Foundations of Geometry - Unit 1

Conjecture

A statement that appears to be correct based on observation but hasn't been proven or disproven.

Induction

A way of thinking that uses observations to form a general rule.

Symmetry

The property of having a line, or axis, that divides into two identical parts. A shape can have more than one line of symmetry.

Conditional Statement

A statement that uses the form in "A" then "B". Meaning if A is true, B is true. Eg: If it rains, I will use my umbrella.

Deduction

A way of thinking that starts with a given set of rules and conditions and figures out what must be true based on what is given.

Contrapositive

A statement in the form, "if not B, then not A"

converse

A statement in the form, "If B, then A"

Venn Diagram

A diagram the uses 2 or more circles or other shapes to represent sets. Elements that belong in more than one set are placed in areas where the shapes overlap.

Inverse

A statement in the form "If not A, then not B".

Syllogism

A form of deductive reasoning that combines 2 or more related conditional statements in order to arrive at a conclusion. Eg: A\=B, B\=C, so A\=C

Common Notion

A statement that is not officially defined but that is understood to be common sense. Eg: No car faces in two directions at the same time.

Corollary

A state that makes sense based on what is already proven. (That is, a theorem)

Definition

A statement that describes of an idea, object, or process.

Indirect Proof

A type of proof that is written in paragraph form, where the contradiction of the statement to be proved is shown to be false, so the statement to be proved is therefore true. Also called proof by contradiction.

Postulate

A statement accepted as true without proof; an axiom

Proof

A logical argument of definitions, theorems, and postulates that leads to the conclusion that a statement is always true.

Theorem

A statement that has already been proven to be true. Eg: Pythagorean Theorem

Two-column proof

A type of proof that has 2 columns: A left hand column for statements, or deductions. As well as a right hand column for the reason of each statement. (That is, a definition, postulate, or theorem.)

Acute Angle

An angle that is measures less than 90 degrees.

Adjacent Angles

Angles that share a vertex and one side. (Essentially meaning next to each other.)

Angle Bisector

A ray that divides an angle into two angles of equal measure.

Complementary

Angle measures that add up to 90 degrees. If two complementary angles are adjacent, they form a right angle.

Congruent

Having the same size or shape. If polygons are congruent, their sides and angles are also congruent.

Endpoint

A point at the end of a ray, line segment, or arc.

Line Segment

A part of a line with endpoints at both ends.

Linear Pair

A pair of adjacent angles that add up to 180 degrees. Linear pairs of angles are called supplementary

Midpoint

The point halfway between the endpoints of a line segment.

Obtuse Angle

An angle which measures over 90 degrees but less than 180 degrees.

Point

It is used to mark and represent locations. It has no length, width, or height.

Ray

A part of a lie that starts as an endpoint, but continues forever in one direction.

Right Angle

An angle that measures exactly 90 degrees.

Straight Angle

An angle whose sides form a line. The measure of a straight angle is 180 degrees.

Supplementary

Having angle measures that add up to 180 degrees. If two supplementary angles are adjacent, they form a straight line.

Vertex

The point where rays or lie segments meet to form an angle.

Segment Addition

A postulate stating if AC + BC = AB, then point C is between points A and B

Zero Angle

An angle that has a measure of 0 degrees ad whose sides overlap to form a ray.

Line

A set of points in a plane that are at equal distance from two points.

Angle Addition Postulate

If point C is i the interior of angle AVB, then angle AVC + CVB = AVB

Angle

The object formed by two rays that share the same endpoint.

Collinear

Lying in a straight line. 3 or more points are collinear if a ***straight*** line can be drawn through them. 2 points are always collinear.

Coplanar

Lying in the same plane. Four or more points are coplanar if a plane contains them.

Infinite

Having no boundary or limit. It goes on forever.

One-dimensional

Having length but width or height.

Plane

Extends forever in all directions. Has no thickness, so it has only 2 dimensions.

Three-Dimensional

Having length, width, and height. Example: A cube.

Two-Dimensional

Having length and width, but no height.

Zero-Dimensional

Having no length, width, or height.

Perpendicular Bisector

A line, ray, or line segment that bisects a line segment at a right angle.

Vertical Angles

A pair of opposite angles formed by intersecting lines. Vertical angles have equal measures.

Perpendicular Lines

Lines that meet to form a right angle.

Alternate Interior Angles

Two angles formed by a line (Called a transversal) that intersects two parallel lines. The angles are on OPPOSING sides of the transversal and inside the parallel lines.

Consecutive Interior Angles

Two angles formed by a line (Called a transversal) that intersects two parallel lines. The angles are on SAME sides of the transversal and inside the parallel lines.

Corresponding Angles

Two nonadjacent angles formed on the same side of a line. (Called a transversal) that intersects two parallel lines, with one angle interior and one angle exterior on the sides.

Intersect

To cross over one another.

Parallel Lines

Lines lying in the same plane without intersecting.

Skew Lines

Lines that are not in the same plane. They do not intersect, but aren’t parallel.

Transversal

A line, ray, or segment that intersects two or more coplanar lines, rays, or segments.