Geometry Core Concepts: Basics, Reasoning, Proofs, and Lines

Comprehensive vocabulary flashcards covering the fundamentals of geometry, logical reasoning and proofs, and parallel and perpendicular line relationships.

Point

An undefined term in geometry that has no dimension and is represented by a dot.

Line

An undefined term in geometry that has one dimension and extends without end in two opposite directions.

Plane

An undefined term in geometry that has two dimensions, is represented by a shape that looks like a floor or a wall, and extends without end.

Collinear Points

Points that lie on the same line.

Coplanar Points

Points that lie in the same plane.

Line Segment

A portion of a line consisting of two endpoints and all points on the line between them.

Ray

A portion of a line that consists of an endpoint and all points on the line on one side of the endpoint.

Opposite Rays

Two rays that have the same endpoint and form a line.

Intersection

The set of points two or more geometric figures have in common.

Postulate

A rule in geometry that is accepted without proof (also called an axiom).

Theorem

A statement or conjecture in geometry that can be proven true.

Coordinate

The real number that corresponds to a point on a line.

Distance

The absolute value of the difference between the coordinates of two points on a line.

Congruent Segments

Line segments that have the same length.

Midpoint

The point that divides a segment into two congruent segments.

Segment Bisector

A point, ray, line, line segment, or plane that intersects a segment at its midpoint.

Polygon

A closed plane figure formed by three or more line segments called sides.

Convex Polygon

A polygon in which no line containing a side contains a point in the interior of the polygon.

Concave Polygon

A polygon that is not convex, meaning a line containing a side contains a point in the interior of the polygon.

Angle

A set of points consisting of two rays with the same endpoint, called the vertex.

Vertex

The common endpoint of the two rays that form an angle.

Acute Angle

An angle with a measure greater than $0^\circ$ and less than $90^\circ$.

Right Angle

An angle with a measure equal to $90^\circ$.

Obtuse Angle

An angle with a measure greater than $90^\circ$ and less than $180^\circ$.

Straight Angle

An angle with a measure equal to $180^\circ$.

Congruent Angles

Angles that have the same measure.

Angle Bisector

A ray that divides an angle into two congruent angles.

Complementary Angles

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

Supplementary Angles

Two angles whose measures have a sum of $180^\circ$.

Adjacent Angles

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

Linear Pair

Two adjacent angles whose noncommon sides are opposite rays.

Vertical Angles

Two angles whose sides form two pairs of opposite rays.

Conditional Statement

A logical statement that has two parts, a hypothesis $p$ and a conclusion $q$, written in if-then form as $p \rightarrow q$.

Hypothesis

The "if" part of a conditional statement written in if-then form.

Conclusion

The "then" part of a conditional statement written in if-then form.

Negation

The opposite of a statement, symbolized by $\sim p$ (read as "not $p$").

Converse

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

Inverse

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

Contrapositive

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

Equivalent Statements

Two statements that are both true or both false.

Truth Table

A table that lists all possible combinations of truth values for a logical statement.

Biconditional Statement

A statement that contains the phrase "if and only if," written as $p \leftrightarrow q$.

Inductive Reasoning

The process of finding a pattern in specific cases and writing a conjecture for the general case.

Conjecture

An unproven statement that is based on observations.

Counterexample

A specific case for which a conjecture is false.

Deductive Reasoning

The process of reasoning logically from given statements or facts to a conclusion.

Law of Detachment

A law of deductive reasoning stating that if the hypothesis of a true conditional statement is true, then the conclusion is true.

Law of Syllogism

A law of deductive reasoning stating that if $p \rightarrow q$ is true and $q \rightarrow r$ is true, then $p \rightarrow r$ is true.

Proof

A logical argument that uses deductive reasoning to show that a statement is true.

Two-Column Proof

A proof arranged in two columns containing numbered statements and corresponding reasons in logical order.

Paragraph Proof

A proof written as sentences in a paragraph to explain the logical flow of an argument.

Flowchart Proof

A proof (or flow proof) that uses boxes and arrows to show the flow of a logical argument.

Parallel Lines

Lines in a plane that are always the same distance apart and do not intersect.

Perpendicular Lines

Lines that intersect at a right angle ($90^\circ$).

Skew Lines

Lines that do not intersect and are not coplanar.

Parallel Planes

Planes that do not intersect.

Transversal

A line that intersects two or more coplanar lines at distinct points.

Corresponding Angles

Two angles that have corresponding positions when two lines are cut by a transversal.

Alternate Interior Angles

Two angles that lie between two lines and on opposite sides of the transversal.

Alternate Exterior Angles

Two angles that lie outside two lines and on opposite sides of the transversal.

Consecutive Interior Angles

Two angles that lie between two lines and on the same side of the transversal.

Slope

The ratio of vertical change (rise) to horizontal change (run) between any two points on a line ($m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$).

Distance from a Point to a Line

The length of the perpendicular segment from the point to the line, which represents the least distance between them.

Perpendicular Bisector

A line, ray, or segment that is perpendicular to a segment at its midpoint.

Directed Line Segment

A line segment that represents moving from one specified endpoint to another endpoint.