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2.8: Two-Column Proofs

Difficulty Level: At Grade Created by: CK-12
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Two-Column Proofs

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, 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 told 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.

 

Writing a Two-Column Proof

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*}.

First of all, 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 corresponding markings, \begin{align*}AB = CD\end{align*}, to the line.

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, if \begin{align*}\overrightarrow{BF}\end{align*} bisects \begin{align*}\angle ABC\end{align*}, then \begin{align*}m \angle ABF=m \angle FBC\end{align*}. Also, because the word “bisect” was used in the given, the definition will probably be used in the proof.

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

 

Proving the Right Angle Theorem 

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.

Proving the Same Angle Supplements Theorem

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

Proving the Vertical Angles Theorem 

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

 

Examples

Example 1

Write a two-column proof for the following:

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}\end{align*} 1. Given
2. \begin{align*} \angle BCA\end{align*} and \begin{align*} \angle DCA\end{align*} are right angles 2. Definition of Perpendicular Lines
3. \begin{align*}\angle BCA \cong \angle DCA\end{align*} 3. Right Angle Theorem
4. \begin{align*} \angle 1 \cong \angle 4\end{align*} 4. Given
5. \begin{align*}\angle 2 \cong \angle 3\end{align*} 5. Subtraction \begin{align*}PoE\end{align*}

Example 2

Write a two-column proof for the following:

Given: \begin{align*}\angle L\end{align*} is supplementary to \begin{align*}\angle M\end{align*}, \begin{align*}\angle P\end{align*} is supplementary to \begin{align*}\angle O\end{align*}, \begin{align*}\angle L \cong \angle O\end{align*}

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

Statement Reason
1. \begin{align*}\angle L\end{align*} is supplementary to \begin{align*}\angle M\end{align*}, \begin{align*}\angle P\end{align*} is supplementary to \begin{align*}\angle O\end{align*} 1. Given
2. \begin{align*}\angle L + \angle M =180\end{align*}, \begin{align*}\angle P + \angle O=180\end{align*} 2. Definition of Supplementary Angles
3. \begin{align*}\angle L + \angle M=\angle P + \angle O\end{align*} 3. Substitution \begin{align*}PoE\end{align*}
4. \begin{align*}\angle L \cong \angle O\end{align*} 4. Given
5. \begin{align*}\angle O + \angle M=\angle P + \angle O\end{align*} 5. Substitution \begin{align*}PoE\end{align*}
6. \begin{align*}\angle M=\angle P\end{align*} 6. Subtraction \begin{align*}PoE\end{align*}

Review

Write a two-column proof for questions 1-5.

  1. Given: \begin{align*}\angle MLN \cong \angle\end{align*} \begin{align*}OLP\end{align*} Prove: \begin{align*}\angle MLO \cong \angle NLP\end{align*}
  2. 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*}
  3. Given: \begin{align*}\angle 1 \cong \angle\end{align*} \begin{align*}4\end{align*} Prove: \begin{align*}\angle 2 \cong \angle 3\end{align*}
  4. Given: \begin{align*}l \bot\end{align*} \begin{align*}m\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 2\end{align*}
  5. Given: \begin{align*}l \bot\end{align*} \begin{align*}m\end{align*} Prove: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are complements

Use the picture for questions 6-15.

Given: \begin{align*}H\end{align*} is the midpoint of \begin{align*}\overline{AE}, \overline{MP}\end{align*} and \begin{align*}\overline{GC}\end{align*}

\begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{GA}\end{align*}

\begin{align*}P\end{align*} is the midpoint of \begin{align*}\overline{CE}\end{align*}

\begin{align*}\overline{AE} \bot \overline{GC}\end{align*}

  1. List two pairs of vertical angles.
  2. List all the pairs of congruent segments.
  3. List two linear pairs that do not have \begin{align*}H\end{align*} as the vertex.
  4. Identify a right angle.
  5. List two pairs of adjacent angles that are NOT linear pairs.
  6. What is the perpendicular bisector of \begin{align*}\overline{AE}\end{align*}?
  7. List two bisectors of \begin{align*}\overline{MP}\end{align*}.
  8. Identify a pair of complementary angles.
  9. If \begin{align*}\overline{GC}\end{align*} is an angle bisector of \begin{align*}\angle AGE\end{align*}, what two angles are congruent?
  10. Fill in the blanks for the proof below.

Given: Picture above and \begin{align*}\angle ACH \cong \angle ECH\end{align*}

Prove: \begin{align*}\overline{CH}\end{align*} is the angle bisector of \begin{align*}\angle ACE\end{align*}

Statement Reason

1. \begin{align*}\angle ACH \cong \angle ECH\end{align*}

\begin{align*}\overline{CH}\end{align*} is on the interior of \begin{align*}\angle ACE\end{align*}

1.
2. \begin{align*}m \angle ACH = m \angle ECH\end{align*} 2.
3. 3. Angle Addition Postulate
4. 4. Substitution PoE
5. \begin{align*}m \angle ACE = 2m \angle ACH\end{align*} 5.
6. 6. Division PoE
7. 7.

Review (Answers)

To view the Review answers, open this PDF file and look for section 2.8. 

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Vocabulary

linear pair

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

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