Vocab:

• Collinear - Lying in one line (usually refers to a set of points).

• Congruent - Of exactly the same size and shape; in other words, of exactly the same dimensions.

• Coplanar - Existing in one plane (usually refers to points or lines).

• Dimension - The number of lines required to span a region in space.

• Line - A collection of points arrayed in a straight formation without limit.

• Noncollinear - Not lying in one line (usually refers to a set of points).

• Noncoplanar - Not existing in one plane (usually refers to points or lines).

• Perpendicular - At a 90 degree angle (something is perpendicular relative to something else).

• Plane - A flat, boundless surface in space.

• Point - A specific location in space.

• Ray - A portion of a line with a fixed endpoint on one end that extends without bound in the other direction.

• Segment - A portion of a line with two endpoints and, thus, finite length.

• Space - The collection of all points.

Geometry is the study of shapes. Every object we see is a shape of some kind. Some are simple, like a triangle, square, or circle. Others seem to be combinations of these simple shapes.The simplest unit of geometry is the point. A collection of points in a certain array makes a line, and collections of lines create shapes, which may exist in a single plane, or in more than one plane in space.

Points:

A point is a way to describe a specific location in space. It is drawn by placing a single dot at the location you want to specify, and denoted in text with a capital letter. Below is pictured the point B. A point has no length, or width. In real life it is not tangible; points are useful for identifying specific locations, but are not objects in themselves. They only appear so when drawn on a page.

Lines:

A line is the infinite set of all points arrayed in a straight formation. A line has no thickness, it has only length, and can be named by any two points on that line. For example, a line can be called line AB, or symbolized.

A line can also be given a single letter as a name, such as p, and be called line p.

To form a line, take any two points, A and B, and draw a straight line through them. The line AB looks like this on paper:

A line extends in both directions without bound; this is why lines are usually depicted with arrows on each end. Its length is infinite, and between any two points on a line, there lie an infinite number of other points. You can choose two points on a line that seem to lie very close to each other, but if you "zoom in" on these points, you can always identify a point halfway between them. Then you can repeat the process with one of the original "close" points and the new halfway point to identify another point in between the two "close" points. This way you can find an infinite number of points between any two points on a line.

Collinear:

Points are called collinear if they lie in the same line. Likewise, points are called non collinear if they lie in different lines. Since a line is determined by two points, any two points are always collinear. When a group of three points is considered, however, they may be noncollinear. Collinearity is a relative term. Points are only collinear or noncollinear when considered with respect to other points. The figure below has a set of noncollinear points on the left, and a set of collinear points on the right.

Line Segments:

A line segment is the portion of a line that lies between two points on that line, points A and B. Whereas a line has infinite length, a line segment has a finite length. A line segment is denoted by segment AB, or the symbol

Line segments of the same length are called congruent. A dash or a set number of dashes is drawn through congruent segments to symbolize their congruence. Here is a figure of a segment:

Rays:

A ray is a cross between a line and a line segment. It extends without bound in one direction, but not the other. It is determined by two points, one being the starting point for the ray, and the other determining the direction of the ray. A ray can be symbolized in the following way:

Below is a figure of a ray:

Just like lines, segments and rays have no thickness, only length. They are intangible, and only used to specify a set of locations in space.

Planes:

A plane is a boundless surface in space. It has length, like a line; it also has width, but not thickness. A plane is denoted by writing "plane P", or just writing "P". On paper, a plane looks something like this:

There are two ways to form a plane. First, a plane can be formed by three noncolinear points. Any number of colinear points form one line, but such a line can lie in an infinite number of distinct planes. See below how different planes can contain the same line.

It takes a third, noncollinear point to form a specific plane. This point fixes the plane in position. The situation is something like a door being shut. Before the door is shut, it swings on hinges, which form a line. The door (a plane) can be opened to an infinite number of different positions, maybe just cracked a few inches, or maybe wide open (figures a, b in the diagram below). When the door is shut however, the wall on the other side of the hinges acts as the noncollinear third point and holds the door in place. At this point, the door represents one distinct plane (figure c).

The second way to form a plane is with a line and a point in that line. There are just two conditions. 1) the line must be perpendicular to the plane being formed (for an explanation of this concept, see Geometric Surfaces, Lines and Planes); 2) the point in the line must also be in the plane being formed. Given a line, a point in that line, and these conditions, a plane is determined.

Dimension:

Dimension is a characteristic of all geometric regions, objects, and spaces. It is roughly the number of directions in which a region or object can be measured. More formally, it is the number of lines required to span a region in space. Examples make dimensions much easier to understand.

A point is zero-dimensional. It has no length, width, thickness, or any other physical means of measurement. It only exists as a symbol to identify a single location in space.

A line is one-dimensional. It has the dimension of length. To put it another way, there is only one way that you can move along a line: lengthwise. In a similar vein, there is no way to move within a point. A point is a single location in itself, whereas a line is a collection of points, or locations.

A plane is two-dimensional. It has length and width. (Technically speaking, the property of width is really only length in a different direction). You can move along a plane in two directions, lengthwise and widthwise. You might think that you can actually move along a plane in an infinite number of directions, but actually every direction in which you move can be broken down into a component of length and a component of width.

A point is not a region in space, it is only a specific location. Therefore it takes zero lines to span it, and it is zero-dimensional.

One line is required to span a line (itself). Therefore a line is one- dimensional.

It requires two lines to span a plane, so therefore a plane is two- dimensional. These two lines represent length and width. Any point in the plane can be expressed as a combination of a certain length and a certain width, depending on the location of the point. The span of a line (or many lines) is the region that contains all the points that can be expressed as combinations of that line (or lines). A geometric space can also be spanned by points or planes.

Space:

All of these lie within space. Space is the collection of all points. It has no shape, and it has no limits. Space is three-dimensional; that is, it has length, width, and a new dimension, height. Again, height is only length in a different direction. Height is a measurement of length perpendicular to length and width. Imagine a box: It has length, width, and height. It encloses space.

A region in space can be either zero, one, two, or three-dimensional. A zero-dimensional region in space is a point. Instead of calling a point a region, because it is spanned by zero lines, it is more understandable to call it a location. Lines and planes are also regions in space. A three-dimensional object, like a ball of clay or a raindrop, also occupies a region in space. So does your shoe, your house, and your finger. Each is a collection of points.These four building blocks of geometry, points, lines, planes, and space, form the basis for all of the geometry you will study in this guide. It is important to understand their properties fully. Depictions of points, lines, and planes on paper (or computer screens) often are misleading because they appear to add dimensions to the basic building blocks. Points appear to have dimension, lines appear to have width, and planes appear to have thickness. It is critically important to remember now and forever the true nature of these basic elements of geometry, so that they don't mislead you in the future when things are much more difficult to visualize.