Point, Line and Plane.
- Two points A and B determine a unique line,
to be denoted by AB.
- (The Distance Axiom). To every pair of distinct points
there corresponds a unique positive number, called their distance.
This distance satisfies the requirement of the next axiom.
- (The Ruler Axiom). Every line can be put in
one-one correspondence with the real numbers so that if
P and Q are two points on the line, then the absolute value
of the difference of the corresponding real numbers is
the distance between them.
- (The Ruler Placement Axiom). Given two points P and Q
on a line, the correspondence with real numbers in
the preceding axiom can be chosen so that P corresponds
to zero and Q corresponds to a positive number.
- There are at least three noncollinear points.
- (The Plane Separation Axiom). Given a line L.
Then the points not on L form two convex sets,
and any line segment AB joining a point A in one set
and a point B in the other must intersect L.
The convex sets are called the half-planes determined by L.
- (The Angle Measurement Axiom). To every
there corresponds a real number between 0 and 180,
to be denoted by mABC,
called the measure of the angle.
- (The Angle Construction Axiom). Given a line AB
and a half-plane H determined by AB, then for every number r
between 0 and 180, there is exactly one ray AP in H
so that mPAB = r.
- (The Angle Addition Axiom). If D is a point
in the interior of BAC, then
- (The Parallel Axiom). Through a given external point,
there is at most one line parallel to a given line.
- (The Area Axiom). To every polygonal region,
there corresponds a unique positive number,
called its area, with the following properties:
(i) congruent triangles have the same area;
(ii) area is additive on disjoint unions; and
(iii) the area of a rectangle is the product of
the lengths of its sides.
- (The Angle Supplement Axiom). If two angles
form a linear pair, then their measures add up to 180.
- SAS Axiom for congruence of triangles.
- SSS Axiom for congruence of triangles.
- ASA Axiom for congruence of triangles.
- (The AA Axiom for Similarity). Two triangles
with two pairs of angles equal are similar.