# 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