Two-Column Proofs. Bill Zahner Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers

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1 Two-Column Proofs Bill Zahner Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers Say Thanks to the Authors Click (No sign in required)

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2 To access a customizable version of this book, as well as other interactive content, visit 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. AUTHORS Bill Zahner Dan Greenberg Lori Jordan Andrew Gloag Victor Cifarelli Jim Sconyers EDITOR Annamaria Farbizio Copyright 2012 CK-12 Foundation, 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 (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/Non- Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License ( 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 Printed: November 3, 2012

3 Concept 1. Two-Column Proofs CONCEPT 1 Two-Column Proofs 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 XY Z is a right angle and that YW bisects XY Z. You are then asked to prove XYW = WY Z. After completing this Concept, you ll be able to create a two-column proof to prove this congruency. Watch This MEDIA Click image to the left for more content. 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. s 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. s 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 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. 1

4 Add the given, AB = CD. Draw the 2-column proof and start with the given information. TABLE 1.1: 1. A,B,C, and D are collinear, in that order. 1. Given 2. AB = CD 2. Given 3. BC = BC 3. Reflexive PoE 4. AB + BC = BC +CD 4. Addition PoE 5. AB + BC = AC 5. Segment Addition Postulate BC +CD = BD 6. AC = BD 6. Substitution or Transitive PoE Example B Write a two-column proof. Given: BF bisects ABC; ABD = CBE Prove: DBF = EBF First, put the appropriate markings on the picture. Recall, that bisect means to cut in half. Therefore, m ABF = m FBC. 2

5 Concept 1. Two-Column Proofs TABLE 1.2: 1. BF bisects ABC, ABD = CBE 1. Given 2. m ABF = m FBC 2. Definition of an Angle Bisector 3. m ABD = m CBE 3. If angles are =, then their measures are equal. 4. m ABF = m ABD + m DBF 4. Angle Addition Postulate m FBC = m EBF + m CBE 5. m ABD + m DBF = m EBF + m CBE 5. Substitution PoE 6. m ABD + m DBF = m EBF + m ABD 6. Substitution PoE 7. m DBF = m EBF 7. Subtraction PoE 8. DBF = EBF 8. 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: A and B are right angles Prove: A = B TABLE 1.3: 1. A and B are right angles 1. Given 2. m A = 90 and m B = Definition of right angles 3. m A = m B 3. Transitive PoE 4. A = B 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. 3

6 Given: A and B are supplementary angles. B and C are supplementary angles. Prove: A = C TABLE 1.4: 1. A and B are supplementary 1. Given B and C are supplementary 2. m A + m B = Definition of supplementary angles m B + m C = m A + m B = m B + m C 3. Substitution PoE 4. m A = m C 4. Subtraction PoE 5. A = C 5. = 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: 1 = 3 TABLE 1.5: 4. Definition of Supplementary Angles 1. Lines k and m intersect 1. Given 2. 1 and 2 are a linear pair 2. Definition of a Linear Pair 2 and 3 are a linear pair 3. 1 and 2 are supplementary 3. Linear Pair Postulate 2 and 3 are supplementary 4. m 1 + m 2 = 180 m 2 + m 3 = m 1 + m 2 = m 2 + m 3 5. Substitution PoE 6. m 1 = m 3 6. Subtraction PoE 7. 1 = 3 7. = angles have = measures Guided Practice 1. 1 = 4 and C and F are right angles. Which angles are congruent and why? 4

7 Concept 1. Two-Column Proofs 2. In the figure 2 = 3 and k p. Each pair below is congruent. State why. a) 1 and 5 b) 1 and 4 c) 2 and 6 d) 6 and 7 3. Write a two-column proof. Given: 1 = 2 and 3 = 4 Prove: 1 = 4 Answers: 1. By the Right Angle Theorem, C = F. Also, 2 = 3 by the Same Angles Supplements Theorem because 1 = 4 and they are linear pairs with these congruent angles. 2. a) Vertical Angles Theorem 5

8 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: 1. 1 = 2 and 3 = 4 1. Given 2. 2 = 3 2. Vertical Angles Theorem 3. 1 = 4 3. Transitive PoC Practice Fill in the blanks in the proofs below. 1. Given: ABC = DEF and GHI = JKL Prove: m ABC + m GHI = m DEF + m JKL TABLE 1.7: Given 2. m ABC = m DEF 2. m GHI = m JKL Addition PoE 4. m ABC + m GHI = m DEF + m JKL Given: M is the midpoint of AN. N is the midpoint MB Prove: AM = NB TABLE 1.8: 1. Given 2. Definition of a midpoint 3. AM = NB 3. Given: AC BD and 1 = 4 Prove: 2 = 3 6

9 Concept 1. Two-Column Proofs TABLE 1.9: AC BD, 1 = m 1 = m lines create right angles 4. m ACB = 90 m ACD = m 1 + m 2 = m ACB 5. m 3 + m 4 = m ACD Substitution 7. m 1 + m 2 = m 3 + m Substitution 9. 9.Subtraction PoE = Given: MLN = OLP Prove: MLO = NLP TABLE 1.10: = angles have = measures Angle Addition Postulate Substitution 5. m MLO = m NLP = angles have = measures 5. Given: AE EC and BE ED Prove: 1 = 3 7

10 TABLE 1.11: lines create right angles 3. m BED = 90 m AEC = Angle Addition Postulate Substitution 6. m 2 + m 3 = m 1 + m Subtraction PoE = angles have = measures 6. Given: L is supplementary to M and P is supplementary to O and L = O Prove: P = M TABLE 1.12: 2. m L = m O Definition of supplementary angles Substitution Substitution Subtraction PoE 7. M = P Given: 1 = 4 Prove: 2 = 3 8

11 Concept 1. Two-Column Proofs TABLE 1.13: 2. m 1 = m Definition of a Linear Pair 4. 1 and 2 are supplementary 4. 3 and 4 are supplementary Definition of supplementary angles 6. m 1 + m 2 = m 3 + m m 1 + m 2 = m 3 + m m 2 = m = Given: C and F are right angles Prove: m C + m F = 180 TABLE 1.14: 2. m C = 90,m F = = m C + m F = Given: l m 9

12 Prove: 1 = 2 TABLE 1.15: 1. l m and 2 are right angles Given: m 1 = 90 Prove: m 2 = 90 TABLE 1.16: 2. 1 and 2 are a linear pair Linear Pair Postulate Definition of supplementary angles Substitution 6. m 2 = Given: l m Prove: 1 and 2 are complements 10

13 Concept 1. Two-Column Proofs TABLE 1.17: lines create right angles 3. m 1 + m 2 = and 2 are complementary Given: l m and 2 = 6 Prove: 6 = 5 TABLE 1.18: 2. m 2 = m = m 5 = m m 5 = m

Triangle Sum Theorem. Bill Zahner Lori Jordan. Say Thanks to the Authors Click (No sign in required)

Triangle Sum Theorem. Bill Zahner Lori Jordan. Say Thanks to the Authors Click (No sign in required) Triangle Sum Theorem 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