A Quick Review of Elementary Euclidean Geometry

0.1 MEASUREMENT AND CONGRUENCE
0.2 PASCH’S AXIOM AND THE CROSSBAR THEOREM
0.3 LINEAR PAIRS AND VERTICAL PAIRS
0.4 TRIANGLE CONGRUENCE CONDITIONS
0.5 THE EXTERIOR ANGLE THEOREM
0.6 PERPENDICULAR LINES AND PARALLEL LINES
0.7 THE PYTHAGOREAN THEOREM
0.8 SIMILAR TRIANGLES
0.9 QUADRILATERALS
0.10 CIRCLES AND INSCRIBED ANGLES
0.11 AREA

This preliminary chapter lays out the basic results from elementary Euclidean
geometry that will be assumed in the course. Those readers who are using this book
as a supplement to a course in the foundations of geometry should omit the chapter
and simply refer to it as needed for a summary of the notation and terminology that
will be used in the remainder of the book. Readers who are using this book as a
stand-alone text in Euclidean geometry should study the chapter carefully because
the material in this chapter will be used in later chapters.

The theorems stated in this chapter are to be assumed without proof; they
may be viewed as an extended set of axioms for the subject of advanced Euclidean
geometry. The results in the exercises in the chapter should be proved using the
theorems stated in the chapter. All the exercises in the chapter are results that will
be needed later in the course.

We will usually refer directly to Euclid’s Elements when we need a result
from elementary Euclidean geometry. Several current editions of the Elements are
listed in the bibliography (see [4], [5], or [10]). The Elements are in the public
domain and are freely available on the world wide web. Euclid’s propositions are
referenced by book number followed by the proposition number within that book.
Thus, for example, Proposition III.36 refers to the 36th proposition in Book III of
the Elements.

0.1 MEASUREMENT AND CONGRUENCE
For each pair of points A and B in the plane there is a nonnegative number AB,
called the distance from A to B. The segment from A to B, denoted consists of A
and B together with all the points between A and B. The length of   is the distance
from A to B. Two segments and are congruent, written AB > ,  if they
have the same length. There is also a ray and a line

FIGURE 0.1: A segment, a ray, and a line

For each triple of points A, B, and C with A ≠ B and A ≠ C there is an angle,
denoted that is defined by The measure of the angle
is a number We will always measure angles in degrees and assume that
The measure is 0° if the two rays and are equal; the
measure is 180° if the rays are opposite; otherwise it is between 0° and 180° . An
angle is acute if its measure is less than 90° , it is right if its measure equals 90° , and
it is obtuse if its measure is greater than 90° . Two angles are congruent if they have
the same measure.

The triangle with vertices A, B, and C consists of the points on the three
segments determined by the three vertices; i.e.,

The segments are called the sides of the triangle ΔABC. Two
triangles are congruent if there is a correspondence between the vertices of the first
triangle and the vertices of the second triangle such that corresponding angles are
congruent and corresponding sides are congruent.

Notation. It is understood that the notation ΔABC ΔDEF means that the two
triangles are congruent under the correspondence A ↔  D, B ↔ E, and C ↔ F. The
assertion that two triangles are congruent is really the assertion that there are six
congruences, three angle congruences and three segment congruences. Specifically,
ΔABC ΔDEF means
  In high school this is often abbreviated
CPCTC (corresponding parts of congruent triangles are congruent).

0.2 PASCH’S AXIOM AND THE CROSSBAR THEOREM
The two results stated in this section specify how one-dimensional lines separate
the two-dimensional plane. Neither of these results is stated explicitly in Euclid’s
Elements. They are the kind of foundational results that Euclid took for granted.
The first statement is named for Moritz Pasch (1843–1930).

Pasch’s Axiom. Let ΔABC be a triangle and let ℓ be a line such that none of the
vertices A, B, and C lie on ℓ.If intersects then ℓ also intersects either or
(but not both).

Let A, B, and C be three noncollinear points. A point P is in the interior of
  if P is on the same side of as C and on the same side of as B.

Note that the interior of is defined provided
It would be reasonable to define the interior of to be the empty set in case
but there is no interior for an angle of measure 180 . The segment
is called a crossbar for

Crossbar Theorem. If D is in the interior of then there is a point G such that
G lies on both and

0.3 LINEAR PAIRS AND VERTICAL PAIRS
Angles and form a linear pair if A, B, and C are collinear and A is
between B and C.

Linear Pair Theorem. If angles and   form a linear pair, then

Two angles whose measures add to 180° are called supplementary angles or
supplements. The Linear Pari Theorem asserts that if two angles form a linear pair,
then they are supplements.

Angles and form a vertical pair (or are vertical angles) if rays
and are opposite and rays and are opposite or if rays and are
opposite and rays and are opposite.

Vertical Angles Theorem. Vertical angles are congruent.

FIGURE 0.2: A linear pair and a vertical pair

The linear pair theorem is not found in the Elements because Euclid did
not use angle measure; instead he simply called two angles ‘‘equal’’ if, in our
terminology, they have the same measure. The vertical angles theorem is Euclid’s
Proposition I.15.

0.4 TRIANGLE CONGRUENCE CONDITIONS
If you have two triangles and you know that three of the parts of one are congruent
to the corresponding parts of the other, then you can usually conclude that the other
three parts are congruent as well. That is the content of the triangle congruence
conditions.

Side-Angle-Side Theorem (SAS). If ΔABC and ΔDEF are two triangles such that
then ΔABC ΔDEF.

Euclid used his ‘‘method of superposition’’ to prove SAS (Proposition I.4),
but it is usually taken to be a postulate in modern treatments of geometry. The next
two results (ASA and AAS) are both contained in Euclid’s Proposition I.26 and the
third (SSS) is Euclid’s Proposition I.8.

Angle-Side-Angle Theorem (ASA). If ΔABC and ΔDEF are two triangles such
that then ΔABC ΔDEF.

Angle-Angle-Side Theorem (AAS). If ΔABC and ΔDEF are two triangles such
that then ΔABC ΔDEF.

Side-Side-Side Theorem (SSS). If ΔABC and ΔDEF are two triangles such that
then ΔABC ΔDEF.

There is no Angle-Side-Side condition, except in the special case in which the
angle is a right angle.

Hypotenuse-Leg Theorem (HL). If ΔABC and ΔDEF are two right triangles with
right angles at the vertices C and F, respectively, then
ΔABC ΔDEF.

EXERCISES
0.4.1. Use SAS to prove the following theorem (Euclid’s Proposition I.5).
Isosceles Triangle Theorem. If ΔABC is a triangle and then

0.4.2. Draw an example of two triangles that satisfy the ASS condition but are not
congruent.
0.4.3. The perpendicular bisector of a segment is a line ℓ such that ℓ intersects at
its midpoint and Prove the following theorem.
Pointwise Characterization of Perpendicular Bisector. A point P lies on the
perpendicular bisector of if and only if PA = PB.
0.4.4. The angle bisector of is a ray such that is between and and
The distance from a point to a line is measured along a
perpendicular. Prove the following theorem.
Pointwise Characterization of Angle Bisector. A point P lies on the bisector of
if and only if P is in the interior of and the distance from P to
equals the distance from P to