Two-Column Proofs Bill Zahner Lori Jordan
Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required)
To access a customizable version of this book, as well as other interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2013 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: August 30, 2013
AUTHORS Bill Zahner Lori Jordan
www.ck12.org
C ONCEPT
Concept 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 for more content.
CK-12 Foundation: Chapter2TwoColumnProofsA
MEDIA Click image to the left for more content.
James Sousa:Introduction toProof UsingProperties 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. • • • • • •
1
www.ck12.org 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.
Draw the two-column proof and start with the given information. From there, we can use deductive reasoning to reach the next statement and what we want to prove. Reasons will be definitions, postulates, properties and previously proven theorems.
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
When you reach what it is that you wanted to prove, you are done.
Example B
Write a two-column proof.
2
www.ck12.org
Concept 1. Two-Column Proofs
−→ 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.
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. 3
www.ck12.org 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 1.4: 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
Watch this video for help with the Examples above.
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
www.ck12.org
Concept 1. Two-Column Proofs
MEDIA Click image to the left for more content.
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
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 1.6: Statement 1. AC⊥BD 2. 6 BCA and 6 DCA are right angles 3. 6 BCA ∼ = 6 DCA 4. 6 1 ∼ =6 4 6 5. 2 ∼ =6 3
Reason 1. Given 2. Definition of a Perpendicular Lines 3. Right Angle Theorem 4. Given 5. Subtraction PoE 5
www.ck12.org 2.
TABLE 1.7: 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 6 5. O + 6 M = 6 P + 6 O 6. 6 M = 6 P Practice
Write a two-column proof for questions 1-5. 1. Given: 6 MLN ∼ = 6 OLPProve: 6 MLO ∼ = 6 NLP
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 6
Reason 1. Given 2. 3. 4. 5. 6.
Definition of Supplementary Angles Substitution PoE Given Substitution PoE Subtraction PoE
www.ck12.org
Concept 1. Two-Column Proofs
5. Given: l⊥mProve: 6 1 and 6 2 are complements
Use the picture for questions 6-15.
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 1.8: Statement 1. 6 ACH ∼ = 6 ECH CH is on the interior of 6 ACE
Reason 1.
7
www.ck12.org
TABLE 1.8: (continued) Statement 2. m6 ACH = m6 ECH 3. 4. 5. m6 ACE = 2m6 ACH 6. 7.
8
Reason 2. 3. Angle Addition Postulate 4. Substitution 5. 6. Division PoE 7.
