Sign In
An error ocurred, try again later!
Postulates About Points, Lines, and Planes
Slide 1 of 0
Reference
Postulates About Points, Lines, and Planes
Rule
Two Point Postulate
Rule
Line-Point Postulate
Rule
Line Intersection Postulate
If two lines intersect, their intersection is exactly one point.
This postulate is accepted without a proof.
Why
Suppose that two lines have two points of intersection. For that to be true, one line would have to bend to cross the other line again.
This is not possible because lines are straight by definition. The only possibility for two lines to have more than one point of intersection is if the lines overlap. If this is true, the two lines would have infinitely many common points — they would be coincidental lines.
Rule
Perpendicular Postulate
Given any straight line ℓ and a point P not on the line, there is exactly one line through P that is perpendicular to ℓ.
At any given point, only one perpendicular line can be constructed through another line. This postulate is accepted without a proof. It is one of the basic truths used when proving and finding further characteristics of perpendicular lines.
Rule
Parallel Postulate
Given any straight line ℓ and a point P not on the line, there is exactly one line through P that is parallel to ℓ.
This postulate is accepted without a proof.
In the case of two lines and their transversal, the lines are parallel only if the sum of their interior angles is 180∘. Based on Euclid's postulate, John Playfair proposed that only one line can be constructed through a given point that will be parallel to a given line. All other lines will eventually intersect with that original line.
Extra
The postulate above, proposed by John Playfair, is an equivalent version of Euclid's Fifth Postulate.
|
Euclid's Fifth Postulate |
|
If a line segment crosses two lines in such a way that the sum of their interior angles on the same side is less than 180∘, then the lines will eventually meet. |
Rule
Plane Intersection Postulate
Rule
Plane-Point Postulate
Rule
Plane-Line Postulate
If a plane contains two points, then it contains the line passing through the points.
This postulate is accepted without a proof.
Why
Consider two points on a plane and the line passing through them.
If the line is not contained in the plane, then there must be a point that lies on the line but not on the plane. Planes have infinite width and length but no height, so for this to be the case, the line must curve away from the plane.
However, a line is always straight — it cannot curve away from the plane. Therefore, if a plane contains two points, it must contain the line passing through them.
Rule
Three Point Postulate
Given any three noncollinear points, there exists exactly one plane that contains them all.
This postulate is accepted without a proof.
Why
Infinitely many planes pass through two given points. Any plane that contains the two points can be rotated about the line that connects the points. This will generate new planes that contain the two points.
Adding a third point allows a single plane to be specified. If two different planes containing the three points existed, they would overlap and, therefore, be indistinguishable. 
This means that there is only one plane that passes through any three noncollinear points.
Exercises
Recommended exercises
To get personalized recommended exercises, please login first.
Loading content
Loading content