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Reference

Postulates About Points, Lines, and Planes

Rule

Two Point Postulate

Through any two points, there exists exactly one line.

This postulate is accepted without a proof.

Rule

Line-Point Postulate

A line contains at least two points.

This postulate is accepted without a proof.

Rule

Perpendicular Postulate

Given a line $ℓ,$ the Perpendicular Postulate states that there is one and only one line perpendicular to $ℓ$ that passes through a certain point $P .$

This postulate is one of the very basic truths used when proving and finding further characteristics of perpendicular lines.

Rule

Parallel Postulate

The parallel postulate states that for a point, $P,$ not on the line, $ℓ,$ there exists exactly one line through $P$ that is parallel to $ℓ .$

There exist several equivalent axioms to the parallel postulate but they all stem from the following statement.

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

One of the other equivalent axioms is the triangle interior angles theorem, which states that the sum of the interior angles of a triangle is always

18 0^{\circ} .

Rule

Plane Intersection Postulate

If two planes intersect, their intersection is a line.

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.

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.

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