In Euclidean geometry, a plane can be defined by two distinct lines, by three non-collinear points, or by one line and a point not on the line.
Plane geometry is the geometry of objects in 2 dimensional space.
A geometric object is planar if it can be entirely contained within one plane. A geometric object is non-planar if it can not be entirely contained within one plane.
Two planes in a 3-space either intersect or are parallel. If two planes are parallel, a line that is perpendicular to the one of the planes is also perpendicular to the other. If two planes intersect, the intersection forms a line. The angle between two intersecting planes is called a dihedral angle. If the dihedral angle between two planes is a right angle, then the planes are perpendicular.
Planes are used in many contexts. Here are some of them.
|Cartesian Plane||A plane divided into four quadrants by two perpendicular number lines. The vertical number line represents a dependent variable and the horizontal number line represents the independent variable.|
|Rectangular coordinate plane|
|Complex Plane||A plane divided into four quadrants by two perpendicular number lines. Complex numbers can be plotted on a complex plane. The vertical number line represents the imaginary part of the complex number, and the horizontal number line represents the real part.|
|Half-plane||A half-plane is a part of a plane cut off by a line.|
|A plane that intersects a reference plane. The reference plane is usually horizontal.|
|Metric plane||Any two dimensional space which has a unit of measure for each dimension and a way to find the distance between any two points in the plane. Cartesian coordinates and Polar coordinates are examples of metric planes.|
|Table 1: Types of planes.|
Some of the postulates and theorems dealing with planes are found in table 2. Click on the blue buttons in the manipulatives and drag them to change the figures.
|Minimum Plane Postulate||A plane contains at least 3 non-collinear points.|
|Unique Plane Postulate||There is exactly one plane that passes through any 3 non-collinear points.|
|Same Plane Postulate||If two points lie in a plane, then the entire line joining those points lies in the same plane.|
|Intersecting Planes Postulate||If two planes intersect, then their intersection is a line.|
|Point, Line, Plane Theorem||If a point lies outside a line, then there is exactly one plane containing both the point and the line.|
|Intersecting Lines Plane Theorem||If two lines intersect, then there is exactly one plane containing both lines.|
|Table 2: Postulates and Theorems of Planes.|
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