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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

Two-Column Proofs

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.

When writing your own two-column proof, keep these 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 given in the problem.
  • Use symbols and abbreviations for words within proofs. For example, \begin{align*}\cong\end{align*} can be used in place of the word congruent. You could also use \begin{align*}\angle\end{align*} for the word angle.

Suppose you are told that \begin{align*}\angle XYZ\end{align*} is a right angle and that \begin{align*}\overrightarrow{YW}\end{align*} bisects \begin{align*}\angle XYZ\end{align*}. You are then asked to prove \begin{align*}\angle XYW \cong \angle WYZ\end{align*}.

 

Examples

Example 1

Write a two-column proof for the following:

If \begin{align*}A, B, C\end{align*}, and \begin{align*}D\end{align*} are points on a line, in the given order, and \begin{align*}AB = CD\end{align*}, then \begin{align*}AC = BD\end{align*}.

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 \begin{align*}A, B, C, D\end{align*} on a line.

Add the given, \begin{align*}AB = CD\end{align*}.

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

Statement Reason
1. \begin{align*}A, B, C\end{align*}, and \begin{align*}D\end{align*} are collinear, in that order. 1. Given
2. \begin{align*}AB = CD\end{align*} 2. Given
3. \begin{align*}BC = BC\end{align*} 3. Reflexive \begin{align*}PoE\end{align*}
4. \begin{align*}AB + BC = BC + CD\end{align*} 4. Addition \begin{align*}PoE\end{align*}

5. \begin{align*}AB + BC = AC\end{align*}

\begin{align*}BC + CD = BD\end{align*}

5. Segment Addition Postulate
6. \begin{align*}AC = BD\end{align*} 6. Substitution or Transitive \begin{align*}PoE\end{align*}

Example 2

Write a two-column proof.

Given: \begin{align*}\overrightarrow{BF}\end{align*} bisects \begin{align*}\angle ABC\end{align*}; \begin{align*}\angle ABD \cong \angle CBE\end{align*}

Prove: \begin{align*}\angle DBF \cong \angle EBF\end{align*}

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

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

4. \begin{align*}m\angle ABF = m\angle ABD + m\angle DBF\end{align*}

\begin{align*}m\angle FBC = m\angle EBF + m\angle CBE\end{align*}

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

Example 3

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: \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are right angles

Prove: \begin{align*}\angle A \cong \angle B\end{align*}

Statement Reason
1. \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are right angles 1. Given
2. \begin{align*}m\angle A = 90^\circ\end{align*} and \begin{align*}m\angle B = 90^\circ\end{align*} 2. Definition of right angles
3. \begin{align*}m\angle A = m\angle B\end{align*} 3. Transitive \begin{align*}PoE\end{align*}
4. \begin{align*}\angle A \cong \angle B\end{align*} 4. \begin{align*}\cong\end{align*} 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 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.

Given: \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary angles. \begin{align*}\angle B\end{align*} and \begin{align*}\angle C\end{align*} are supplementary angles.

Prove: \begin{align*}\angle A \cong \angle C\end{align*}

Statement Reason

1. \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary

\begin{align*}\angle B\end{align*} and \begin{align*}\angle C\end{align*} are supplementary

1. Given

2. \begin{align*}m\angle A + m\angle B =180^\circ\end{align*}

\begin{align*}m\angle B + m\angle C = 180^\circ\end{align*}

2. Definition of supplementary angles
3. \begin{align*}m\angle A + m\angle B = m\angle B + m\angle C\end{align*} 3. Substitution \begin{align*}PoE\end{align*}
4. \begin{align*}m\angle A = m\angle C\end{align*} 4. Subtraction \begin{align*}PoE\end{align*}
5. \begin{align*}\angle A \cong \angle C\end{align*} 5. \begin{align*}\cong\end{align*} angles have = measures

Example 5

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

Given: Lines \begin{align*}k\end{align*} and \begin{align*}m\end{align*} intersect.

Prove: \begin{align*}\angle 1 \cong \angle 3\end{align*}

Statement Reason
1. Lines \begin{align*}k\end{align*} and \begin{align*}m\end{align*} intersect 1. Given

2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair

\begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are a linear pair

2. Definition of a Linear Pair

3. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are supplementary

\begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are supplementary

3. Linear Pair Postulate

4. \begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}

\begin{align*}m\angle 2 + m\angle 3 = 180^\circ\end{align*}

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

Example 6

\begin{align*}\angle 1 \cong \angle 4\end{align*} and \begin{align*}\angle C\end{align*} and \begin{align*}\angle F\end{align*} are right angles.

Which angles are congruent and why?

By the Right Angle Theorem, \begin{align*}\angle C \cong \angle F\end{align*}. Also, \begin{align*}\angle 2 \cong \angle 3\end{align*} by the Same Angles Supplements Theorem because \begin{align*}\angle 1 \cong \angle 4\end{align*} and they are linear pairs with these congruent angles.

Review

Fill in the blanks in the proofs below.

  1. Given: \begin{align*}\angle ABC \cong \angle DEF\end{align*} and \begin{align*}\angle GHI \cong \angle JKL\end{align*}

Prove: \begin{align*}m\angle ABC + m \angle GHI = m \angle DEF + m\angle JKL\end{align*}

Statement Reason
1. 1. Given

2. \begin{align*}m\angle ABC = m\angle DEF\end{align*}

\begin{align*}m\angle GHI = m\angle JKL\end{align*}

2.
3. 3. Addition \begin{align*}PoE\end{align*}
4. \begin{align*}m\angle ABC + m\angle GHI = m\angle DEF + m\angle JKL\end{align*} 4.
  1. Given: \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{AN}\end{align*}. \begin{align*}N\end{align*} is the midpoint \begin{align*}\overline{MB}\end{align*}

Prove: \begin{align*}AM = NB\end{align*}

Statement Reason
1. Given
2. Definition of a midpoint
3. \begin{align*}AM = NB\end{align*}
  1. Given: \begin{align*}\overline{AC} \bot \overline{BD}\end{align*} and \begin{align*}\angle 1 \cong \angle 4\end{align*}

Prove: \begin{align*}\angle 2 \cong \angle 3\end{align*}

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

4. \begin{align*}m\angle ACB = 90^\circ\end{align*}

\begin{align*}m\angle ACD = 90^\circ \end{align*}

4.

5. \begin{align*}m\angle 1 + m\angle 2 = m\angle ACB\end{align*}

\begin{align*}m\angle 3 + m\angle 4 = m\angle ACD \end{align*}

5.
6. 6. Substitution
7. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 \end{align*} 7.
8. 8. Substitution
9. 9.Subtraction \begin{align*}PoE\end{align*}
10. \begin{align*}\angle 2 \cong \angle 3 \end{align*} 10.
  1. Given: \begin{align*}\angle MLN \cong \angle OLP\end{align*}

Prove: \begin{align*}\angle MLO \cong \angle NLP\end{align*}

Statement Reason
1. 1.
2. 2. \begin{align*}\cong\end{align*} angles have = measures
3. 3. Angle Addition Postulate
4. 4. Substitution
5. \begin{align*}m\angle MLO = m\angle NLP \end{align*} 5.
6. 6. \begin{align*}\cong\end{align*} angles have = measures
  1. Given: \begin{align*}\overline{AE} \bot \overline{EC}\end{align*} and \begin{align*}\overline{BE} \bot \overline{ED}\end{align*}

Prove: \begin{align*}\angle 1 \cong \angle 3\end{align*}

Statement Reason
1. 1.
2. 2. \begin{align*}\bot\end{align*} lines create right angles

3. \begin{align*}m\angle BED = 90^\circ\end{align*}

\begin{align*}m\angle AEC = 90^\circ \end{align*}

3.
4. 4. Angle Addition Postulate
5. 5. Substitution
6. \begin{align*}m\angle 2 + m\angle 3 = m\angle 1 + m\angle 3\end{align*} 6.
7. 7. Subtraction \begin{align*}PoE\end{align*}
8. 8. \begin{align*}\cong\end{align*} angles have = measures
  1. Given: \begin{align*}\angle L\end{align*} is supplementary to \begin{align*}\angle M\end{align*} and \begin{align*}\angle P\end{align*} is supplementary to \begin{align*}\angle O\end{align*} and \begin{align*}\angle L \cong \angle O\end{align*}

Prove: \begin{align*}\angle P \cong \angle M\end{align*}

Statement Reason
1. 1.
2. \begin{align*}m\angle L = m\angle O \end{align*} 2.
3. 3. Definition of supplementary angles
4. 4. Substitution
5. 5. Substitution
6. 6. Subtraction \begin{align*}PoE\end{align*}
7. \begin{align*}\angle M \cong \angle P \end{align*} 7.
  1. Given: \begin{align*}\angle 1 \cong \angle 4\end{align*}

Prove: \begin{align*}\angle 2 \cong \angle 3\end{align*}

Statement Reason
1. 1.
2. \begin{align*}m\angle 1 = m\angle 4 \end{align*} 2.
3. 3. Definition of a Linear Pair

4. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are supplementary

\begin{align*}\angle 3\end{align*} and \begin{align*}\angle 4\end{align*} are supplementary

4.
5. 5. Definition of supplementary angles
6. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 \end{align*} 6.
7. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 1 \end{align*} 7.
8. \begin{align*}m\angle 2 = m\angle 3 \end{align*} 8.
9. \begin{align*}\angle 2 \cong \angle 3 \end{align*} 9.
  1. Given: \begin{align*}\angle C\end{align*} and \begin{align*}\angle F\end{align*} are right angles

Prove: \begin{align*}m\angle C + m\angle F = 180^\circ\end{align*}

Statement Reason
1. 1.
2. \begin{align*}m\angle C = 90^\circ, m\angle F = 90^\circ \end{align*} 2.
3. \begin{align*}90^\circ + 90^\circ = 180^\circ \end{align*} 3.
4. \begin{align*}m\angle C + m\angle F = 180^\circ \end{align*} 4.
  1. Given: \begin{align*}l \bot m\end{align*}

Prove: \begin{align*}\angle 1\cong \angle 2\end{align*}

Statement Reason
1. \begin{align*}l \bot m\end{align*} 1.
2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are right angles 2.
3. 3.
  1. Given: \begin{align*}m\angle 1 = 90^\circ\end{align*}

Prove: \begin{align*}m\angle 2 = 90^\circ\end{align*}

Statement Reason
1. 1.
2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair 2.
3. 3. Linear Pair Postulate
4. 4. Definition of supplementary angles
5. 5. Substitution
6. \begin{align*}m\angle 2 = 90^\circ \end{align*} 6.
  1. Given: \begin{align*}l \bot m\end{align*}

Prove: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are complements

Statement Reason
1. 1.
2. 2. \begin{align*}\bot\end{align*} lines create right angles
3. \begin{align*}m\angle 1 + m\angle 2 = 90^\circ \end{align*} 3.
4. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are complementary 4.
  1. Given: \begin{align*}l \bot m\end{align*} and \begin{align*}\angle 2 \cong \angle 6\end{align*}

Prove: \begin{align*}\angle 6 \cong \angle 5\end{align*}

Statement Reason
1. 1.
2. \begin{align*}m\angle 2 = m\angle 6 \end{align*} 2.
3. \begin{align*}\angle 5 \cong \angle 2\end{align*} 3.
4. \begin{align*}m\angle 5 = m\angle 2\end{align*} 4.
5. \begin{align*}m\angle 5 = m\angle 6\end{align*} 5.

Review (Answers)

To see the Review answers, open this PDF file and look for section 2.7. 

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Vocabulary

two column proof

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

linear pair

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

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