Course: 2.7 两色证明 | The Way To Learn

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两色证明

Two-Column Proofs
::两色证明

A two-column proof is one common way to organize a proof in geometry. Two-column proofs always have two columns: one for statements and one for reasons. The best way to understand two-column proofs is to read through examples.
::两栏证明是组织几何证据的一种常见方法。两栏证明总是有两栏:一栏用于说明,一栏用于理由。理解两栏证明的最佳方法是通过实例阅读。

When writing your own two-column proof, keep these things in mind:
::写出自己的两栏证明时, 请记住:

Number each step.
::每个步骤的数目。

Start with the given information.
::以给定的信息开始。

Statements with the same reason can be combined into one step . It is up to you.
::理由相同的语句可以合并为一步,由你决定。

Draw a picture and mark it with the given information.
::绘制图片并用给定的信息标记它。

You must have a reason for EVERY statement.
::你必须有一个理由每一个声明。

The order of the statements in the proof is not always fixed, but make sure the order makes logical sense.
::证据中的语句顺序并不总是固定的,但要确保顺序合乎逻辑。

Reasons will be definitions, postulates, properties and previously proven theorems. “Given” is only used as a reason if the information in the statement column was given in the problem.
::理由将是定义、假设、属性和以前证明的理论。 “给予”只有在说明栏中的信息是在问题中提供的时才被用作理由。

Use symbols and abbreviations for words within proofs. For example, $\begin{aligned} ≅ \end{aligned}$ can be used in place of the word congruent . You could also use $\begin{aligned} ∠ \end{aligned}$ for the word angle .
::使用符号和缩略语表示在校对中的单词。例如, \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Suppose you are told that $\begin{aligned} ∠ X Y Z \end{aligned}$ is a right angle and that $\begin{aligned} \vec{Y W} \end{aligned}$ bisects $\begin{aligned} ∠ X Y Z \end{aligned}$ . You are then asked to prove $\begin{aligned} ∠ X Y W ≅ ∠ W Y Z \end{aligned}$ .
::假设你被告知 XYZ是一个正确的角度和YWQ 的两块 XYZ。然后你被要求证明 XYWWWWYZ。

Examples
::实例

Example 1
::例1

Write a two-column proof for the following:
::为下列事项写两栏证明:

If $\begin{aligned} A, B, C \end{aligned}$ , and $\begin{aligned} D \end{aligned}$ are points on a line , in the given order, and $\begin{aligned} A B = C D \end{aligned}$ , then $\begin{aligned} A C = B D \end{aligned}$ .
::如果A、B、C和D是线上的点,则按给定顺序排列,AB=CD,然后是AC=BD。

When the statement is given in this way, the “if” part is the given and the “then” part is what we are trying to prove.
::当以这种方式作出声明时,“如果”部分是给的,“当时”部分是我们试图证明的。

Always start with drawing a picture of what you are given.
::总是从绘制给定内容的图片开始。

Plot the points in the order $\begin{aligned} A, B, C, D \end{aligned}$ on a line.
::按A、B、C、D顺序排列点数。

Add the given, $\begin{aligned} A B = C D \end{aligned}$ .
::添加给定的,AB=CD。

Draw the two-column proof and start with the given information.
::绘制两栏证明,并用给定的信息开始。

*Statement*	*Reason*
1. $\begin{aligned} A, B, C \end{aligned}$ , and $\begin{aligned} D \end{aligned}$ are collinear , in that order.	1. Given
2. $\begin{aligned} A B = C D \end{aligned}$	2. Given
3. $\begin{aligned} B C = B C \end{aligned}$	3. Reflexive $\begin{aligned} P o E \end{aligned}$
4. $\begin{aligned} A B + B C = B C + C D \end{aligned}$	4. Addition $\begin{aligned} P o E \end{aligned}$
5. $\begin{aligned} A B + B C = A C \end{aligned}$ ::5. AB+BC=AC $\begin{aligned} B C + C D = B D \end{aligned}$ ::BC+CD=BD (BC+CD=BD)	5. Segment Addition Postulate
6. $\begin{aligned} A C = B D \end{aligned}$	6. Substitution or Transitive $\begin{aligned} P o E \end{aligned}$

Example 2
::例2

Write a two-column proof.
::写两栏证明书

Given : $\begin{aligned} \vec{B F} \end{aligned}$ bisects $\begin{aligned} ∠ A B C \end{aligned}$ ; $\begin{aligned} ∠ A B D ≅ ∠ C B E \end{aligned}$
::参考:BF-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-B-B-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-B-B-AB-AB-AB-AB-AB-AB-AB-B-B-B-B-B-B-B-B-B-B-B-B-B-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-AB-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-B-

Prove : $\begin{aligned} ∠ D B F ≅ ∠ E B F \end{aligned}$
::证明: DBF_EBF

First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, $\begin{aligned} m ∠ A B F = m ∠ F B C \end{aligned}$ .
::首先,在照片上贴上适当的标记。回顾,这个两部分的意思是“切成两半”。因此,MABF=mFBC。

*Statement*	*Reason*
1. $\begin{aligned} \vec{B F} \end{aligned}$ bisects $\begin{aligned} ∠ A B C, ∠ A B D ≅ ∠ C B E \end{aligned}$	1. Given
2. $\begin{aligned} m ∠ A B F = m ∠ F B C \end{aligned}$	2. Definition of an Angle Bisector
3. $\begin{aligned} m ∠ A B D = m ∠ C B E \end{aligned}$	3. If angles are $\begin{aligned} ≅ \end{aligned}$ , then their measures are equal.
4. $\begin{aligned} m ∠ A B F = m ∠ A B D + m ∠ D B F \end{aligned}$ ::4. mABF=mABD+mDBF $\begin{aligned} m ∠ F B C = m ∠ E B F + m ∠ C B E \end{aligned}$ ::mFBC=mEBF+mCBE FBC=mEBF+mCBE FBC=mEBF+mCBE	4. Angle Addition Postulate
5. $\begin{aligned} m ∠ A B D + m ∠ D B F = m ∠ E B F + m ∠ C B E \end{aligned}$	5. Substitution $\begin{aligned} P o E \end{aligned}$
6. $\begin{aligned} m ∠ A B D + m ∠ D B F = m ∠ E B F + m ∠ A B D \end{aligned}$	6. Substitution $\begin{aligned} P o E \end{aligned}$
7. $\begin{aligned} m ∠ D B F = m ∠ E B F \end{aligned}$	7. Subtraction $\begin{aligned} P o E \end{aligned}$
8. $\begin{aligned} ∠ D B F ≅ ∠ E B F \end{aligned}$	8. If measures are equal, the angles are $\begin{aligned} ≅ \end{aligned}$ .

Example 3
::例3

The Right Angle Theorem states that if two angles are right angles, then the angles are congruent. Prove this theorem .
::右角角定理指出,如果两个角度是右角,那么角度是相同的。证明这个定理。

To prove this theorem, set up your own drawing and name some angles so that you have specific angles to talk about.
::为了证明这个定理, 设置您自己的绘图, 并指定一些角度, 这样您就可以有具体的角度来讨论。

Given : $\begin{aligned} ∠ A \end{aligned}$ and $\begin{aligned} ∠ B \end{aligned}$ are right angles
::给出: A 和 B 是正确角度

Prove : $\begin{aligned} ∠ A ≅ ∠ B \end{aligned}$
::证明: @AB

*Statement*	*Reason*
1. $\begin{aligned} ∠ A \end{aligned}$ and $\begin{aligned} ∠ B \end{aligned}$ are right angles	1. Given
2. $\begin{aligned} m ∠ A = 90^{\circ} \end{aligned}$ and $\begin{aligned} m ∠ B = 90^{\circ} \end{aligned}$	2. Definition of right angles
3. $\begin{aligned} m ∠ A = m ∠ B \end{aligned}$	3. Transitive $\begin{aligned} P o E \end{aligned}$
4. $\begin{aligned} ∠ A ≅ ∠ B \end{aligned}$	4. $\begin{aligned} ≅ \end{aligned}$ angles have = measures

Any time right angles are mentioned in a proof, you will need to use this theorem to say the angles are congruent.
::每当在证据中提及右角度时, 您需要使用此定理来表示角度一致。

Example 4
::例4

The Same Angle Supplements Theorem states that if two angles are supplementary to the same angle then the two angles are congruent. Prove this theorem.
::同一角补充理论指出,如果两个角度是同一角度的补充,那么这两个角度是相同的。证明这个理论。

Given : $\begin{aligned} ∠ A \end{aligned}$ and $\begin{aligned} ∠ B \end{aligned}$ are . $\begin{aligned} ∠ B \end{aligned}$ and $\begin{aligned} ∠ C \end{aligned}$ are supplementary angles.
::说明:A和B是.B和C是补充角度。

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

*Statement*	*Reason*
1. $\begin{aligned} ∠ A \end{aligned}$ and $\begin{aligned} ∠ B \end{aligned}$ are supplementary ::1. A和B是补充 $\begin{aligned} ∠ B \end{aligned}$ and $\begin{aligned} ∠ C \end{aligned}$ are supplementary ::B和C是补充	1. Given
2. $\begin{aligned} m ∠ A + m ∠ B = 180^{\circ} \end{aligned}$ ::2. mA+mB=180 $\begin{aligned} m ∠ B + m ∠ C = 180^{\circ} \end{aligned}$ ::mB+mC=180\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\C\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\	2. Definition of supplementary angles
3. $\begin{aligned} m ∠ A + m ∠ B = m ∠ B + m ∠ C \end{aligned}$	3. Substitution $\begin{aligned} P o E \end{aligned}$
4. $\begin{aligned} m ∠ A = m ∠ C \end{aligned}$	4. Subtraction $\begin{aligned} P o E \end{aligned}$
5. $\begin{aligned} ∠ A ≅ ∠ C \end{aligned}$	5. $\begin{aligned} ≅ \end{aligned}$ angles have = measures

Example 5
::例5

The Vertical Angles Theorem states that are congruent. Prove this theorem.
::垂直角定理显示相似。证明这个定理。

Given : Lines $\begin{aligned} k \end{aligned}$ and $\begin{aligned} m \end{aligned}$ intersect.
::K线和m线交叉。

Prove: $\begin{aligned} ∠ 1 ≅ ∠ 3 \end{aligned}$
::证明:% 1 @% 3

*Statement*	*Reason*
1. Lines $\begin{aligned} k \end{aligned}$ and $\begin{aligned} m \end{aligned}$ intersect	1. Given
2. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are a linear pair ::2. 1和2是线性对 $\begin{aligned} ∠ 2 \end{aligned}$ and $\begin{aligned} ∠ 3 \end{aligned}$ are a linear pair ::2和3是线性对	2. Definition of a Linear Pair
3. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are supplementary ::3.1和2是补充性的 $\begin{aligned} ∠ 2 \end{aligned}$ and $\begin{aligned} ∠ 3 \end{aligned}$ are supplementary ::2和3是补充	3. Linear Pair Postulate
4. $\begin{aligned} m ∠ 1 + m ∠ 2 = 180^{\circ} \end{aligned}$ ::m1+m2=180 $\begin{aligned} m ∠ 2 + m ∠ 3 = 180^{\circ} \end{aligned}$ ::m2+m3=180	4. Definition of Supplementary Angles
5. $\begin{aligned} m ∠ 1 + m ∠ 2 = m ∠ 2 + m ∠ 3 \end{aligned}$	5. Substitution $\begin{aligned} P o E \end{aligned}$
6. $\begin{aligned} m ∠ 1 = m ∠ 3 \end{aligned}$	6. Subtraction $\begin{aligned} P o E \end{aligned}$
7. $\begin{aligned} ∠ 1 ≅ ∠ 3 \end{aligned}$	7. $\begin{aligned} ≅ \end{aligned}$ angles have = measures

Example 6
::例6

$\begin{aligned} ∠ 1 ≅ ∠ 4 \end{aligned}$ and $\begin{aligned} ∠ C \end{aligned}$ and $\begin{aligned} ∠ F \end{aligned}$ are right angles.
::14 和 C 和 F 是正确角度。

Which angles are congruent and why?
::哪个角度是一致的,为什么?

By the Right Angle Theorem, $\begin{aligned} ∠ C ≅ ∠ F \end{aligned}$ . Also, $\begin{aligned} ∠ 2 ≅ ∠ 3 \end{aligned}$ by the Same Angles Supplements Theorem because $\begin{aligned} ∠ 1 ≅ ∠ 4 \end{aligned}$ and they are linear pairs with these congruent angles.
::以右角角定理为依归, 也以同角定理为依归, 因为它们是线性对子,

Review
::回顾

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

Given: $\begin{aligned} ∠ A B C ≅ ∠ D E F \end{aligned}$ and $\begin{aligned} ∠ G H I ≅ ∠ J K L \end{aligned}$
::来源:@ABCDEF和GHIJKL

Prove: $\begin{aligned} m ∠ A B C + m ∠ G H I = m ∠ D E F + m ∠ J K L \end{aligned}$
::证明:mABC+mGHI=mDEF+mJKL

*Statement*	*Reason*
1.	1. Given
2. $\begin{aligned} m ∠ A B C = m ∠ D E F \end{aligned}$ ::2. mABC=mDEF $\begin{aligned} m ∠ G H I = m ∠ J K L \end{aligned}$ ::	2.
3.	3. Addition $\begin{aligned} P o E \end{aligned}$
4. $\begin{aligned} m ∠ A B C + m ∠ G H I = m ∠ D E F + m ∠ J K L \end{aligned}$	4.

Given: $\begin{aligned} M \end{aligned}$ is the midpoint of $\begin{aligned} \bar{A N} \end{aligned}$ . $\begin{aligned} N \end{aligned}$ is the midpoint $\begin{aligned} \bar{M B} \end{aligned}$
::表示:M是 AN的中点。N是中点MB

Prove : $\begin{aligned} A M = N B \end{aligned}$
::证明:AM=NB

*Statement*	*Reason*
1.	Given
2.	Definition of a midpoint
3. $\begin{aligned} A M = N B \end{aligned}$

Given: $\begin{aligned} \bar{A C} ⊥ \bar{B D} \end{aligned}$ and $\begin{aligned} ∠ 1 ≅ ∠ 4 \end{aligned}$
::依据:AC =BD 和14

Prove : $\begin{aligned} ∠ 2 ≅ ∠ 3 \end{aligned}$
::证明:% 2 @% 3

*Statement*	*Reason*
1. $\begin{aligned} \bar{A C} ⊥ \bar{B D}, ∠ 1 ≅ ∠ 4 \end{aligned}$	1.
2. $\begin{aligned} m ∠ 1 = m ∠ 4 \end{aligned}$	2.
3.	3. $\begin{aligned} ⊥ \end{aligned}$ lines create right angles
4. $\begin{aligned} m ∠ A C B = 90^{\circ} \end{aligned}$ ::4. mACB=90___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ $\begin{aligned} m ∠ A C D = 90^{\circ} \end{aligned}$ ::máACD=90	4.
5. $\begin{aligned} m ∠ 1 + m ∠ 2 = m ∠ A C B \end{aligned}$ ::m1+m2=mACB $\begin{aligned} m ∠ 3 + m ∠ 4 = m ∠ A C D \end{aligned}$ ::3+m4=mACD	5.
6.	6. Substitution
7. $\begin{aligned} m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4 \end{aligned}$	7.
8.	8. Substitution
9.	9.Subtraction $\begin{aligned} P o E \end{aligned}$
10. $\begin{aligned} ∠ 2 ≅ ∠ 3 \end{aligned}$	10.

Given: $\begin{aligned} ∠ M L N ≅ ∠ O L P \end{aligned}$
::参照:*MLNOLP

Prove : $\begin{aligned} ∠ M L O ≅ ∠ N L P \end{aligned}$
::证明:

*Statement*	*Reason*
1.	1.
2.	2. $\begin{aligned} ≅ \end{aligned}$ angles have = measures
3.	3. Angle Addition Postulate
4.	4. Substitution
5. $\begin{aligned} m ∠ M L O = m ∠ N L P \end{aligned}$	5.
6.	6. $\begin{aligned} ≅ \end{aligned}$ angles have = measures

Given: $\begin{aligned} \bar{A E} ⊥ \bar{E C} \end{aligned}$ and $\begin{aligned} \bar{B E} ⊥ \bar{E D} \end{aligned}$
::以: AE EC 并被锁定

Prove : $\begin{aligned} ∠ 1 ≅ ∠ 3 \end{aligned}$
::证明:

*Statement*	*Reason*
1.	1.
2.	2. $\begin{aligned} ⊥ \end{aligned}$ lines create right angles
3. $\begin{aligned} m ∠ B E D = 90^{\circ} \end{aligned}$ ::3. mBED=90 $\begin{aligned} m ∠ A E C = 90^{\circ} \end{aligned}$ ::mAEC=90	3.
4.	4. Angle Addition Postulate
5.	5. Substitution
6. $\begin{aligned} m ∠ 2 + m ∠ 3 = m ∠ 1 + m ∠ 3 \end{aligned}$	6.
7.	7. Subtraction $\begin{aligned} P o E \end{aligned}$
8.	8. $\begin{aligned} ≅ \end{aligned}$ angles have = measures

Given: $\begin{aligned} ∠ L \end{aligned}$ is supplementary to $\begin{aligned} ∠ M \end{aligned}$ and $\begin{aligned} ∠ P \end{aligned}$ is supplementary to $\begin{aligned} ∠ O \end{aligned}$ and $\begin{aligned} ∠ L ≅ ∠ O \end{aligned}$
::因此:L是M的补充,P是O和LO的补充。

Prove : $\begin{aligned} ∠ P ≅ ∠ M \end{aligned}$
::证明:

*Statement*	*Reason*
1.	1.
2. $\begin{aligned} m ∠ L = m ∠ O \end{aligned}$	2.
3.	3. Definition of supplementary angles
4.	4. Substitution
5.	5. Substitution
6.	6. Subtraction $\begin{aligned} P o E \end{aligned}$
7. $\begin{aligned} ∠ M ≅ ∠ P \end{aligned}$	7.

Given: $\begin{aligned} ∠ 1 ≅ ∠ 4 \end{aligned}$
::百分比:%1%4

Prove : $\begin{aligned} ∠ 2 ≅ ∠ 3 \end{aligned}$
::证明:% 2 @% 3

*Statement*	*Reason*
1.	1.
2. $\begin{aligned} m ∠ 1 = m ∠ 4 \end{aligned}$	2.
3.	3. Definition of a Linear Pair
4. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are supplementary ::4.1和2是补充性的 $\begin{aligned} ∠ 3 \end{aligned}$ and $\begin{aligned} ∠ 4 \end{aligned}$ are supplementary ::3和4是补充	4.
5.	5. Definition of supplementary angles
6. $\begin{aligned} m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 4 \end{aligned}$	6.
7. $\begin{aligned} m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 1 \end{aligned}$	7.
8. $\begin{aligned} m ∠ 2 = m ∠ 3 \end{aligned}$	8.
9. $\begin{aligned} ∠ 2 ≅ ∠ 3 \end{aligned}$	9.

Given: $\begin{aligned} ∠ C \end{aligned}$ and $\begin{aligned} ∠ F \end{aligned}$ are right angles
::给定: C 和F 是正确角度

Prove : $\begin{aligned} m ∠ C + m ∠ F = 180^{\circ} \end{aligned}$
::证明: mC+mF=180{____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

*Statement*	*Reason*
1.	1.
2. $\begin{aligned} m ∠ C = 90^{\circ}, m ∠ F = 90^{\circ} \end{aligned}$	2.
3. $\begin{aligned} 90^{\circ} + 90^{\circ} = 180^{\circ} \end{aligned}$	3.
4. $\begin{aligned} m ∠ C + m ∠ F = 180^{\circ} \end{aligned}$	4.

Given: $\begin{aligned} l ⊥ m \end{aligned}$
::来源:百分比

Prove : $\begin{aligned} ∠ 1 ≅ ∠ 2 \end{aligned}$
::证明:%1 @%2

*Statement*	*Reason*
1. $\begin{aligned} l ⊥ m \end{aligned}$	1.
2. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are right angles	2.
3.	3.

Given: $\begin{aligned} m ∠ 1 = 90^{\circ} \end{aligned}$
::百分比: m1=90

Prove : $\begin{aligned} m ∠ 2 = 90^{\circ} \end{aligned}$
::证明: m2=90

*Statement*	*Reason*
1.	1.
2. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are a linear pair	2.
3.	3. Linear Pair Postulate
4.	4. Definition of supplementary angles
5.	5. Substitution
6. $\begin{aligned} m ∠ 2 = 90^{\circ} \end{aligned}$	6.

Given: $\begin{aligned} l ⊥ m \end{aligned}$
::来源:百分比

Prove : $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are complements
::证明:%1和%2是补充

*Statement*	*Reason*
1.	1.
2.	2. $\begin{aligned} ⊥ \end{aligned}$ lines create right angles
3. $\begin{aligned} m ∠ 1 + m ∠ 2 = 90^{\circ} \end{aligned}$	3.
4. $\begin{aligned} ∠ 1 \end{aligned}$ and $\begin{aligned} ∠ 2 \end{aligned}$ are complementary	4.

Given: $\begin{aligned} l ⊥ m \end{aligned}$ and $\begin{aligned} ∠ 2 ≅ ∠ 6 \end{aligned}$
::来源: lm 和 26

Prove : $\begin{aligned} ∠ 6 ≅ ∠ 5 \end{aligned}$
::证明:% 6 @% 5

*Statement*	*Reason*
1.	1.
2. $\begin{aligned} m ∠ 2 = m ∠ 6 \end{aligned}$	2.
3. $\begin{aligned} ∠ 5 ≅ ∠ 2 \end{aligned}$	3.
4. $\begin{aligned} m ∠ 5 = m ∠ 2 \end{aligned}$	4.
5. $\begin{aligned} m ∠ 5 = m ∠ 6 \end{aligned}$	5.

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

两色证明

Two-Column Proofs ::两色证明

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)