA=56, 那么 mC=56在对面角定理旁。

\begin{aligned} m ∠ A + m ∠ B & = 180^{\circ} by the Consecutive Angles Theorem. \\ 56^{\circ} + m ∠ B & = 180^{\circ} \\ m ∠ B & = 124^{\circ} m ∠ D = 124^{\circ} because it is an opposite angle to ∠ B . \end{aligned}

::mA+mB=180by the contracticive angles Theorem. 56mB=180B=124mD=124B 因为它与B反角。

Example 4
::例4

Find the values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::查找 x 和 y 的值。

Remember that opposite sides of a parallelogram are congruent. Set up equations and solve.
::记住平行图的对立面是相似的。设置方程式和解析。

\begin{aligned} 6 x - 7 & = 2 x + 9 & y + 3 = 12 \\ 4 x & = 16 & y = 9 \\ x & = 4 \end{aligned}

::6x-7=2x+9y+3=124x=16y=9x=4

Example 5
::例5

Prove the Opposite Sides Theorem.
::证明对面侧定理

Given : $\begin{aligned} A B C D \end{aligned}$ is a parallelogram with diagonal $\begin{aligned} \bar{B D} \end{aligned}$
::参考:ABCD是和对角 BD 平行的图形

Prove : $\begin{aligned} \bar{A B} ≅ \bar{D C}, \bar{A D} ≅ \bar{B C} \end{aligned}$
::证明:AB=DC======================================================================================================

*Statement*	*Reason*
1. $\begin{aligned} A B C D \end{aligned}$ is a parallelogram with diagonal $\begin{aligned} \bar{B D} \end{aligned}$	1. Given
2. $\begin{aligned} \bar{A B} ‖ \bar{D C}, \bar{A D} ‖ \bar{B C} \end{aligned}$	2. Definition of a parallelogram
3. $\begin{aligned} ∠ A B D ≅ ∠ B D C, ∠ A D B ≅ ∠ D B C \end{aligned}$	3. Alternate Interior Angles Theorem
4. $\begin{aligned} \bar{D B} ≅ \bar{D B} \end{aligned}$	4. Reflexive PoC
5. $\begin{aligned} △ A B D ≅ △ C D B \end{aligned}$	5. ASA
6. $\begin{aligned} \bar{A B} ≅ \bar{D C}, \bar{A D} ≅ \bar{B C} \end{aligned}$	6.

The proof of the Opposite Angles Theorem is almost identical.
::对面角理论的证明几乎完全相同。

Review
::回顾

$\begin{aligned} A B C D \end{aligned}$ is a parallelogram. Fill in the blanks below.
::ABCD 是一个平行图。填充下面的空白。

If $\begin{aligned} A B = 6 \end{aligned}$ , then $\begin{aligned} C D = \end{aligned}$ ______.
::如果AB=6,那么CD=___________________。

If $\begin{aligned} A E = 4 \end{aligned}$ , then $\begin{aligned} A C = \end{aligned}$ ______.
::如果AE=4,AC=____________________________________________________________________________________________________________________________________________________________________

If $\begin{aligned} m ∠ A D C = 80^{\circ}, m ∠ D A B \end{aligned}$ = ______.
::如果MADC=80,mDAB=_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

If $\begin{aligned} m ∠ B A C = 45^{\circ}, m ∠ A C D \end{aligned}$ = ______.
::如果bAC=45,mACD=____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

If $\begin{aligned} m ∠ C B D = 62^{\circ}, m ∠ A D B \end{aligned}$ = ______.
::如果是CBD=62,那么是ABD=_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

If $\begin{aligned} D B = 16 \end{aligned}$ , then $\begin{aligned} D E \end{aligned}$ = ______.
::如果DB=16,那么DE=____。

If $\begin{aligned} m ∠ B = 72^{\circ} \end{aligned}$ in parallelogram $\begin{aligned} A B C D \end{aligned}$ , find the other three angles.
::如果 mB=72 以平行的 ABCD 表示, 请找到另外三个角度。

If $\begin{aligned} m ∠ S = 143^{\circ} \end{aligned}$ in parallelogram $\begin{aligned} P Q R S \end{aligned}$ , find the other three angles.
::如果 mS=143 以平行方形 PQRS 表示,请找到另外三个角度。

If $\begin{aligned} \bar{A B} ⊥ \bar{B C} \end{aligned}$ in parallelogram $\begin{aligned} A B C D \end{aligned}$ , find the measure of all four angles.
::如果AB BC 以平行的 ABCD 表示, 找到所有四个角度的测量值。

If $\begin{aligned} m ∠ F = x^{\circ} \end{aligned}$ in parallelogram $\begin{aligned} E F G H \end{aligned}$ , find the other three angles.
::如果 mF=x 以平行图 EFGH 表示, 请找到另外三个角度。

For questions 11-18, find the values of the variable(s). All the figures below are parallelograms.
::对于问题11-18,请查看变量的值。以下所有数字均为平行图。

Use the parallelogram $\begin{aligned} W A V E \end{aligned}$ to find:
::使用平行图 WAVE 查找 :

$\begin{aligned} m ∠ A W E \end{aligned}$
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$\begin{aligned} m ∠ E S V \end{aligned}$
::máESV( máESV)

$\begin{aligned} m ∠ W E A \end{aligned}$
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$\begin{aligned} m ∠ A V W \end{aligned}$
::aAVW

Find the point of intersection of the diagonals to see if $\begin{aligned} E F G H \end{aligned}$ is a parallelogram.
::查找对角线的交叉点, 看看 EFGH 是否是一个平行图。

$\begin{aligned} E (- 1, 3), F (3, 4), G (5, - 1), H (1, - 2) \end{aligned}$
::E(-1,3,3),F(3,4),G(5)-1,H(1,-2)

$\begin{aligned} E (3, - 2), F (7, 0), G (9, - 4), H (5, - 4) \end{aligned}$
::E(3,-2),F(7,7,0,G(9,-4),H(5,-4)

$\begin{aligned} E (- 6, 3), F (2, 5), G (6, - 3), H (- 4, - 5) \end{aligned}$
::E(-6,6,3),F(2,5),G(6,3),H(-4,5)

$\begin{aligned} E (- 2, - 2), F (- 4, - 6), G (- 6, - 4), H (- 4, 0) \end{aligned}$
::E(-2,-2,2),F(-4,-6),G(-6,-4,4),H(-4,0)

Fill in the blanks in the proofs below.
::填充以下证据中的空白。

Opposite Angles Theorem
::对面角定理

Given : $\begin{aligned} A B C D \end{aligned}$ is a parallelogram with diagonal $\begin{aligned} \bar{B D} \end{aligned}$
::参考:ABCD是和对角 BD 平行的图形

Prove : $\begin{aligned} ∠ A ≅ ∠ C \end{aligned}$
::证明:

*Statement*	*Reason*
1.	1. Given
2. $\begin{aligned} \bar{A B} ‖ \bar{D C}, \bar{A D} ‖ \bar{B C} \end{aligned}$	2.
3.	3. Alternate Interior Angles Theorem
4.	4. Reflexive PoC
5. $\begin{aligned} △ A B D ≅ △ C D B \end{aligned}$	5.
6. $\begin{aligned} ∠ A ≅ ∠ C \end{aligned}$	6.

Parallelogram Diagonals Theorem
::对角对角定理

Given : $\begin{aligned} A B C D \end{aligned}$ is a parallelogram with diagonals $\begin{aligned} \bar{B D} \end{aligned}$ and $\begin{aligned} \bar{A C} \end{aligned}$
::参考:ABCD是和对角BD和AC的平行图形

Prove : $\begin{aligned} \bar{A E} ≅ \bar{E C}, \bar{D E} ≅ \bar{E B} \end{aligned}$
::证明:AEEC,DEEEEE

*Statement*	*Reason*
1.	1.
2.	2. Definition of a parallelogram
3.	3. Alternate Interior Angles Theorem
4. $\begin{aligned} \bar{A B} ≅ \bar{D C} \end{aligned}$	4.
5.	5.
6. $\begin{aligned} \bar{A E} ≅ \bar{E C}, \bar{D E} ≅ \bar{E B} \end{aligned}$	6.

Find $\begin{aligned} x, y^{\circ}, \end{aligned}$ and $\begin{aligned} z^{\circ} \end{aligned}$ . (The two quadrilaterals with the same side are parallelograms.)
::查找 x, y 和 z 。 (具有相同侧面的两个四边形是平行图。 )

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

平行图

Parallelograms ::平行图

Facts about Parallelograms ::平行图事实

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)

Parallelograms
::平行图

Facts about Parallelograms
::平行图事实

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)