Geometry Foundations: Logic, Proofs, and Geometric Measurement

Vocabulary flashcards covering key definitions, logical reasoning concepts, conditional statements, laws of deduction, algebraic properties of equality, and geometric postulates.

Collinear Points

Points that lie on the same straight line.

Statement

A sentence that is either true or false, but cannot be both.

Negation

A statement that has the opposite meaning and truth value of the original statement, represented by the symbol $\sim$.

Conjunction

A compound statement formed by joining two or more statements with the word "and" ($p \land q$), which is true only when all individual statements are true.

Disjunction

A compound statement formed by joining two or more statements with the word "or" ($p \lor q$), which is true if at least one of the individual statements is true.

Conditional Statement

An if-then statement ($p \rightarrow q$) that relates a hypothesis to a conclusion.

Hypothesis

The phrase immediately following the word "if" in a conditional statement.

Conclusion

The phrase immediately following the word "then" in a conditional statement.

Converse

A statement formed by reversing the hypothesis and the conclusion of a conditional statement ($q \rightarrow p$).

Inverse

A statement formed by negating both the hypothesis and the conclusion of a conditional statement ($\sim p \rightarrow \sim q$).

Contrapositive

A statement formed by negating and reversing both the hypothesis and the conclusion of a conditional statement ($\sim q \rightarrow \sim p$).

Biconditional Statement

The combination of a conditional statement and its converse, written as $p \leftrightarrow q$ and read as "$p$ if and only if $q$".

Counterexample

A specific case or example showing that a statement is false by demonstrating that the hypothesis can be true while the conclusion is false.

Law of Detachment

A rule of deductive reasoning stating that if a conditional statement ($p \rightarrow q$) is true and its hypothesis ($p$) is true, then its conclusion ($q$) must also be true.

Law of Syllogism

A rule of deductive reasoning stating that if two conditionals ($p \rightarrow q$ and $q \rightarrow r$) are true, then the conditional statement $p \rightarrow r$ is also true.

Reflexive Property of Equality

A property stating that any quantity is equal to itself, such as $AB = AB$ for segment lengths or $m\angle A = m\angle A$ for angle measures.

Symmetric Property of Equality

A property stating that if one quantity equals another, their order may be reversed: if $AB = CD$, then $CD = AB$, or if $m\angle A = m\angle B$, then $m\angle B = m\angle A$.

Transitive Property of Equality

A property stating that if two quantities each equal a third quantity, they are equal to each other: if $AB = CD$ and $CD = EF$, then $AB = EF$.

Angle Addition Postulate

A postulate stating that if ray $BD$ lies between ray $BA$ and ray $BC$, then $m\angle ABD + m\angle DBC = m\angle ABC$.

Complementary Angles

Two angles whose measures have a sum of $90^\circ$.

Midpoint

A point that divides a line segment into two congruent segments.