1.1

Question 1

Points

Answer 1

We’ll use Euclid’s definition. Sayeth Euclid: “A
point is that which has no part.” We name
points with capital letters. We picture them with
dots.

Question 2

Definitions, Diagrams, Names

Answer 2

In this section I will give you definitions for a set
of related geometrical terms. They are “point”,
“line”, “collinear”, “segment”, “ray”, “polygon”
and “intersection”.

Question 3

Everything Else

Answer 3

Everything else that we will study in this class –
lines, triangles, quadrilaterals, circles and all the
rest – are collections of points. They are, as it
were, points in the plural. They are many points.

Question 4

Lines

Answer 4

Again we’ll follow Euclid. Sayeth Euclid: “A line is
limitless, breadthless length”. We name lines
with lowercase letters – “a”, “b”, “c” etc.
Typically we don’t place arrows on the ends of a
picture of a line; instead context makes clear
that we mean a line.

Question 5

Postulates

Answer 5

So we now have a pair of definitions, the definition
of “point” and of “line”. We will find that we must
also make assumptions about the behavior of the
objects we’ve defined and about their relationships.
We call these assumptions “postulates”.

Do not be dismayed by the word “assumption”. I do
not ask to you take them on faith. Instead you’ll find
them obviously true.

Question 6

The Line Existence Postulate

Answer 6

The first of our postulates is the Line Existence
Postulate. It says:

For any pair of points, a line exists that passes
through them. Moreover that line is unique;
that is, only one line passes through a pair of
points.

Question 7

A Second Way to Name Lines

Answer 7

Since the line through a pair of points is unique,
we may name a line by a pair of points that lie
on it. For instance, we may name the line below
line AB.

Question 8

An Obvious Point

Answer 8

Of course I don’t mean to say that only A and B
will do. Any pair of points on the line will work
just as well. Thus the line below may be named
AB, AC or BC.

Question 9

A Second Obvious Point

Answer 9

The order of letters in a line name is irrelevant.
The line through points A and B is the same as
the line through points B and A. Symbolically:AB = BA

Question 10

Collinear Defined

Answer 10

We say that two or more points are collinear
when they all lie on a single line.

Question 11

Line Segment Defined

Answer 11

A line segment is two points on a line and all
points on that line between them. We call those
two points the end-points of the segment.

Question 12

Segment Names

Answer 12

To name a line segment, we write the names of
its two endpoints side-by-side and then place
the segment symbol on top. Order is irrelevant.

Name: AB or BA

Question 13

Between Defined

Answer 13

We say that a point is between two others when
it lies in the interior of the segment they form.

Point O is between
points M and N. Point P
is not between points
M and N, nor is N
between M and O.

Question 14

Ray Defined

Answer 14

A ray is a point on a line and all points on it that
lie on one side of that point. A ray has a single
end-point.

Question 15

How to Name a Ray

Answer 15

First write the name of its endpoint and then
the name of any second point on it. Over the
two, place the ray symbol. Order matters!

Question 16

Polygon Defined

Answer 16

A polygon is a collection of segments joined at
their endpoints. We call those segments sides;
we call the points where they join the vertices.

Sides: AB, BC, CD, DA
Vertices: A, B, C, D

Question 17

Polygon Types

Answer 17

You‘re no doubt familiar with the usual names: a
triangle is a three-sided polygon, a quadrilateral
is a four-sided polygon, etc.

We can also use names of the form “n-gon”. So a
triangle can be called a 3-gon, a quadrilateral a
4-gon, etc.

Question 17

Polygon Names

Answer 17

To name a polygon, we choose a vertex and a
direction; and then we travel around the
polygon from that vertex in that direction and
list off vertex names as we find them.

Valid names: ABCD, CDAB,
DCBA
Invalid name: BDCA

Question 18

Intersection Defined

Answer 18

The intersection of two objects is the set of
points that lie on both; that is, it’s what’s
common to them. (Yes, I mean all of it, however
much that is.) Below the intersection of lines a
and b below is the point Z.