# 2.6: Chapter 2 Review

Difficulty Level: At Grade Created by: CK-12

## Symbol Toolbox

\begin{align*}\rightarrow\end{align*} if-then

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

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

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

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

Keywords

Inductive Reasoning
The study of patterns and relationships is a part of mathematics. The conclusions made from looking at patterns are called conjectures. Looking for patterns and making conjectures is a part of inductive reasoning, where a rule or statement is assumed true because specific cases or examples are true.
Conjecture
The study of patterns and relationships is a part of mathematics. The conclusions made from looking at patterns are called conjectures.
Counterexample
We can disprove a conjecture or theory by coming up with a counterexample. Called proof by contradiction, only one counterexample is needed to disprove a conjecture or theory (no number of examples will prove a conjecture). The counterexample can be a drawing, statement, or number.
Conditional Statement (If-Then Statement)
Geometry uses conditional statements that can be symbolically written as \begin{align*}p \rightarrow q\end{align*} (read as “if \begin{align*}p\end{align*}, then \begin{align*}q\end{align*}”). “If” is the hypothesis, and “then” is the conclusion.
Hypothesis
The conditional statement is false when the hypothesis is true and the conclusion is false.
Conclusion
The second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.
Converse
A statement where the hypothesis and conclusion of a conditional statement are switched.
Inverse
A statement where the hypothesis and conclusion of a conditional statement are negated.
Contrapositive
A statement where the hypothesis and conclusion of a conditional statement are exchanged and negated.
Biconditional Statement
If \begin{align*}p \rightarrow \ q\end{align*} is true and \begin{align*}q \rightarrow \ p\end{align*} is true, it can be written as \begin{align*}p \rightarrow \ q\end{align*}.
If \begin{align*}p\end{align*} is not true, then we cannot conclude \begin{align*}q\end{align*} is true.
If we are given \begin{align*}q\end{align*}, we cannot make a conculsion. We cannot conclude \begin{align*}p\end{align*} is true.
Logic
The study of reasoning.
Deductive Reasoning
Uses logic and facts to prove the relationship is always true.
Law of Detachment
The Law of Detachment states: If \begin{align*}p \ q\end{align*} is true and \begin{align*}p\end{align*} is true, then \begin{align*}q\end{align*} is true.
If \begin{align*}p\end{align*} is not true, then we cannot conclude \begin{align*}q\end{align*} is true
If we are given \begin{align*}q\end{align*}, we cannot make a conclusion. We cannot conclude \begin{align*}p\end{align*} is true.
Law of Contrapositive
If the conditional statement is true, the converse and inverse may or may not be true. However, the contrapositive of a true statement is always true. The contrapositive is logically equivalent to the original conditional statement.
Law of Syllogism
The Law of Syllogism states: If \begin{align*}p \rightarrow \ q\end{align*} and \begin{align*}q \rightarrow \ r\end{align*} are true, then \begin{align*}p \rightarrow \ r\end{align*} is true.
Right Angle Theorem
If two angles are right angles, then the angles are congruent.
Same Angle Supplements Theorem
If two angles are supplementary to the same angle (or to congruent angles), then the angles are congruent.
Same Angle Complements Theorem
If two angles are complementary to the same angle (or to congruent angles), then the angles are congruent.
Reflexive Property of Equality
\begin{align*}a = a\end{align*}.
Symmetric Property of Equality
\begin{align*}a = b\end{align*} and \begin{align*}b = a\end{align*}.
Transitive Property of Equality
\begin{align*}a = b\end{align*} and \begin{align*}b = c\end{align*}, then \begin{align*}a = c\end{align*}.
Substitution Property of Equality
If \begin{align*}a = b\end{align*}, then \begin{align*}b\end{align*} can be used in place of \begin{align*}a\end{align*} and vise versa.
If \begin{align*}a = b\end{align*}, then \begin{align*}a + c = b + c\end{align*}.
Subtraction Property of Equality
If \begin{align*}a = b\end{align*}, then \begin{align*}a - c = b - c\end{align*}.
Multiplication Property of Equality
If \begin{align*}a = b\end{align*}, then \begin{align*}ac = bc\end{align*}.
Division Property of Equality
If \begin{align*}a = b\end{align*}, then \begin{align*}a \div \ c = b \div \ c\end{align*}.
Distributive Property
\begin{align*}a(b + c) = ab + ac\end{align*}.
Reflexive Property of Congruence
For Line Segments \begin{align*}\overline{AB} \cong \overline{AB}\end{align*} For Angles \begin{align*}\overline{AB} \cong \angle ABC \cong \angle CBA\end{align*}
Symmetric Property of Congruence
For Line Segments If \begin{align*}\overline{AB} \cong CD\end{align*}, then \begin{align*}\overline{CD} \cong \overline{AB}\end{align*} For Angles \begin{align*}\overline{CD} \cong \overline{AB}\end{align*} If \begin{align*}\angle ABC \cong \angle DFF \cong \angle ABC\end{align*}
Transitive Property of Congruence
For Line Segments If \begin{align*}\overline{AB} \cong \overline{CD}\end{align*} and \begin{align*}\overline{CD} \cong \overline{EF}\end{align*}, then \begin{align*}\overline{AB} \cong \overline{EF}\end{align*} For Angles If \begin{align*}\angle ABC \cong \angle DEF\end{align*} and \begin{align*}\angle DEF \cong \angle GHI\end{align*}, then \begin{align*}\angle ABC \cong \angle GHI\end{align*}

## Review

Match the definition or description with the correct word.

1. \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
2. An educated guess — B. Inductive Reasoning
3. \begin{align*}6(2a + 1) = 12a +12\end{align*} — C. Inverse
4. 2, 4, 8, 16, 32,... — D. Transitive Property of Equality
5. \begin{align*}\overline{AB} \cong \overline{CD}\end{align*} and \begin{align*}\overline{CD} \cong \overline{AB}\end{align*} — E. Counterexample
6. \begin{align*}\sim p \rightarrow \sim q\end{align*} — F. Conjecture
7. Conclusions drawn from facts. — G. Deductive Reasoning
8. 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
9. \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
10. 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.

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