Download Two-Column Proofs

Two-Column Proofs
Bill Zahner
Lori Jordan
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Printed: January 4, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. Two-Column Proofs
1
Two-Column Proofs
Here you’ll learn how to write a two-column geometric proof.
What if you wanted to prove that two angles are congruent? After completing this Concept, you’ll be able to formally
prove geometric ideas with two-column proofs.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136909
CK-12 Foundation: Chapter2TwoColumnProofsA
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/1376
James Sousa: Introduction to Proof Using Properties of Congruence
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 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, ∼
= can be used in place of the word
6
congruent. You could also use for the word angle.
•
•
•
•
•
•
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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.
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 A, B,C, D on a line.
Add the corresponding markings, AB = CD, to the line.
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, if BF bisects
6 ABC, then m6 ABF = m6 FBC. Also, because the word “bisect” was used in the given, the definition will probably
be used in the proof.
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Chapter 1. Two-Column Proofs
TABLE 1.1:
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 1.2:
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.
Given: 6 A and 6 B are supplementary angles. 6 B and 6 C are supplementary angles.
Prove: 6 A ∼
=6 C
TABLE 1.3:
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
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Example E
The Vertical Angles Theorem states that vertical angles are congruent. Prove this theorem.
Given: Lines k and m intersect.
∼6 3
Prove: 6 1 =
TABLE 1.4:
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
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
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136910
CK-12 Foundation: Chapter2TwoColumnProofsB
Vocabulary
A two-column proof is one common way to organize a proof in geometry. Two-column proofs always have two
columns: statements and reasons.
Guided Practice
1. Write a two-column proof for the following:
Given: AC⊥BD and 6 1 ∼
=6 4
Prove: 6 2 ∼
=6 3
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Chapter 1. Two-Column Proofs
2. Write a two-column proof for the following:
Given: 6 L is supplementary to 6 M, 6 P is supplementary to 6 O, 6 L ∼
=6 O
Prove: 6 P ∼
=6 M
Answers:
1.
TABLE 1.5:
Statement
1. AC⊥BD
2. 6 BCA and 6 DCA are right angles
3. 6 BCA ∼
= 6 DCA
∼
4. 6 1 = 6 4
5. 6 2 ∼
=6 3
Reason
1. Given
2. Definition of a Perpendicular Lines
3. Right Angle Theorem
4. Given
5. Subtraction PoE
2.
TABLE 1.6:
Statement
1. 6 L is supplementary to 6 M, 6 P is supplementary to
6 O
2. 6 L + 6 M = 180, 6 P + 6 O = 180
3. 6 L + 6 M = 6 P + 6 O
4. 6 L ∼
=6 O
5. 6 O + 6 M = 6 P + 6 O
6. 6 M = 6 P
Reason
1. Given
2.
3.
4.
5.
6.
Definition of Supplementary Angles
Substitution PoE
Given
Substitution PoE
Subtraction PoE
Explore More
Write a two-column proof for questions 1-5.
1. Given: 6 MLN ∼
= 6 OLPProve: 6 MLO ∼
= 6 NLP
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2. Given: AE⊥EC and BE⊥EDProve: 6 1 ∼
=6 3
3. Given: 6 1 ∼
= 6 4Prove: 6 2 ∼
=6 3
4. Given: l⊥mProve: 6 1 ∼
=6 2
5. Given: l⊥mProve: 6 1 and 6 2 are complements
Use the picture for questions 6-15.
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Chapter 1. Two-Column Proofs
Given: H is the midpoint of AE, MP and GC
M is the midpoint of GA
P is the midpoint of CE
AE⊥GC
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
List two pairs of vertical angles.
List all the pairs of congruent segments.
List two linear pairs that do not have H as the vertex.
List a right angle.
List two pairs of adjacent angles that are NOT linear pairs.
What is the perpendicular bisector of AE?
List two bisectors of MP.
List a pair of complementary angles.
If GC is an angle bisector of 6 AGE, what two angles are congruent?
Fill in the blanks for the proof below.
Given: Picture above and 6 ACH ∼
= 6 ECH
Prove: CH is the angle bisector of 6 ACE
TABLE 1.7:
Statement
1. 6 ACH ∼
= 6 ECH
CH is on the interior of 6 ACE
2. m6 ACH = m6 ECH
3.
4.
5. m6 ACE = 2m6 ACH
6.
7.
Reason
1.
2.
3. Angle Addition Postulate
4. Substitution
5.
6. Division PoE
7.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 2.8.
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