Two-Column Proofs
Bill Zahner
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
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Printed: November 3, 2012
AUTHORS
Bill Zahner
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Two-Column Proofs
1
Two-Column Proofs
6
Here you’ll learn how to create two-column proofs with statements and reasons for each step you take in proving a
geometric statement.
−→
∼
Suppose you are told that 6 XY Z is a right angle and that YW bisects 6 XY Z. You are then asked to prove 6 XYW =
WY Z. 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 ColumnProofs
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 6 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.
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Add the given, AB = CD.
Draw the 2-column proof and start with the given information.
TABLE 1.1:
Statement
1. A, B,C, and D are collinear, in that order.
2. AB = CD
3. BC = BC
4. AB + BC = BC +CD
5. AB + BC = AC
BC +CD = BD
6. AC = BD
Reason
1. Given
2. Given
3. Reflexive PoE
4. Addition PoE
5. Segment Addition Postulate
6. Substitution or Transitive PoE
Example B
Write a two-column proof.
−→
Given: BF bisects 6 ABC; 6 ABD ∼
= 6 CBE
Prove: 6 DBF ∼
= 6 EBF
First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, m6 ABF =
m6 FBC.
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Concept 1. Two-Column Proofs
TABLE 1.2:
Statement
−→
1. BF bisects 6 ABC, 6 ABD ∼
= 6 CBE
2. m6 ABF = m6 FBC
3. m6 ABD = m6 CBE
4. m6 ABF = m6 ABD + m6 DBF
m6 FBC = m6 EBF + m6 CBE
5. m6 ABD + m6 DBF = m6 EBF + m6 CBE
6. m6 ABD + m6 DBF = m6 EBF + m6 ABD
7. m6 DBF = m6 EBF
8. 6 DBF ∼
= 6 EBF
Reason
1. Given
2. Definition of an Angle Bisector
3. If angles are ∼
=, then their measures are equal.
4. Angle Addition Postulate
5.
6.
7.
8.
Substitution PoE
Substitution PoE
Subtraction PoE
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: 6 A and 6 B are right angles
Prove: 6 A ∼
=6 B
TABLE 1.3:
Statement
1. 6 A and 6 B are right angles
2. m6 A = 90◦ and m6 B = 90◦
3. m6 A = m6 B
4. 6 A ∼
=6 B
Reason
1. Given
2. Definition of right angles
3. Transitive PoE
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.
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Given: 6 A and 6 B are supplementary angles. 6 B and 6 C are supplementary angles.
Prove: 6 A ∼
=6 C
TABLE 1.4:
Statement
1. 6 A and 6 B are supplementary
6 B and 6 C are supplementary
2. m6 A + m6 B = 180◦
m6 B + m6 C = 180◦
3. m6 A + m6 B = m6 B + m6 C
4. m6 A = m6 C
5. 6 A ∼
=6 C
Reason
1. Given
2. Definition of supplementary angles
3.
4.
5.
Substitution PoE
Subtraction PoE
∼
= 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: 6 1 ∼
=6 3
TABLE 1.5:
Statement
1. Lines k and m intersect
2. 6 1 and 6 2 are a linear pair
6 2 and 6 3 are a linear pair
3. 6 1 and 6 2 are supplementary
6 2 and 6 3 are supplementary
4. m6 1 + m6 2 = 180◦
m6 2 + m6 3 = 180◦
5. m6 1 + m6 2 = m6 2 + m6 3
6. m6 1 = m6 3
7. 6 1 ∼
=6 3
Guided Practice
1. 6 1 ∼
= 6 4 and 6 C and 6 F are right angles.
Which angles are congruent and why?
4
Reason
1. Given
2. Definition of a Linear Pair
3. Linear Pair Postulate
4. Definition of Supplementary Angles
5.
6.
7.
Substitution PoE
Subtraction PoE
∼
= angles have = measures
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Concept 1. Two-Column Proofs
2. In the figure 6 2 ∼
= 6 3 and k⊥p.
Each pair below is congruent. State why.
a) 6 1 and 6 5
b) 6 1 and 6 4
c) 6 2 and 6 6
d) 6 6 and 6 7
3. Write a two-column proof.
Given: 6 1 ∼
= 6 2 and 6 3 ∼
=6 4
Prove: 6 1 ∼
=6 4
Answers:
1. By the Right Angle Theorem, 6 C ∼
= 6 F. Also, 6 2 ∼
= 6 3 by the Same Angles Supplements Theorem because
6 1∼
6
4
and
they
are
linear
pairs
with
these
congruent
angles.
=
2. a) Vertical Angles Theorem
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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.
TABLE 1.6:
Statement
1. 6 1 ∼
= 6 2 and 6 3 ∼
=6 4
∼
2. 6 2 = 6 3
3. 6 1 ∼
=6 4
Reason
1. Given
2. Vertical Angles Theorem
3. Transitive PoC
Practice
Fill in the blanks in the proofs below.
1. Given: 6 ABC ∼
= 6 DEF and 6 GHI ∼
= 6 JKL
Prove: m6 ABC + m6 GHI = m6 DEF + m6 JKL
TABLE 1.7:
Statement
1.
2. m6 ABC = m6 DEF
m6 GHI = m6 JKL
3.
4. m6 ABC + m6 GHI = m6 DEF + m6 JKL
Reason
1. Given
2.
3. Addition PoE
4.
2. Given: M is the midpoint of AN. N is the midpoint MB
Prove: AM = NB
TABLE 1.8:
Statement
1.
2.
3. AM = NB
3. Given: AC⊥BD and 6 1 ∼
=6 4
Prove: 6 2 ∼
=6 3
6
Reason
Given
Definition of a midpoint
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Concept 1. Two-Column Proofs
TABLE 1.9:
Statement
1. AC⊥BD, 6 1 ∼
=6 4
6
6
2. m 1 = m 4
3.
4. m6 ACB = 90◦
m6 ACD = 90◦
5. m6 1 + m6 2 = m6 ACB
m6 3 + m6 4 = m6 ACD
6.
7. m6 1 + m6 2 = m6 3 + m6 4
8.
9.
10. 6 2 ∼
=6 3
Reason
1.
2.
3. ⊥ lines create right angles
4.
5.
6. Substitution
7.
8. Substitution
9.Subtraction PoE
10.
4. Given: 6 MLN ∼
= 6 OLP
Prove: 6 MLO ∼
= 6 NLP
TABLE 1.10:
Statement
1.
2.
3.
4.
5. m6 MLO = m6 NLP
6.
Reason
1.
2. ∼
= angles have = measures
3. Angle Addition Postulate
4. Substitution
5.
6. ∼
= angles have = measures
5. Given: AE⊥EC and BE⊥ED
Prove: 6 1 ∼
=6 3
7
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TABLE 1.11:
Statement
1.
2.
3. m6 BED = 90◦
m6 AEC = 90◦
4.
5.
6. m6 2 + m6 3 = m6 1 + m6 3
7.
8.
Reason
1.
2. ⊥ lines create right angles
3.
4.
5.
6.
7.
8.
Angle Addition Postulate
Substitution
Subtraction PoE
∼
= angles have = measures
6. Given: 6 L is supplementary to 6 M and 6 P is supplementary to 6 O and 6 L ∼
=6 O
Prove: 6 P ∼
=6 M
TABLE 1.12:
Statement
1.
2. m6 L = m6 O
3.
4.
5.
6.
7. 6 M ∼
=6 P
7. Given: 6 1 ∼
=6 4
Prove: 6 2 ∼
=6 3
8
Reason
1.
2.
3. Definition of supplementary angles
4. Substitution
5. Substitution
6. Subtraction PoE
7.
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Concept 1. Two-Column Proofs
TABLE 1.13:
Statement
1.
2. m6 1 = m6 4
3.
4. 6 1 and 6 2 are supplementary
6 3 and 6 4 are supplementary
5.
6. m6 1 + m6 2 = m6 3 + m6 4
7. m6 1 + m6 2 = m6 3 + m6 1
8. m6 2 = m6 3
9. 6 2 ∼
=6 3
Reason
1.
2.
3. Definition of a Linear Pair
4.
5. Definition of supplementary angles
6.
7.
8.
9.
8. Given: 6 C and 6 F are right angles
Prove: m6 C + m6 F = 180◦
TABLE 1.14:
Statement
1.
2. m6 C = 90◦ , m6 F = 90◦
3. 90◦ + 90◦ = 180◦
4. m6 C + m6 F = 180◦
Reason
1.
2.
3.
4.
9. Given: l⊥m
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Prove: 6 1 ∼
=6 2
TABLE 1.15:
Statement
1. l⊥m
2. 6 1 and 6 2 are right angles
3.
Reason
1.
2.
3.
10. Given: m6 1 = 90◦
Prove: m6 2 = 90◦
TABLE 1.16:
Statement
1.
2. 6 1 and 6 2 are a linear pair
3.
4.
5.
6. m6 2 = 90◦
11. Given: l⊥m
Prove: 6 1 and 6 2 are complements
10
Reason
1.
2.
3. Linear Pair Postulate
4. Definition of supplementary angles
5. Substitution
6.
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Concept 1. Two-Column Proofs
TABLE 1.17:
Statement
1.
2.
3. m6 1 + m6 2 = 90◦
4. 6 1 and 6 2 are complementary
Reason
1.
2. ⊥ lines create right angles
3.
4.
12. Given: l⊥m and 6 2 ∼
=6 6
Prove: 6 6 ∼
=6 5
TABLE 1.18:
Statement
1.
2. m6 2 = m6 6
3. 6 5 ∼
=6 2
6
4. m 5 = m6 2
5. m6 5 = m6 6
Reason
1.
2.
3.
4.
5.
11