Geometry Quiz Review: Planes, Rays, Coordinates, Distance, and Segment Addition Postulate

Intersection of Planes

The intersection of two distinct planes in three-dimensional space is defined as the set of all points that both planes share in common. When two non-parallel planes intersect, their meeting boundary forms a straight line that extends infinitely in two directions.

Quiz concepts regarding plane intersections include two specific procedural skills. First, given two intersecting planes, one must be able to name the line where the two planes cross. Second, given a specific line, one must be able to name two distinct planes that cross at that given line.

Planes are designated either by a single capital script letter or by listing three non-collinear points that lie on the plane. Lines are designated either by a single lowercase italicized letter or by specifying two points on the line.

Opposite Rays

A ray is defined as a portion of a line that begins at a single point, called the endpoint or initial point, and extends infinitely in one direction.

Opposite rays are two rays that share the exact same endpoint and extend in completely opposite directions, together forming a single continuous straight line. For two rays to be considered opposite rays, they must be collinear and share their starting point. For instance, if point $B$ lies between point $A$ and point $C$ on a straight line, the ray beginning at $B$ and passing through $A$, and the ray beginning at $B$ and passing through $C$, form a pair of opposite rays.

Naming Lines and Planes

Lines and planes follow fundamental geometric conventions for identification and notation.

A line is a one-dimensional figure that extends without end in two opposite directions. A line is named by selecting any two distinct points on the line in any order, or by using a single lowercase italicized letter assigned to the line.

A plane is a two-dimensional flat surface extending endlessly in all directions. A plane is named by identifying three non-collinear points on the plane—meaning three points that do not all lie on a single straight line—or by using a single capital script letter.

Coordinate

A coordinate is a real number assigned to a specific point on a line or coordinate grid that identifies its exact numerical position.

In a one-dimensional coordinate system, every single point on a number line corresponds to exactly one real number coordinate. Coordinates provide the algebraic basis for finding distance and calculating lengths between geometric points.

Distance on a Number Line and Absolute Value

The distance between two points on a number line is defined as the spatial length between their coordinates.

Distance is strictly and always POSITIVE (or zero when finding the distance from a point to itself). Distance can never be a negative number.

To find the distance between two points on a number line with coordinates $a$ and $b$, calculate the absolute value of the difference between those coordinates:

$\text{Distance} = |a - b|$

Because absolute value evaluates the absolute magnitude of a real number regardless of its sign, subtracting in either order yields the identical positive distance value:

$\text{Distance} = |b - a|$

Segment Addition Postulate

The Segment Addition Postulate defines the quantitative relationship between adjacent line segments located along the same straight line.

The postulate states that if point $B$ lies on line segment $AC$ between endpoint $A$ and endpoint $C$, then the sum of the lengths of the two smaller adjacent segments equals the total length of the largest segment.

$\text{Segment} + \text{Segment} = \text{Largest Segment}$

$AB + BC = AC$

In this formulation, $AB$ represents the measure or distance from point $A$ to point $B$, $BC$ represents the measure or distance from point $B$ to point $C$, and $AC$ represents the full distance from endpoint $A$ to endpoint $C$. This postulate applies if and only if points $A$, $B$, and $C$ are collinear and point $B$ is situated between $A$ and $C$.