Mini-lesson on Points, Lines, and Planes
(Math 214 Week 2 Chapter 9)
POINTS
A point indicates position—it has no dimensions. To draw a
point, we generally draw a small, filled-in circle such as the
graphic to the right. We use capital letters of the alphabet
to name and label a point. For example, the point to the right
is Point P. We usually use the letters P, R, Q, or A, B, C,
and X, Y, Z to name points; however, any capital letter of the
alphabet may be used.
P
LINES
A line only possesses length. Lines go forever and ever in
both directions but have no height. To draw a line, we draw a
line that is very thin (to represent no thickness) and we
place arrows on each end of the line to signify that lines go
forever and ever in both directions (see the graphic to the
right). Another definition of a line is “an infinite
collection of collinear points,” but this definition is
somewhat circular.
Lines can be named and labeled two different ways:
1. We use a lowercase, cursive letter of the alphabet to
name and label a line. We usually use the letters l, m,
and n; however, any lowercase, script letter of the
alphabet may be used. The graphic to the right shows
line m.
2. We can identify any two points on the line to name and
label a line. We name the line by the two points. We use
either the word “line” or the double-headed arrow above
the two named points to signify the line. Note that any
two points identify a unique line. Thus, the graphic to
the right shows line PQ or PQ.
A line segment is a portion of a line with a distinct start
and a distinct end. Thus, a line segment has two endpoints. We
name a line segment by listing the the two endpoints. We use
either the word “segment” or a segment line above the two
endpoints. Thus, the graphic to the right shows line segment
PQ or PQ.
m
Q
P
Q
P
A ray is a portion of a line with a distinct start but no end. Some people
define a ray as half of a line but this is not entirely correct because a
line goes on forever and ever (to infinity) in both directions. There is no
such number as “half of infinity.” We name a ray by the one endpoint and any
other point on the ray. We place a ray symbol (a one-ended arrow) above the
two points. Be careful when writing the ray. In line PQ above, we could have
written line QP, and we would be referring to the same line.
In line segment PQ above, we could have written line segment
QP and we would be referring to the same line. However, ray
PQ and ray QP are entirely different rays. It is best to use
the ray symbol above a ray’s name so it is clear which of the
two points the ray’s endpoint is. Thus, the graphic to the
right shows ray PQ or PQ.
Q
P
An angle is where two rays have the same endpoint. Angles are named by a
point on one ray, the shared endpoint, and a point on the other ray. The
symbol for an angle is . Thus, the graphic to the right
shows angle QPR or QPR. Remember that the point listed in
Q
the middle of the three points is the vertex of the angle or
the endpoints of both rays. If you see the angle symbol and
P
only one letter, that is another way to write the angle
(using just the endpoint); however, this is a sloppy way to
R
identify an angle because an infinite number of rays emanate
from each endpoint. So, a sloppy way of writing the angle to
the right is P.
Angles are the subject of much study in algebra and geometry. You will
encounter terms such as
1. complementary
2. supplementary
3. acute
4. right
5. obtuse
6. straight
7. parallel (lines)
8. perpendicular (lines)
PLANES
A plane is a flat surface consisting of infinitely many points. To draw a
plane, we generally draw a parallelogram such as the graphic
to the right. Many people think of a sheet of paper when they
want to visualize a plane. A sheet of paper is a good
representation of a plane, as long as we remember that a plane
has no width and goes forever and ever in all directions.
We use lowercase, letters of the Greek alphabet to name and
label a plane. For example, the plane to the right is plane
alpha (α).
α
A unique plane is created by
1. any three points not in a straight line. Such a plane can be named by
those three (non-collinear) points.
2. a line and a point not on that line.
3. two intersecting lines.
4. two parallel lines.
Intersection
It is helpful to know what happens when two of the geometric objects (lines
and/or planes) intersect ().
1. Two planes intersect to form a line (unless they are parallel planes).
2. A line and a plane intersect to form a point (unless they are parallel
or unless the line is contained ON the plane).
3. Two lines on the same plane intersect to form a point (unless they are
parallel).
Using this Information
Refer to this diagram to answer the following questions:
1. How many planes do you see? What are the names of those
planes?
2. How many lines do you see? What are the names of those
lines?
3. How many points do you see? What are the names of those
points?
4. α  β
5. β  ADF
6. AF  BC
7. AF  BC
8. AF  BC
9. AF  BC
10. CEF  CF
11. BD  BA
12. BD  BA
(note that this is UNION, not INTERSECTION)