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Two-Column Proofs

Geometric proofs are among the building blocks of math. Learn how to use two column proofs to assert and prove the validity of a statement.

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Two-Column Proofs

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 two-column proof to prove this congruency.

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CK-12 Two Column Proofs

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 when writing your own two column proof, keep 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 two-column 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 2-column 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 two-column 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 two-column 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

Practice

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.

Vocabulary

two column proof

two column proof

A common way to organize a proof in geometry. Two column proofs always have two columns- statements and reasons.
linear pair

linear pair

Two angles form a linear pair if they are supplementary and adjacent.

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