Honors Geometry: Logic, Conditionals, and Geometric Proofs

Vocabulary flashcards covering symbolic logic, conditional statements, biconditionals, algebraic properties, angle/segment postulates, and geometric proof justifications.

Statement

A declarative sentence that is either true (T) or false (F), but not both. Represented symbolically by variables such as $p$, $q$, or $r$.

Negation

A statement that has the opposite truth value of a given statement $p$. Represented symbolically as $∳ p$ and read as 'Not $p$'.

Conjunction

A compound statement formed by joining two statements with the word 'and'. Represented symbolically as $p \land q$ ('$p$ and $q$'), which is true ONLY when both $p$ and $q$ are true.

Disjunction

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

Conditional Statement

A logical statement written in if-then form, represented symbolically as $p \rightarrow q$ ('If $p$, then $q$' or '$p$ implies $q$'). It is false ONLY when a true hypothesis leads to a false conclusion.

Hypothesis

The prerequisite condition statement in a conditional statement $p \rightarrow q$, which directly follows the word 'if' (represented by $p$).

Conclusion

The result statement in a conditional statement $p \rightarrow q$, which directly follows the word 'then' (represented by $q$).

Converse

A related conditional statement formed by switching the hypothesis and conclusion of the original conditional statement. Represented symbolically as $q \rightarrow p$ ('If $q$, then $p$').

Inverse

A related conditional statement formed by negating both the hypothesis and conclusion of the original conditional statement. Represented symbolically ~p → ~q

'If not p, then not q' Negates both hypothesis and conclusion.

Contrapositive

A related conditional statement formed by both switching and negating the hypothesis and conclusion of the original conditional statement. Represented symbolically ~q → ~p

If not q, then not p' Switches and negates hypothesis & conclusion.

Logically Equivalent Statements

Statements that have the exact same truth values in every line of a truth table, such as a conditional statement and its contrapositive, or a converse and its inverse.

Biconditional Statement

The conjunction of a conditional statement and its converse, written using the phrase 'if and only if' (iff) and represented symbolically as $p \leftrightarrow q$. It is true ONLY when both $p$ and $q$ have the same truth value.

Geometric Proof

A logical argument/chain in which each statement made is supported by a justification accepted as true, starting with given information and ending with the statement to be proven.

Two-Column Proof

A proof format (also called a T-chart proof) organized into two columns, where mathematical facts or steps are listed under 'Statements' and their justifications are listed under 'Reasons'.

Paragraph Proof

A proof format that presents statements and reasons in clear, connected sentences using mathematical words to explain the logical progression.

Reflexive Property

A property stating that a quantity or geometric figure is equal or congruent to itself (e.g., $a = a$, $AB \cong AB$, or $\angle A \cong \angle A$).

Symmetric Property

A property stating that if $a = b$, then $b = a$ (or if $AB \cong CD$, then $CD \cong AB$; if $\angle A \cong \angle B$, then $\angle B \cong \angle A$).

Transitive Property

A property stating that if $a = b$ and $b = c$, then $a = c$ (or if $AB \cong CD$ and $CD \cong EF$, then $AB \cong EF$; if $\angle A \cong \angle B$ and $\angle B \cong \angle C$, then $\angle A \cong \angle C$).

Substitution Property

A property stating that equal quantities may replace one another in any expression or equation.

Segment Addition Postulate

A postulate stating that if three points $A$, $B$, and $C$ are collinear and $B$ is between $A$ and $C$, then $AB + BC = AC$.

Angle Addition Postulate

A postulate stating that the measure of a whole angle equals the sum of its non-overlapping parts.

Linear Pair Postulate

A postulate stating that if two angles form a linear pair, then they are supplementary ($m\angle 1 + m\angle 2 = 180^\circ$).

Vertical Angles Congruence Theorem

A theorem stating that vertical angles are congruent ($\angle 1 \cong \angle 3$).

Congruent Supplements Theorem

A theorem stating that angles supplementary to the same angle (or to congruent angles) are congruent.

Congruent Complements Theorem

A theorem stating that angles complementary to the same angle (or to congruent angles) are congruent.

Right Angles Congruence Theorem

A theorem stating that all right angles are congruent.

Definition of Midpoint

A geometric definition stating that a midpoint divides a line segment into two congruent segments.

Definition of Segment Bisector

A geometric definition stating that a segment, ray, line, or plane intersects a segment at its midpoint.

Definition of Angle Bisector

A geometric definition stating that an angle bisector divides an angle into two congruent angles.