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**Subsections**

What is a triangle? Is a triangle a physical
object made up of 3 straight pieces of wood or metal or somesuch, joined
at the corners, or is it an ideal object consisting of lines that have no
width lying in a plane that has no thickness?

Historically, all lengths and
angles are somehow constructible. That is, they are abstract objects that,
in some sense are capable of being realized as physical objects. We will
take a somewhat different point of view; we will assume familiarity with
real numbers, and with the correspondence between real numbers and points on
a line.

The plane, lines, points, length and distance,
angle measure.

- Points lie on lines.
- If two distinct points lie on a line, then the length of the line segment
between these points is well defined.
- A point separates a line into two half-lines.
- Two distinct points on the same line separate it into a line segment,
which has a length (the length is a positive number), and two half-lines.
- A line separates the plane into two half-planes, which are regions (in
modern terms, a region is a connected open subset of the plane).
- Two distinct lines either meet at a
point, or are disjoint, in which case they are
*parallel*.

These are the first few; a few more will
follow. The reader should be aware that the numbering of the axioms, as
well as the theorems, propositions, etc. is unique to these notes. It is
also possible to have the same notion of planar geometry with a slightly
different collection of axioms, but these are what we shall use.

**Axiom 1**
*Two distinct lines intersect in at most one point.*

**Axiom 2**
*Any two distinct points lie on a line.*

**Axiom 3**
*If two lines intersect in a point, they separate the
plane into 4 regions, called **sectors*, and they define an angle in
each of these sectors; the sum of the measures of any two adjacent angles is
.

**Axiom 4**
*If two lines do not intersect, they divide the plane
into three regions, with exactly one of them, the one between the two lines,
having both lines on its boundary. *

**Exercise 1.1**
*Using the above axioms, show that given any two distinct points, there is
exactly one line that contains them both.*

If and are distinct points, then the (unique) line on which
they lie is denoted by . The line segment between and is
also denoted by ; this should cause no confusion.
The length of the line segment is denoted by .
If and are distinct lines or line segments, the angle between them
is denoted by
, and its measure is denoted by
. We may use to denote an angle which has the point
at its vertex, if it is clear from the context which angle is
being referred to. It is also not unusual to use Greek letters such as
, , , , etc. to denote both angles and
their measures.

**Theorem 1.1** (Vertical angles)

*Vertical angles are equal. That is, if two distinct lines
intersect at a point, the measure of the angles of any two non-adjacent
sectors is equal.*

**Exercise 1.2**
*Prove Thm. 1.1. You may use any of the axioms above,
along with logical axioms and results for real numbers (since angle
measure is a real number).*

(more will follow)
- Any line segment can be extended in either direction, or in both
directions.
- If , then there is a point on the line , where
lies between and , so that
(see also Axiom ).
- If and are points on the line , and we are
given an angle
, where
, then we can
construct points and , one on each side of the line , so that
.
(see also Axiom ).

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Scott Sutherland
2002-12-18