Geometry Module 3 - Foundations of Proofs and Mathematical Logic

Vocabulary flashcards reviewing core definitions, geometric postulates, algebraic properties, angle relationships, and logic terms for Geometry Module 3.

Geometric Proof

A logical argument that formally demonstrates the truth of a statement using statements and associated reasons.

Linear Pair Postulate

A postulate stating that if two angles form a linear pair, then the sum of their measures is $180^\times$ ($m\angle 1 + m\angle 2 = 180^\times$).

Substitution Property

An algebraic property stating that if $a + x = c$ and $x = y$, then $a + y = c$.

Angle Addition Postulate

A postulate stating that if an angle is divided into two adjacent angles, then the sum of their measures equals the measure of the combined angle ($m\angle 1 + m\angle 2 = m\angle 3$).

Reflexive Property

A property stating that an angle or segment is equal to itself (e.g., $m\angle A = m\angle A$ or $BC = BC$).

Complementary Angles

Two angles whose measures sum to $90^\times$.

Supplementary Angles

Two angles whose measures sum to $180^\times$.

Adjacent Angles

Two angles that share a common vertex and a common side but have no interior points in common.

Congruent Segments

Segments that have the exact same measure.

Right Angle

An angle with a measure of $90^\times$.

Angle Bisector

A ray that divides an angle into two equal adjacent angles.

Midpoint

A point that divides a segment into two equal segments.

Parallel Lines

Two lines in the same plane that do not intersect.

Skew Lines

Two lines that do not intersect and are not coplanar.

Perpendicular Lines

Two lines that meet to form right angles.

Corresponding Angles

Two non-adjacent angles on the same side of the transversal, one interior and one exterior.

Alternate Interior Angles

Two non-adjacent interior angles on opposite sides of the transversal.

Alternate Exterior Angles

Two non-adjacent exterior angles on opposite sides of the transversal.

Same-Side Interior Angles

Two non-adjacent interior angles on the same side of the transversal.

Statement

Any sentence that is either true ($T$) or false ($F$), but not both.

Negation

A statement with the opposite meaning and opposite truth value of a given statement $p$, written as $\sim p$ ('not $p$').

Conjunction

A compound statement formed by joining two statements with 'and' (written as $p \land q$), which is true only when both $p$ and $q$ are true.

Disjunction

A compound statement formed by joining two statements with 'or' (written as $p \lor q$), which is true when either $p$ or $q$ (or both) are true.

Conditional Statement

A statement of the form 'If $p$, then $q$' (written as $p \rightarrow q$), which is false only when the hypothesis $p$ is true and the conclusion $q$ is false.

Hypothesis

The '$p$' statement in a conditional statement 'If $p$, then $q$'.

Conclusion

The '$q$' statement in a conditional statement 'If $p$, then $q$'.

Biconditional Statement

A statement of the form '$p$ if and only if $q$' (written as $p \leftrightarrow q$), which is true when $p$ and $q$ have the exact same truth value.

Converse

A statement formed by exchanging the hypothesis and conclusion of a conditional statement, written as $q \rightarrow p$.

Inverse

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

Contrapositive

A statement formed by negating both the hypothesis and the conclusion of the converse of a conditional statement, written as $\sim q \rightarrow \sim p$.