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Angles
Vertex
Side 2
Two rays with a common endpoint.
Model
Item
An angle is
Two rays with
a common end
point.
Notation
 ABC
B
1
The parts are the sides ( rays ) , the vertex ( common point),
interior space, and exterior space.
Angles are classified by rotation of the rays.
Zero degrees
90 degrees
Straight Angle: 180 degrees.
Obtuse angles: between 90 and 180 degrees.
Acute angles are < 90 degrees
Types of Angles
Acute angles are less than 90 degrees
Right angles are equal to 90 degrees.
[ Looks like letter L ]
Obtuse angles are greater than 90 degrees but less than 180 degrees.
Straight angles look like lines and are equal to 180 degrees.
F
N
R
C
O
A
U
T
A
T
P
T
Types of Angles
Angles are differentiated by the quantify of rotation of the
rays as if they were hands of a clock. No rotation is zero
degrees and totally straight is 180 degrees.
S
T
90 degrees
R
Zero
degrees
45 degrees
Types of Angles
Smallest
Zero
Small
Acute
Middle
Right
Large
Obtuse
Largest
Straight
Measuring Angles
The Protractor
Measuring Angles
The Protractor
The smaller number is for the acute
angles and the larger number is for
the obtuse angles.
Notice, the numbers add up to 180.
500
1400
0
40
0
25
0
35
600
600
350
0
35
0
57
530
Adjacent Angles
1
2
Same vertex,
Common ray,
and no common interior
Non-Adjacent Angles
2
1
Not the same endpoint.
Non-Adjacent Angles
B
A
T
G
 BAT and
 BAG
Overlapping Interiors is not allowed.
A
4
3
E
5
6
D
2
1
L
9 8
T
7
S
How Many Angles ?
2+ 1=3
How Many Angles ?
3+ 2+ 1=6
How Many Angles ?
4 + 3 + 2 + 1 = 10
Did you see the pattern?
2+1=3
3+2+1=6
4 + 3 + 2 + 1 = 10
Total angles = sum of countdown of the smallest angle totals.
0
50
Vertex Position
One ray must be horizontal.
Reading a protractor
Protractor Postulate
For
AB on a given plane, choose any point O between A and B.
Consider
OA
and
OB
and all the rays that can be drawn
from O on one side of
A
O
AB.
B
Protractor Postulate
These rays can be paired with the real numbers from 0 to 180
in such a way that:
OA
A
is paired with 0 and
0
O
OB with 180.
180
B
Protractor Postulate
These rays can be paired with the real numbers from 0 to 180
in such a way that:
If
and OQ is paired with y,
OP is paired with x
then
m  POQ  x  y
AB
P
X
Q
AB
Y
OB
A
0
O
180
B
Protractor Postulate
These rays can be paired with the real numbers from 0 to 180
in such a way that:
If
and OQ is paired with y,
OP is paired with x
then
m  POQ  x  y  100  150
= 50
P
Example
100
Q
AB
AB
150
OB
A
0
O
180
B
Example 2
70
120
C
Top Scale
T
A
m  CAT 
m  CAT  110  60 or 60 110  500
Bottom Scale
70  120 or 120  70  500
Angle Addition Postulate
A
B
O
C
If point B lies in the interior of
 AOC then
m  AOB  m  BOC  m  AOC
And
Angle Addition Postulate
B
A
O
C
If  AOC is a straight angle and
B is any point not on
AC
then
m  AOB  m  BOC 180
Note:
The angle addition postulate is just like
the segment addition postulate.
When the two angles form a straight line
then they are called linear pairs.
Euclid referred to this concept as …
“The sum of the parts equals the whole.”
Angle Addition Applications
A
B
310
O
220
C
m  AOB  530
Example 2
A
5x +13
m  AOB  4 x  1
B
m  BOC  22
4x +1
O
m  AOC  5 x  13
220
C
4x +1 +22 = 5x +13
4x +23 = 5x +13
10 = x
Find the values of the angles.
m  AOB  4(10)  1
m  AOB  41
m  AOC  5(10)  13
Substitute back into expressions.
m  AOC  63
Summary
Angles are 2 rays with a common end point.
There are 4 types of angles:
Acute – less than 900
F
N
C
T
O
P
Right = 900
A
R
U
T
Obtuse – between 900 and 1800
Straight =
1800
A
T
Summary 2
Angles can be indicated by numbers,
the vertex, or by 3 letter of which
the middle letter is the vertex.
Angles are measured with a protractor.
The Protractor Postulate establishes
measuring angles with a protractor.
The Angle Addition Postulate establishes
the sum of two adjacent angles is indeed the
sum of the two angles.
C’est fini.
Good day and good luck.