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Reference

Postulates About Points, Lines, and Planes

Rule

Two Point Postulate

Through any two points, there exists exactly one line.

Line passing through A and B

This postulate is accepted without a proof.
Rule

Line-Point Postulate

A line contains at least two points.

A line l with two points that lie on it

This postulate is accepted without a proof.
Rule

Line Intersection Postulate

If two lines intersect, their intersection is exactly one point.

Intersection of two lines

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.

Two lines with two points of intersection

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.

Two overlapping lines

However, coincidental lines cannot be distinguished from each other, so they cannot be considered two different lines. Therefore, any two intersecting lines have just one point of intersection. The lines are first infinitely far from each other, then get closer and closer until they intersect. After this point, they move farther and farther away from each other again.

Two intersecting 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.

Line l, point P not on l, and a perpendicular intersecting 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.

Line l, point P not on l, and a parallel line through P
This postulate is accepted without a proof.

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.

The applet visualize this postulate. If the sum of ∠ A and ∠ B is less than 180^(∘), then according to the postulate, if these lines are extended, they will intersect eventually.

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.

Rule

Plane Intersection Postulate

If two planes intersect, their intersection is a line.

Intersection of two planes
This postulate is accepted without a proof.
Rule

Plane-Point Postulate

Any plane contains at least three noncollinear points.

Plane with three points and a line through two of them
This postulate is accepted without a proof.
Rule

Plane-Line Postulate

If a plane contains two points, then it contains the line passing through the points.

A plane containing two points and the line through them

This postulate is accepted without a proof.

Why
Consider two points on a plane and the line passing through them.

A plane containing two points and the line 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.

A line through two points on a plane and a third point not contained in 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.

Three points and a plane through them

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.

Two planes through the line containing points A and B

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.

One plane through noncollinear points A, B, and the third unnamed point

This means that there is only one plane that passes through any three noncollinear points.
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