<meta http-equiv="refresh" content="1; url=/nojavascript/"> Two-Column Proofs ( Read ) | Geometry | CK-12 Foundation
Dismiss
Skip Navigation
You are viewing an older version of this Concept. Go to the latest version.

Two-Column Proofs

%
Best Score
Practice Two-Column Proofs
Practice
Best Score
%
Practice Now
Two-Column Proofs
 0  0  0 Share To Groups

Suppose you are told that \angle XYZ is a right angle and that \overrightarrow{YW} bisects \angle XYZ . You are then asked to prove \angle XYW \cong \angle WYZ . After completing this Concept, you'll be able to create a two-column proof to prove this congruency.

Watch This

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, \cong can be used in place of the word congruent . You could also use \angle for the word angle.

Example A

Write a two-column proof for the following:

If A, B, C , and D are points on a line, in the given order, and AB = CD , then 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.

Always start with drawing a picture of what you are given.

Plot the points in the order A, B, C, D on a line.

Add the given, AB = CD .

Draw the 2-column proof and start with the given information.

Statement Reason
1. A, B, C , and D are collinear, in that order. 1. Given
2. AB = CD 2. Given
3. BC = BC 3. Reflexive PoE
4. AB + BC = BC + CD 4. Addition PoE

5. AB + BC = AC

BC + CD = BD

5. Segment Addition Postulate
6. AC = BD 6. Substitution or Transitive PoE

Example B

Write a two-column proof.

Given : \overrightarrow{BF} bisects \angle ABC ; \angle ABD \cong \angle CBE

Prove : \angle DBF \cong \angle EBF

First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, m \angle ABF = m \angle FBC .

Statement Reason
1. \overrightarrow{BF} bisects \angle ABC, \angle ABD \cong \angle CBE 1. Given
2. m \angle ABF = m\angle FBC 2. Definition of an Angle Bisector
3. m\angle ABD = m\angle CBE 3. If angles are \cong , then their measures are equal.

4. m\angle ABF = m\angle ABD + m\angle DBF

m\angle FBC = m\angle EBF + m\angle CBE

4. Angle Addition Postulate
5. m\angle ABD + m\angle DBF = m\angle EBF + m\angle CBE 5. Substitution PoE
6. m\angle ABD + m\angle DBF = m\angle EBF + m\angle ABD 6. Substitution PoE
7. m\angle DBF = m\angle EBF 7. Subtraction PoE
8. \angle DBF \cong \angle EBF 8. If measures are equal, the angles are \cong .

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 : \angle A and \angle B are right angles

Prove : \angle A \cong \angle B

Statement Reason
1. \angle A and \angle B are right angles 1. Given
2. m\angle A = 90^\circ and m\angle B = 90^\circ 2. Definition of right angles
3. m\angle A = m\angle B 3. Transitive PoE
4. \angle A \cong \angle B 4. \cong 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 : \angle A and \angle B are supplementary angles. \angle B and \angle C are supplementary angles.

Prove : \angle A \cong \angle C

Statement Reason

1. \angle A and \angle B are supplementary

\angle B and \angle C are supplementary

1. Given

2. m\angle A + m\angle B =180^\circ

m\angle B + m\angle C = 180^\circ

2. Definition of supplementary angles
3. m\angle A + m\angle B = m\angle B + m\angle C 3. Substitution PoE
4. m\angle A = m\angle C 4. Subtraction PoE
5. \angle A \cong \angle C 5. \cong angles have = measures

Example E

The Vertical Angles Theorem states that vertical angles are congruent. Prove this theorem.

Given : Lines k and m intersect.

Prove: \angle 1 \cong \angle 3

Statement Reason
1. Lines k and m intersect 1. Given

2. \angle 1 and \angle 2 are a linear pair

\angle 2 and \angle 3 are a linear pair

2. Definition of a Linear Pair

3. \angle 1 and \angle 2 are supplementary

\angle 2 and \angle 3 are supplementary

3. Linear Pair Postulate

4. m\angle 1 + m\angle 2 = 180^\circ

m\angle 2 + m\angle 3 = 180^\circ

4. Definition of Supplementary Angles
5. m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 5. Substitution PoE
6. m\angle 1 = m\angle 3 6. Subtraction PoE
7. \angle 1 \cong  \angle 3 7. \cong angles have = measures

Guided Practice

1. \angle 1 \cong  \angle 4 and \angle C and \angle F are right angles.

Which angles are congruent and why?

2. In the figure \angle 2 \cong \angle 3 and k \bot p .

Each pair below is congruent. State why.

a) \angle 1 and \angle 5

b) \angle 1 and \angle 4

c) \angle 2 and \angle 6

d) \angle 6 and \angle 7

3. Write a two-column proof.

Given : \angle 1 \cong  \angle 2 and \angle 3 \cong  \angle 4

Prove : \angle 1 \cong \angle 4

Answers:

1. By the Right Angle Theorem, \angle C \cong \angle F . Also, \angle 2 \cong \angle 3 by the Same Angles Supplements Theorem because \angle 1 \cong \angle 4 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. \angle 1 \cong \angle 2 and \angle 3 \cong \angle 4 1. Given
2. \angle 2 \cong \angle 3 2. Vertical Angles Theorem
3. \angle 1 \cong \angle 4 3. Transitive PoC

Practice

Fill in the blanks in the proofs below.

1. Given : \angle ABC \cong \angle DEF and \angle GHI \cong \angle JKL

Prove: m\angle ABC + m \angle GHI = m \angle DEF + m\angle JKL

Statement Reason
1. 1. Given

2. m\angle ABC = m\angle DEF

m\angle GHI = m\angle JKL

2.
3. 3. Addition PoE
4. m\angle ABC + m\angle GHI = m\angle DEF + m\angle JKL 4.

2. Given : M is the midpoint of \overline{AN} . N is the midpoint \overline{MB}

Prove : AM = NB

Statement Reason
1. Given
2. Definition of a midpoint
3. AM = NB

3. Given : \overline{AC} \bot \overline{BD} and \angle 1 \cong \angle 4

Prove : \angle 2 \cong \angle 3

Statement Reason
1. \overline{AC} \bot \overline{BD}, \angle 1 \cong  \angle 4 1.
2. m\angle 1 = m\angle 4 2.
3. 3. \bot lines create right angles

4. m\angle ACB = 90^\circ

m\angle ACD = 90^\circ

4.

5. m\angle 1 + m\angle 2 = m\angle ACB

m\angle 3 + m\angle 4 = m\angle ACD

5.
6. 6. Substitution
7. m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4  7.
8. 8. Substitution
9. 9.Subtraction PoE
10. \angle 2 \cong  \angle 3 10.

4. Given : \angle MLN \cong \angle OLP

Prove : \angle MLO \cong  \angle NLP

Statement Reason
1. 1.
2. 2. \cong angles have = measures
3. 3. Angle Addition Postulate
4. 4. Substitution
5. m\angle MLO = m\angle NLP 5.
6. 6. \cong angles have = measures

5. Given : \overline{AE} \bot \overline{EC} and \overline{BE} \bot \overline{ED}

Prove : \angle 1 \cong \angle 3

Statement Reason
1. 1.
2. 2. \bot lines create right angles

3. m\angle BED = 90^\circ

m\angle AEC = 90^\circ

3.
4. 4. Angle Addition Postulate
5. 5. Substitution
6. m\angle 2 + m\angle 3 = m\angle 1 + m\angle 3 6.
7. 7. Subtraction PoE
8. 8. \cong angles have = measures

6. Given : \angle L is supplementary to \angle M and \angle P is supplementary to \angle O and \angle L \cong \angle O

Prove : \angle P \cong \angle M

Statement Reason
1. 1.
2. m\angle L = m\angle O 2.
3. 3. Definition of supplementary angles
4. 4. Substitution
5. 5. Substitution
6. 6. Subtraction PoE
7. \angle M \cong  \angle P 7.

7. Given : \angle 1 \cong \angle 4

Prove : \angle 2 \cong \angle 3

Statement Reason
1. 1.
2. m\angle 1 = m\angle 4 2.
3. 3. Definition of a Linear Pair

4. \angle 1 and \angle 2 are supplementary

\angle 3 and \angle 4 are supplementary

4.
5. 5. Definition of supplementary angles
6. m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 6.
7. m\angle 1 + m\angle 2 = m\angle 3 + m\angle 1 7.
8. m\angle 2 = m\angle 3 8.
9. \angle 2 \cong  \angle 3 9.

8. Given : \angle C and \angle F are right angles

Prove : m\angle C + m\angle F = 180^\circ

Statement Reason
1. 1.
2. m\angle C = 90^\circ, m\angle F = 90^\circ 2.
3. 90^\circ + 90^\circ = 180^\circ 3.
4. m\angle C + m\angle F = 180^\circ 4.

9. Given : l \bot m

Prove : \angle 1\cong  \angle 2

Statement Reason
1. l \bot m 1.
2. \angle 1 and \angle 2 are right angles 2.
3. 3.

10. Given : m\angle 1 = 90^\circ

Prove : m\angle 2 = 90^\circ

Statement Reason
1. 1.
2. \angle 1 and \angle 2 are a linear pair 2.
3. 3. Linear Pair Postulate
4. 4. Definition of supplementary angles
5. 5. Substitution
6. m\angle 2 = 90^\circ 6.

11. Given : l \bot m

Prove : \angle 1 and \angle 2 are complements

Statement Reason
1. 1.
2. 2. \bot lines create right angles
3. m\angle 1 + m\angle 2 = 90^\circ 3.
4. \angle 1 and \angle 2 are complementary 4.

12. Given : l \bot m and \angle 2 \cong  \angle 6

Prove : \angle 6 \cong  \angle 5

Statement Reason
1. 1.
2. m\angle 2 = m\angle 6 2.
3. \angle 5 \cong  \angle 2 3.
4. m\angle 5 = m\angle 2 4.
5. m\angle 5 = m\angle 6 5.

Image Attributions

Reviews

Email Verified
Well done! You've successfully verified the email address .
OK
Please wait...
Please wait...
ShareThis Copy and Paste

Original text