*Sharon Kim*

# Properties and Postulates of Geometric Figures

## Postulates

**Postulates** are basic truths that do not require formal proofs to prove that they are true. Instead, these are used to prove other theorems to be true. Geometry is actually built on just a few basic truths or postulates. These deal with the very basic point, line, and plane.

Why do these postulates matter? They are important for you to know because these form the building blocks of geometry. Without knowing these, you won't really know or understand how geometry works. Once you know and understand these postulates, you can get a better and easier feel for geometry. Watch and see if you can easily remember all of them.

## Points

A point is simply a dot. In geometry, a point has no dimensions. It has no height, width, or length. The postulates that refer to points talk about how they form lines and planes.

One postulate says that given any two points, there is exactly one line that will pass through both points. You can remember this postulate easily by drawing two points and you will see that there is only one line you can draw to connect the two points together.

Another postulate says that for any three non-collinear points, there will be exactly one plane that will pass through all three points. 'Non-collinear' means that the points are not all on the same line. To remember this, picture any three points in space and picture placing a giant flat piece of paper so that the sheet touches all three points. You will see that there is only one way to do that.

Yet another postulate tells us that for both lines and planes, there will be at least one point that is not on the line or the plane, respectively. If the line belongs in one particular plane, there will be at least one point not on the line but that also belongs in the plane. It is similar for the plane. If the plane belongs in one space, there will be a point in the same space but is not in the plane. You can remember this by just picturing a stick or a sheet of paper. Can you find any other point that you can point to that is either not on the stick or the sheet of paper?

## Lines

A line is any straight line or mark that extends forever. The postulates that deal with lines talk about how they are linked with a number line as well as how they behave in relation to planes.

The postulate that mentions the number line says that any line can be a number line. Any point on the line can be a 0 and any other point can be a 1. Remember this one by thinking of how you draw a number line. What do you start with? And also what is the second word in the word 'number line?'

Another postulate regarding lines says that if the two points that make up a line belong to a particular plane, then the line also belongs in the plane. To remember this one, think about two points and the line that connects them. What happens when you use a sheet of paper to connect the two dots? Doesn't the sheet of paper also contain the line?

## Planes

A plane is like a flat sheet of paper that extends forever. It's like a giant whiteboard on which you can draw anything you want and where you won't ever run out of space. The postulate that talks about planes talks about how they behave with each other.

This postulate tells us that if we have two planes that intersect each other, the intersection will be a line. Remember this one by picturing one sheet of paper cutting into another sheet of paper. The place where they end up cutting each other will be a line.

## Lesson Summary

So, what have we learned? We've learned that geometry is based on **postulates**, basic truths that do not require formal proof. These postulates are the basis or the evidence for other theorems. They are the building blocks of geometry. They deal with the basic shapes of a point, line, and plane in geometry.

The postulates about lines tell us that two points make a line while three non-collinear points make a plane. Another postulate tells us for both lines and planes there will be at least one point that is not on the line or the plane. For lines, the postulates tell us that any line can be a number line and if the two points that make up the number belong in one plane, the line also belongs in the plane. For planes, the postulates tell us the place where two planes intersect will always be a line.

## Lesson Outcome

Completing this video lesson could provide you with the knowledge needed to:

- Describe the basic mathematical postulates that deal with points, lines and planes in geometry
- Provide examples of a geometric plane
- Understand the meaning of non-collinear

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