Geometry Reasoning and Conditional Statements

Vocabulary flashcards covering types of related conditional statements, logic laws, reasoning types, and mathematical proofs from Geometry Semester 1.

Conditional Statement

A logical statement that relates individual statements that can be written in the form "if $p$, then $q$", where the "if" part is the hypothesis statement ($p$) and the "then" part is the conclusion statement ($q$).

Hypothesis Statement

The "if" part ($p$) of a conditional statement.

Conclusion Statement

The "then" part ($q$) of a conditional statement.

Negation

A statement having the opposite meaning of the original statement by using the word "not" (symbolized as $\text{not } p$ or $\sim p$).

Converse

The statement formed by switching the hypothesis and the conclusion of a conditional statement ("if $q$, then $p$", or $q \rightarrow p$).

Inverse

The statement formed by negating both the hypothesis and conclusion of a conditional statement ("if not $p$, then not $q$", or $\sim p \rightarrow \sim q$).

Contrapositive

The statement formed by negating and switching both the hypothesis and the conclusion of a conditional statement ("if not $q$, then not $p$", or $\sim q \rightarrow \sim p$).

Biconditional

A statement used only when the conditional and converse statements are both true; it separates the hypothesis and conclusion using the phrase "if and only if" (sometimes abbreviated as "iff", symbolized as $p \leftrightarrow q$).

Conjecture

An unproven statement or general rule believed to be true based on an observed pattern.

Counterexample

An example used to prove a conjecture false (with the prefix "counter" meaning "in opposition to or go against").

Deductive Reasoning

A type of reasoning where facts and information are used to reach a logical conclusion (e.g., two-column proofs, Law of Syllogism, Law of Detachment).

Inductive Reasoning

A type of reasoning process based on observing a pattern to form a general rule to determine what will happen.

Law of Detachment

A logic law that states if a conditional statement and its hypothesis are both true, therefore the conclusion is true (given $p \rightarrow q$ and $p$ are both "T", therefore $q$ is "T"). The scenario must relate to the hypothesis to make a conclusion.

Law of Syllogism

A logic law that states given two true conditional statements that overlap, they can be combined into one conditional statement (given $p \rightarrow q$ and $q \rightarrow r$, therefore $p \rightarrow r$).

Mathematical Proof (or Proof)

A logical argument where a conjecture (or conditional statement) is taken, starting from the hypothesis, and step-by-step statements and reasons are given to eventually show the conclusion is true.

Two-Column Proof

A format for writing a proof that uses two columns: the first column for statements, and the second column for reasons.