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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 (read as “if , then ”). “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 is true and is true, it can be written as .
- If is not true, then we cannot conclude is true.
- If we are given , we cannot make a conculsion. We cannot conclude 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 is true and is true, then is true.
- If is not true, then we cannot conclude is true
- If we are given , we cannot make a conclusion. We cannot conclude 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 and are true, then 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
- .
- Symmetric Property of Equality
- and .
- Transitive Property of Equality
- and , then .
- Substitution Property of Equality
- If , then can be used in place of and vise versa.
- Addition Property of Equality
- If , then .
- Subtraction Property of Equality
- If , then .
- Multiplication Property of Equality
- If , then .
- Division Property of Equality
- If , then .
- Distributive Property
- .
- Reflexive Property of Congruence
- For Line Segments For Angles
- Symmetric Property of Congruence
- For Line Segments If , then For Angles If
- Transitive Property of Congruence
- For Line Segments If and , then For Angles If and , then
Review
Match the definition or description with the correct word.
- and , then — A. Law of Contrapositive
- An educated guess — B. Inductive Reasoning
- — C. Inverse
- 2, 4, 8, 16, 32,... — D. Transitive Property of Equality
- and — E. Counterexample
- — F. Conjecture
- Conclusions drawn from facts. — G. Deductive Reasoning
- If I study, I will get an “” on the test. I did not get an . Therefore, I didn’t study. — H. Distributive Property
- and are right angles, therefore . — 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, 2012Last Modified:
Aug 21, 2014
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