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.
- Addition Property of Equality
- 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.
- \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.
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angle pairs
biconditional statements
CK.MAT.ENG.SE.1.Geometry-Basic.2
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CK.MAT.ENG.SE.2.Geometry.2
conclusion
conditional statements
congruence
conjecture
conjectures
contrapositive
converse
counterexamples
deductive reasoning
hypothesis
if-then statement
if-then statements
inductive and deductive reasoning
inductive reasoning
inverse
Law of Contrapositive
Law of Detachment
Law of Syllogism
logic
patterns
proof
proofs
reasoning
Segments
theorems
truth tables
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Date Created:
Feb 22, 2012
Last Modified:
Feb 03, 2016
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