MATH 3181 - 001

for Plane Geometry

The Euclidean plane will be denoted by **E**^{2}
and the hyperbolic plane will be **H**^{2}.
If we are referring to either of them, without specificity, we
will use the notation **P**.

The plane is a set of points (undefined terms) and a line is a
subset of the plane subject to the axioms that follow. If _{} is
a line and *A* is a point in that subset, we say that *A
lies on _{}* or

**Axiom 1:** If *A* and *B* are distinct points, then there
is one and only one line that passes through them both.

**Axiom 2:** There is a function *|PQ|*
defined for all pairs of points *P,Q* such that

and

if and only if P=Q. This function satisfies the following properties:

for all triples of points *A*, *B*,and *C*.

This axiom makes the plane into a *metric space* and *|PQ|*
is called the *distance* between *P* and *Q*.

**Axiom 3:** For each line , there is a
one-to-one function, _{}, such that if *A*
and *B* are any points on _{}, then
|*AB*| = |*x(A) - x(B)*|.

This mapping is not unique because we can use this same mapping
and add a constant. On the other hand, it does say that each
of our lines is *isometric* to the set of real numbers and,
hence, are all isometric to one another.

This mapping _{} is a *coordinatization*
of the line. That is, it defines a coordinate system on the line
_{}. There is a
point that corresponds to 0 and there are points
that correspond to the positive real numbers and points that correspond
to the negative real numbers. If A and B are points of _{},
then the segment *AB* of _{} is the set of points

and the number |*AB*| is its *length*.
The points *A* and *B* are its *endpoints*. When we have a
coordinatization, we say that the line _{} is a
*directed line*. If _{},
the set of points *A* such that _{}
is a *ray* _{} with *Z*
as its *origin* or *initial point*. The set of points
*A* such that _{}
is called the *opposite ray*.

**Axiom 4:** If _{} is any line,
there are two corresponding subsets *HP*_{1} and *HP*_{2},
called *half-planes*, such that the sets , *HP*_{1},
and *HP*_{2} are disjoint, and the union of these sets and
_{} is all of **P**, such that:

- If
*P*and*Q*are in the same half-plane, the segment*PQ*contains no point of_{}, while - if
*P*and*Q*are in opposite half-planes, the segment*PQ*contains exactly one point of_{}.

We say that P and Q are on the same side of or on opposite sides
of _{}.

Two rays with a common origin, *A*, constitute
an *angle* with A as its *vertex*. Other authors insist
that the two rays are not opposite rays, but no such restriction
is made here. If the two rays are not the same ray nor opposite
rays, then the interior of the angle is the set of all points
on the same side of _{} as *C*
and on the same side of _{} as *B*.
The interior of the triangle ABC is the intersection of the interiors
of its three angles. If two lines intersect at the point A then
the two angles that share a common side are called *supplementary
angles* and the two angles that do not share a common side
are called *vertical angles*.

**Axiom 5:** (Measures of angles) For each
angle _{} there is a number _{}
in the interval _{}, called the (radian)
*measure* of the angle such that

- If the two rays comprising the angle are the same ray, the
measure is 0; if they are opposite rays, the measure is
_{}; - the sum of the measures of an angle and its supplement is
_{}; - if
_{}is in the interior of_{}then_{}; - if a ray from the point
*A*lies in a line then in each half-plane bounded by_{}, the set of rays from A is in a one-to-one correspondence with the set of real numbers in_{}; - if the ray in part (d) contains the point P, then the angle measure depends continuously on P.

Two segments with the equal lengths are called congruent segments and two angles with equal measures are called congruent angles. A right angle is an angle that is congruent with its supplement.

Given three non-collinear points A, B, and C the triangle ABC is the union of the three segments AB, BC, and AC. The points A, B, and C are called the vertices, the segments are called the sides and the three angles defined by these segments are called its angles.

**Axiom 6:** (SAS Axiom) If two sides and
the included angle of a first triangle are congruent respectively
to two sides and the included angle of a second triangle, then
the triangles are congruent. That is, the remaining sides are
congruent and the remaining angles are congruent in pairs.

**Parallel Axiom:** There are two distinct choices for the
Parallel Axiom:

(Euclidean Parallel Axiom) Given any line and any point not on that line, there is only one line through the given point that never intersects the given line.

(Hyperbolic Parallel Axiom) There exists a line and a point not
on that line such that there are at least two lines through that
point that do not intersect the given line.

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Last updated 8/27/96 by

David Royster david.royster@uky.edu