Solved two-column proof examples with explanations

📚 Quick Study Guide

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• Definition: A two-column proof is a method to prove a mathematical statement by listing statements and justifications in two columns.
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• Statements: These are the logical steps in the proof.
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• Justifications: These are the reasons why each statement is true, based on definitions, postulates, or previously proven theorems.
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• Common Justifications:
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  • Definition of midpoint
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  • Division Property of Equality
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  • Reflexive Property
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  • Addition Property of Equality
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  • Substitution Property
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  • Definition of Angle Bisector
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• Tips: Start with what you know and work towards what you need to prove. Look for key words that indicate specific definitions or theorems.

Practice Quiz

1. Question 1: Which of the following is NOT a valid justification in a two-column proof?
   1. A) Definition of Congruence
   2. B) Vertical Angles Theorem
   3. C) Feeling Lucky
   4. D) Reflexive Property
2. Question 2: Given: $AB = CD$. $BC = BC$. Prove: $AC = BD$. What is the justification for the statement $AC = AB + BC$ and $BD = BC + CD$?
   1. A) Addition Property of Equality
   2. B) Substitution Property
   3. C) Segment Addition Postulate
   4. D) Reflexive Property
3. Question 3: Given: $\angle A \cong \angle B$, $\angle B \cong \angle C$. Prove: $\angle A \cong \angle C$. What is the justification for the statement that leads to the proof?
   1. A) Definition of Congruence
   2. B) Transitive Property of Congruence
   3. C) Reflexive Property of Congruence
   4. D) Symmetric Property of Congruence
4. Question 4: Which property states that $a = a$?
   1. A) Symmetric Property
   2. B) Transitive Property
   3. C) Reflexive Property
   4. D) Substitution Property
5. Question 5: Given: $2(x + 3) = 10$. Prove: $x = 2$. What is the justification for the step $2x + 6 = 10$?
   1. A) Addition Property of Equality
   2. B) Distributive Property
   3. C) Division Property of Equality
   4. D) Combining Like Terms
6. Question 6: If $\angle ABC$ is bisected by $\overrightarrow{BD}$, what definition justifies that $\angle ABD \cong \angle DBC$?
   1. A) Definition of Supplementary Angles
   2. B) Definition of Complementary Angles
   3. C) Definition of Angle Bisector
   4. D) Definition of Congruent Angles
7. Question 7: Given: $x + y = 5$ and $y = 2$. Prove: $x = 3$. What is the justification for substituting $2$ for $y$ in the first equation?
   1. A) Addition Property of Equality
   2. B) Substitution Property
   3. C) Transitive Property
   4. D) Reflexive Property