Chapter 1 - Honors Geometry - Definitions/Undefined

Point

(UNDEFINED). A location in space. The point has no thickness. It is infinitely small. All figures in geometry are made up of points.

How to denote a point

Capital printed letter

Line

(UNDEFINED). Extends in opposite directions without ending. It has no thickness and can not be measured. Made up of an infinite number of points.

How to denote a line

1. Lower-case cursive letter
2. Use two points that are on the line

Plane

(UNDEFINED). A flat surface that goes on forever. It has no thickness.

How to denote a plane

1. Capital Cursive letter
2. Any 3 (or more) non-collinear points

Space

The set of all points.

How is space represented?

Represented by 4 non-coplanar points.

Collinear points

Points all in one line.

Coplanar points

Points all in one plane.

Intersection

The set of all points that are in both (or all) figures.

Undefined Terms

They are intuitive ideas. Basis of all geometry. Point, line, plane.

Definitions

Based off of "buying in" completely to the undefined terms.

Postulated

Statements that are accepted without proof.

Theorms

Can be proved. Based on postulates, definitions, and undefined terms.

Between

For N to be "between" M and P, all three points must be collinear and N must be in the middle.

Segment

2 points and all points between them.

Ray

Segment XY and all points Z such that Y is between X and Z.

Opposite Rays

Collinear rays that share exactly one point.

Another word for postulates

Axioms

Segment Addition Postulate

If B is between A and C, then AB + BC = AC

Midpoint of a Segment

Point that divides a segment into two congruent segments.

Congruent

Objects that have the same exact size and shape.

Congruent Segments

Segments are congruent if and only if their measurements are equal.

Equal vs Congruent

Lengths are EQUAL
Segments are CONGRUENT

Segment Bisector

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

Angle

A figure formed by 2 rays that have the same endpoint

The 2 rays are called the

sides of an angle

Their common endpoint is called the

vertex of an angle

Point on Interior

A point is on the interior if and only if it lies on a segment whose endpoints are on the angle, but the point is not.

Acute Angle

An angle whose measure is between 0 and 90

Right Angle

An angle whose measure is 90

Obtuse Angle

An angle whose measure is between 90 and 180

Straight Angle

An angle whose measure is 180

Angle Addition Postulate

If point K lies in the interior of

Congruent Angles

Angles that have equal measures.

Adjacent Angles

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

Linear Pair

Adjacent Angles whose non-common sides form opposite rays.

Angle Bisector

The ray that divides an angle into congruent adjacent angles.

Postulate on Line, Plane, and Space

A line contains at least 2 points.
A plane contains at least 3 non-collinear points.
Space contains at least 4 non-coplanar points.

Through any 2 points, there is EXACTLY ONE line

Through any 3 points, there is at least 1 plane.
Through any 3 non-collinear points, there is EXACTLY ONE plane.

If 2 points are on a plane, then the line that contains the points is in that plane.

If 2 planes intersect, then their intersection is a line.

If 2 lines intersect, then they intersect in EXACTLY ONE point.

Through a line and a point not on that line, there is EXACTLY ONE plane.

If 2 lines intersect, then EXACTLY ONE plane contains the lines.

Existence vs Uniqueness

Existence = At least one
Uniqueness = EXACTLY ONE