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Basic Geometry
Math 1-Unit 1A
Geometry
• The word geometry comes from Greek
words meaning “to measure the Earth”
• Basically, Geometry is the study of shapes
and is one of the oldest branches of
mathematics
The Greeks and Euclid
• Our modern understanding of geometry
began with the Greeks over 2000 years ago.
• The Greeks felt the need to go beyond
merely knowing certain facts to being able
to prove why they were true.
• Around 350 B.C., Euclid of Alexandria
wrote The Elements, in which he recorded
systematically all that was known about
Geometry at that time.
Undefined Terms?
•
The terms points, lines, and planes are the foundations of
geometry, but…
•
•
point, line, and plane are all what we call undefined terms.
•
How can that be?
•
Well, any definition we could give them would depend on the
definition of some other mathematical idea that these three terms help
define. In other words, the definition would be circular!
Point
• Has no dimension
• Usually represented by a small dot
A
The above is called point A. Note the point
is represented with a capital letter.
Line
• Extend in one dimension.
• Represented with straight line with two arrowheads to
indicate that the line extends without end in two directions.
This is Line l, (using the lower case
script letter) or symbolically we call it
AB
l
A
B
NOTICE: The arrowheads are in
both directions on the symbol
Plane
•Extend in two dimensions.
•Represented by a slanted 4 sided figure, but you
must envision it extends without end, even though
the representation has edges.
A
B
M
C
This is Plane M or plane ABC (be
sure to only use three of the
points when naming a plane)
Undefined Concepts
• Collinear points are points that lie on the same line.
l
B
C
A
Points A, B and C are collinear.
Undefined Concepts
• Coplanar points are points that lie on the same plane.
A
B
C
Points A, B and C are coplanar.
Line Segment
•Let’s look at the idea of a point in between two other points on a line.
Here is line AB, or recall
symbolically AB
A
B
The line segment does not
extend without end. It has
endpoints, in this case A and
B. The segment contains all
the points on the line
between A and B
This is segment AB
Notice the difference in
the symbolic notation!
Ray
Let’s look at a ray:
A is called the initial
point
A
B
Ray AB extends in
one direction
without end.
•Symbolized by AB
The initial point is
always the first
letter in naming a
ray. Notice the
difference in
symbols from both
a line and segment.
Symbol alert!
•Not all symbols are created equal!
AB
is the same as
BA
A
B
AB
is the same as
BA
A
B
BUT…
Symbol alert!!
The ray is different!
AB
is not the same as
Initial point 1st
BA
A
B
A
B
AB
BA
Notice that the initial point is listed first in the symbol. Also note
that the symbolic ray always has the arrowhead on the right
regardless of the direction of the ray.
Opposite Rays
•If C is between A and B,
A
C
then CA and
B
CB are opposite rays.
C is the common initial point for the rays!
Angles
•Rays are important because they help us define something very
important in geometry…Angles!
•An angle consists of two different rays that have the same initial
point. The rays are sides of the angles. The initial point is called the
vertex.
vertex
B
sides
A
C
Notation: We denote an angle with
three points and  symbol. The
middle point is always the vertex.
We can also name the angle with
just the vertex point. This angle can
be denoted as:
BAC , CAB, or A
Classifying Angles
•Angles are classified as acute, right, obtuse, and straight,
according to their measures. Angles have measures greater
than 0° and less or equal to 180°.
A
Straight angle
A
A
A
Acute angle
Right angle
Obtuse angle
0°< m  A < 90°
m  A = 90°
90°< m  A < 180°
m
A = 180°

Adjacent angles are “side by side” and share a
common ray.
15º
45º
These are examples of adjacent angles.
80º
45º
35º
55º
130º
85º
20º
50º
These angles are NOT adjacent.
100º
50º
35º
35º
55º
45º
Supplementary angles add up to 180º.
40º
120º
60º
Adjacent and Supplementary
Angles
140º
Supplementary Angles
but not Adjacent
Complementary angles add up to 90º.
30º
40º
50º
60º
Adjacent and Complementary
Angles
Complementary Angles
but not Adjacent
Naming Geometric Figures
• Parallel: Two lines in the same plane that never intersect and have
the same slope.
• Represented by symbol “||”
(line 1 || line2)
Transversal- Is a line that intersects two
lines in the same plane
1
Transversal
4
2
6
3
7
5
8
Vertical Angels- Are opposite angles that
share a vertex
Vertical angles are shown with the same color
1
4
2
6
3
7
5
8
When 2 lines intersect, they make vertical angles.
75º
105º
105º
75º
Vertical angles are opposite one
another.
75º
105º
105º
75º
Vertical angles are opposite one
another.
75º
105º
105º
75º
Corresponding Angels- Are in the same
location and on the same side of the
transversal
Corresponding angles are shown with the same
color-They both are on top of the line cut by the
transversal
1
4
2
6
3
7
5
8
Alternate Interior Angles- Are the angles on the
inside of the two lines but on opposite sides of
the transversal
Alternate Interior angles are shown with the same
color
1
4
2
6
3
7
5
8
Alternate Exterior Angles- Are the angles on the
outside of the two lines but on opposite sides of
the transversal
Alternate Interior angles are shown with the same
color
1
4
2
6
3
7
5
8
Same side interior Angles- Are the angles on the
inside of the two lines and on the same side of
the transversal
Same side Interior angles are shown with the same
color
1
4
2
6
3
7
5
8
Basic Terms & Definitions
• A ray starts at a point (called the endpoint) and extends
indefinitely in one direction.
A
B
AB
• A line segment is part of a line and has
two endpoints.
A
B
AB
• An angle is formed by two rays with the same endpoint.
side
vertex
side
• An angle is measured in degrees.
The angle formed by a circle has a
measure of 360 degrees.
• A right angle has a measure of 90 degrees.
• A straight angle has a measure of 180
degrees.
Practice Time!
Directions:
Identify each pair of angles as
vertical, supplementary,
complementary,
or none of the above.
#1
120º
60º
#1
120º
60º
Supplementary Angles
#2
30º
60º
#2
30º
60º
Complementary Angles
#3
75º
75º
#3
Vertical Angles
75º
75º
#4
40º
60º
#4
40º
60º
None of the above
#5
60º
60º
#5
60º
60º
Vertical Angles
#6
135º
45º
#6
135º
45º
Supplementary Angles
#7
25º
65º
#7
25º
65º
Complementary Angles
#8
90º
50º
#8
90º
50º
None of the above
Directions:
Determine the missing angle.
#1
?º
45º
#1
135º
45º
#2
?º
65º
#2
25º
65º
#3
?º
35º
#3
35º
35º
#4
?º
50º
#4
130º
50º
#5
?º
140º
#5
140º
140º
#6
?º
40º
#6
50º
40º
More Terms Defined
• Congruent
– Two or more objects that are the exact same size and
shape.
– Symbol for congruence is ≅
• Bisect
– Dividing and object into two congruent (equal) parts
More Terms Defined
• Midpoint
– The point that divides two objects into two congruent
parts
• Distance
– The amount of space between to objects or people
More Terms Defined
• Proof
– The process used to demonstrate truth of a statement
• Postulate
– A statement that is assumed to be true without proof.
Postulates are the basic structure from which theorems
are derived
• Theorem
– Is a statement that can be demonstrated to be true