Download Foundations of Geometry

September 24, 2010
Foundations of Geometry
Important Vocabulary
Undefined Terms: a term that is used without a specific example or
description. We use these terms to describe other terms in geometry.
1) Point: a location. A point has neither shape nor size. When we name a
point, we name it by a capital letter.
Example:
W
2) Line: made up of points and has no thickness or width. There is exactly
one line through any two points. A line can either be named by two points on
that line or a lowercase script letter.
Example:
Y
k
X
3) Plane: a flat surface that extends indefinitely in all directions. There is
exactly one plane through any three points that are not on the same line. A
plane is named by a capital script letter or by any three points on that plane.
Example:
A
W
X
Y
September 24, 2010
Fundamental Definitions in Geometry
Definitions Related to Points:
· Space: the set of all points.
· Figure: any set of points.
· Plane Figure: all points lie on the same plane.
· Space Figure: a 3-D figure, sketched beyond the plane.
· Intersection: the set of all points common to two or more figures.
· Collinear Points: points that lie on the same line.
· Non-Collinear Points: points that do not lie on the same line.
· Coplanar Figures: figures that lie on the same plane.
· Non-Coplanar Figures: figues that do not lie on the same plane.
Definitions Related to Line Segments:
1) Line Segment (segment): a part of a line that begins at one point and ends at
another.
Example:
A
B
2) Length of a Segment: distance of a segment between its two endpoints.
3) Congruent Segments: segments that have the same exact distance.
Example:
5 cm
A
B
5 cm
C
D
4) Midpoint: divides a segment into two congruent segments.
Example:
2 in
C
2 in
M
D
September 24, 2010
Basic Postulates for
Points, Lines, and Planes
Postulates: statements that are considered true, even without proof.
· Unique Line Postulate: through any two points is exactly one line.
· Line Intersection Postulate: if lines intersect, then they intersect in
exactly one point.
· Plane Intersection Postulate: through any three non-collinear points,
there is exactly one plane.
Definitions Related to Rays
1) Ray: a part of a line that begins at one point and extends infinitely
(without end) in one direction. We name a ray by its endpoint and one other
point on that ray.
Example:
A
J
2) Endpoint of a Ray: the point at which the ray extends from.
3) Opposite Rays: two rays that meet together to form a line. Opposite
rays share a common endpoint and extend in opposite directions.
Example:
A
C
B
4) Angle: a figure formed by two rays with a common endpoint. Each ray is
the side of the angle and each endpoint is thevertex of the angle.
***A vertex of an angle is the place where the two sides of the angle
meet***
W
Example:
X
Y
September 24, 2010
Definitions Related to Angle Measure
1) Congruent Angles: Angles that are equal in measure.
A
Example:
T
C
O
E
M
2) Acute Angle: an angle whose measure is greater than 0
degrees, but less than 90 degrees.
L
Example:
Y
N
3) Obtuse Angle: an angle whose measure is greater than 90
degrees, but less than 180 degrees.
Example:
N
V
I
4) Right Angle: an angle whose measure is exactly 90 degrees.
Example:
O
M
A
5) Straight Angle: an angle whose measure is exactly 180
degrees.
Example:
P
T
A
Angle Pairs
1) Complementary Angles: Two angles whose measure has a
sum of 90 degrees. Each angle is said to be thecomplement of
each other.
Example:
Q
T
R
S
2) Supplementary Angles: Two angles whose measure has a
sum of 180 degrees. Each angle is said to be thesupplement of
each other.
W
Example:
D
R
E
3) Adjacent Angles: Two coplanar angles that share a common
side and a comment vertex, but have no interior points.
Example:
N
K
51
I
ο
28
ο
C
4) Linear Pair: When the two uncommon sides of adjacent
angles are opposite rays. A linear pair creates aline.
Example:
Y
A
N
D
5) Linear Pair Postulate: If two angles form a linear pair, then
they are supplementary.
6) Vertical Angles: Two angles that are directly opposite each
other. A pair of vertical angles are congruent.
Examples:
T
S
<SEP & <TEH are
vertical angles
P
E
H
September 24, 2010
Different Types of Lines
1) Parallel Lines: coplanar lines that never intersect. The
symbol for parallel lines is: ||
k
Example:
l
2) Perpendicular Lines: coplanar lines that intersect to form
right angles. The symbol for perpendicular lines is: ┴
Example:
m
h
3) Skew Lines: lines that are not on the same plane.
Example: