Suppose you are told that is a right angle and that bisects . You are then asked to prove . After completing this Concept, you'll be able to create a twocolumn proof to prove this congruency.
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Guidance
A two column proof is one common way to organize a proof in geometry. Two column proofs always have two columns statements and reasons. The best way to understand two column proofs is to read through examples.
When writing your own two column proof, these keep things in mind:
 Number each step.
 Start with the given information.
 Statements with the same reason can be combined into one step . It is up to you.
 Draw a picture and mark it with the given information.
 You must have a reason for EVERY statement.
 The order of the statements in the proof is not always fixed, but make sure the order makes logical sense.
 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 told in the problem.
 Use symbols and abbreviations for words within proofs. For example, can be used in place of the word congruent . You could also use for the word angle.
Example A
Write a twocolumn proof for the following:
If , and are points on a line, in the given order, and , then .
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.
Always start with drawing a picture of what you are given.
Plot the points in the order on a line.
Add the given, .
Draw the 2column proof and start with the given information.
Statement  Reason 

1. , and are collinear, in that order.  1. Given 
2.  2. Given 
3.  3. Reflexive 
4.  4. Addition 
5.

5. Segment Addition Postulate 
6.  6. Substitution or Transitive 
Example B
Write a twocolumn proof.
Given : bisects ;
Prove :
First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, .
Statement  Reason 

1. bisects  1. Given 
2.  2. Definition of an Angle Bisector 
3.  3. If angles are , then their measures are equal. 
4.

4. Angle Addition Postulate 
5.  5. Substitution 
6.  6. Substitution 
7.  7. Subtraction 
8.  8. If measures are equal, the angles are . 
Example C
The Right Angle Theorem states that if two angles are right angles, then the angles are congruent. Prove this theorem.
To prove this theorem, set up your own drawing and name some angles so that you have specific angles to talk about.
Given : and are right angles
Prove :
Statement  Reason 

1. and are right angles  1. Given 
2. and  2. Definition of right angles 
3.  3. Transitive 
4.  4. 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.
Example D
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.
Given : and are supplementary angles. and are supplementary angles.
Prove :
Statement  Reason 

1. and are supplementary and are supplementary 
1. Given 
2.

2. Definition of supplementary angles 
3.  3. Substitution 
4.  4. Subtraction 
5.  5. angles have = measures 
Example E
The Vertical Angles Theorem states that vertical angles are congruent. Prove this theorem.
Given : Lines and intersect.
Prove:
Statement  Reason 

1. Lines and intersect  1. Given 
2. and are a linear pair and are a linear pair 
2. Definition of a Linear Pair 
3. and are supplementary and are supplementary 
3. Linear Pair Postulate 
4.

4. Definition of Supplementary Angles 
5.  5. Substitution 
6.  6. Subtraction 
7.  7. angles have = measures 
Guided Practice
1. and and are right angles.
Which angles are congruent and why?
2. In the figure and .
Each pair below is congruent. State why.
a) and
b) and
c) and
d) and
3. Write a twocolumn proof.
Given : and
Prove :
Answers:
1. By the Right Angle Theorem, . Also, by the Same Angles Supplements Theorem because and they are linear pairs with these congruent angles.
2. a) Vertical Angles Theorem
b) Same Angles Complements Theorem
c) Vertical Angles Theorem
d) Vertical Angles Theorem followed by the Transitive Property
3. Follow the format from the examples.
Statement  Reason 

1. and  1. Given 
2.  2. Vertical Angles Theorem 
3.  3. Transitive 
Explore More
Fill in the blanks in the proofs below.
1. Given : and
Prove:
Statement  Reason 

1.  1. Given 
2.

2. 
3.  3. Addition 
4.  4. 
2. Given : is the midpoint of . is the midpoint
Prove :
Statement  Reason 

1.  Given 
2.  Definition of a midpoint 
3. 
3. Given : and
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3.  3. lines create right angles 
4.

4. 
5.

5. 
6.  6. Substitution 
7.  7. 
8.  8. Substitution 
9.  9.Subtraction 
10.  10. 
4. Given :
Prove :
Statement  Reason 

1.  1. 
2.  2. angles have = measures 
3.  3. Angle Addition Postulate 
4.  4. Substitution 
5.  5. 
6.  6. angles have = measures 
5. Given : and
Prove :
Statement  Reason 

1.  1. 
2.  2. lines create right angles 
3.

3. 
4.  4. Angle Addition Postulate 
5.  5. Substitution 
6.  6. 
7.  7. Subtraction 
8.  8. angles have = measures 
6. Given : is supplementary to and is supplementary to and
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3.  3. Definition of supplementary angles 
4.  4. Substitution 
5.  5. Substitution 
6.  6. Subtraction 
7.  7. 
7. Given :
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3.  3. Definition of a Linear Pair 
4. and are supplementary and are supplementary 
4. 
5.  5. Definition of supplementary angles 
6.  6. 
7.  7. 
8.  8. 
9.  9. 
8. Given : and are right angles
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3.  3. 
4.  4. 
9. Given :
Prove :
Statement  Reason 

1.  1. 
2. and are right angles  2. 
3.  3. 
10. Given :
Prove :
Statement  Reason 

1.  1. 
2. and are a linear pair  2. 
3.  3. Linear Pair Postulate 
4.  4. Definition of supplementary angles 
5.  5. Substitution 
6.  6. 
11. Given :
Prove : and are complements
Statement  Reason 

1.  1. 
2.  2. lines create right angles 
3.  3. 
4. and are complementary  4. 
12. Given : and
Prove :
Statement  Reason 

1.  1. 
2.  2. 
3.  3. 
4.  4. 
5.  5. 