# Chapter 2: Reasoning and Proof

**At Grade**Created by: CK-12

## Introduction

This chapter explains how to reason and how to use reasoning to prove theorems about angle pairs and segments. This chapter also introduces the properties of congruence, which will also be used in proofs. Subsequent chapters will combine what you have learned in Chapters 1 and 2 and build upon them.

- 2.1.
## Conjectures and Counterexamples

- 2.2.
## If-Then Statements

- 2.3.
## Converse, Inverse, and Contrapositive

- 2.4.
## Inductive Reasoning from Patterns

- 2.5.
## Deductive Reasoning

- 2.6.
## Truth Tables

- 2.7.
## Properties of Equality and Congruence

- 2.8.
## Two-Column Proofs

### Chapter Summary

## Summary

This chapter teaches students how to make conjectures and provide counterexamples. From there, it focuses on rewriting statements in if-then form and finding converses, inverses, and contrapositives. Two types of reasoning, inductive and deductive, are explored. Finally, the properties of equality and congruence are reviewed and practice for completing two-column proofs is provided.

### Symbol Toolbox for Chapter

\begin{align*}\rightarrow\end{align*}

\begin{align*}\land\end{align*}

\begin{align*}\therefore\end{align*} therefore

\begin{align*}\sim\end{align*} not

\begin{align*}\lor\end{align*} or

### Chapter Keywords

- Inductive Reasoning
- Conjecture
- Counterexample
- Conditional Statement (If-Then Statement)
- Hypothesis
- Conclusion
- Converse
- Inverse
- Contrapositive
- Biconditional Statement
- Logic
- Deductive Reasoning
- Law of Detachment
- Law of Contrapositive
- Law of Syllogism
- Right Angle Theorem
- Same Angle Supplements Theorem
- Same Angle Complements Theorem
- Reflexive Property of Equality
- Symmetric Property of Equality
- Transitive Property of Equality
- Substitution Property of Equality
- Addition Property of Equality
- Subtraction Property of Equality
- Multiplication Property of Equality
- Division Property of Equality
- Distributive Property
- Reflexive Property
- of Congruence
- Symmetric Property
- of Congruence
- Transitive Property of Congruence

### Chapter Review

Match the definition or description with the correct word.

- \begin{align*}5 = x\end{align*} and \begin{align*}y + 4 = x\end{align*}, then \begin{align*}5 = y +4\end{align*} — A. Law of Contrapositive
- An educated guess — B. Inductive Reasoning
- \begin{align*}6(2a + 1) = 12a +12\end{align*} — C. Inverse
- 2, 4, 8, 16, 32,... — D. Transitive Property of Equality
- \begin{align*}\overline{AB} \cong \overline{CD}\end{align*} and \begin{align*}\overline{CD} \cong \overline{AB}\end{align*} — E. Counterexample
- \begin{align*}\sim p \rightarrow \sim q\end{align*} — F. Conjecture
- Conclusions drawn from facts. — G. Deductive Reasoning
- If I study, I will get an “\begin{align*}A\end{align*}” on the test. I did not get an \begin{align*}A\end{align*}. Therefore, I didn’t study. — H. Distributive Property
- \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are right angles, therefore \begin{align*}\angle A \cong \angle B\end{align*}. — I. Symmetric Property of Congruence
- 2 disproves the statement: “All prime numbers are odd.” — J. Right Angle Theorem — K. Definition of Right Angles

### Texas Instruments Resources

*In the CK-12 Texas Instruments Geometry FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9687.*