Download Geometry Definitions

Geometry Definitions
Marie Bruley
Math E-Geometry
Points
• Three Points C, M, Q
• A point is the most fundamental object in
geometry. It is represented by a dot and
named by a capital letter. A point represents
position only; it has zero size (that is, zero
length, zero width, and zero height).
Lines
• A line (straight line) It extends forever in two
opposite directions. A line has infinite length, zero
width, and zero height. The symbol ↔ written on
top of two letters is used to name a line. A line may
also be named by one small letter l.
Types of Lines
• PARALLEL lines- two lines that are always the
same distance apart, and will never intersect.
Parallel can be abbreviated as ||. An example
of parallel lines is on the Italian flag. Lines a
and b on the flag are parallel.
Types of Lines
• PERPENDICULAR LINES –
two lines that intersect
and form angles
measuring exactly 90
degrees, like the edges of
a building. If an angle
measures 90 degrees, a
square is place where the
lines intersect to show
that it is a right angle.
Perpendicular is often
abbreviated as _|_.
• Line a _|_ b reads as line a
is perpendicular to line b.
Plane
• A plane may be considered as a set of points
forming a flat surface.
Line Segment
• We may think of a line segment as a "straight"
line that we might draw with a ruler on a
piece of paper. A line segment does not
extend forever, but has two distinct endpoints.
We write the name of a line segment with
endpoints A and B as AB . Note how there are
no arrow heads on the line over AB such as
when we denote a line or a ray.
B
A
Intersecting lines
• Intersecting lines come together at a point.
Example point M.
Ray
• We may think of a ray as a "straight" line that
begins at a certain point and extends forever
in one direction.
• The point where the ray begins is known as its
endpoint.
• We write the name of a ray with endpoint A
and passing through a point B as .
Angles
• Two rays that share the same endpoint form
an angle. The point where the rays intersect is
called the vertex of the angle. The two rays
are called the sides of the angle.
Angles, Cont.
• We can name an angle by using a point on
each ray and the vertex. The angle below may
be named angle ABC or as angle CBA; you may
also see this written as ABC or as CBA.
Degrees: Measuring Angles
• We measure
the size of an
angle using
degrees.
• The
Protractor
Postulate
Types of Angles
• Acute Angle:
Measures between
0° and 90°
• Right Angle:
Measure of 90°
• Obtuse Angle:
Measure between
90° and 180°
• Straight Angle:
Measure of 180°
Angle Relationships
• Complementary Angles: Two
angles are called complementary
angles if the sum of their degree
measurements equals 90 degrees.
• Supplementary Angles: Two angles
are called supplementary angles if
the sum of their degree
measurements equals 180
degrees.
Angle Relationships
• Congruent Angles: Angles
with equal measures.
• Adjacent Angles: Share a
vertex and a common side
but no interior points.
• Bisector of an angle: a ray
that divides the angle into
two congruent angles. In
this picture OY is the
angle bisector.
2
13
3
1
14
4
5
9
7
6
8
12
10
11
Postulates
• A statement that is accepted without proof.
• Usually these have been observed to be true
but cannot be proven using a logic argument.
Postulates Relating Points, Lines,
and Planes
• Postulate 5: A line contains at least two points; a plane
contains at least three points not all in one line; space
contains at least four points not all in one plane.
Postulates Relating Points, Lines,
and Planes
• Postulate 6: Through any two points there is
exactly one line.
Postulates Relating Points, Lines,
and Planes
• Postulate 7: Through any three points there is
at least one plane (if collinear), and through
any three non-collinear points there is exactly
one plane.
Postulates Relating Points, Lines,
and Planes
• Postulate 8: If two points are in a plane, then
the line that contains the points is in that
plane.
A
.
B
.
Postulates Relating Points, Lines,
and Planes
• Postulate 9: If two planes intersect, then their
intersection is a line.
Theorems
• Theorems are
statements that have
been proven using a
logic argument.
• Many theorems follow
directly from the
postulates.
Theorems Relating Points, Lines,
and Planes
• Theorem 1-1: If two lines intersect, then they
intersect in exactly one point.
• Theorem 1-2: Through a line and a point not
in the line there is exactly one plane.
• Theorem 1-3: If two lines intersect, the
exactly one plane contains the lines.