ANGLES

1
ANGLES

• DEF of an Angle An angle is FORMED BY TWO RAYS
  WITH A COMMON ENDPOINT.
• The Rays are the sides of the angle.
• The endpoints of the rays are the vertex

2
NAMING ANGLES

• USE 3 LETTERS
• ONE FOR THE VERTEX
• ONE FOR A POINT ON EACH SIDE OF THE ANGLE

The MIDDLE LETTER OF the NAME ALWAYS VERTEX
vertex
?ABC
3
ALTERNATIVE ANGLE NAMES

• USE 1 LETTER (THE VERTEX) OR
• 1 NUMBER (THE VERTEX)
• FOR THE NAME.
• WE SAY

4
vertex
?4 or ?B
4
Name the angles in this the diagram
1
2
How does your answer change if I ask for the
distinct angles in the diagram?
5
ANGLE NOTATION

• - ANGLES ARE MEASURED IN DEGREES.
• - WHEN WE WANT TO STATE THE MEASURE OF AN ANGLE
  WE USE LOWER CASE m IN FRONT OF THE NAME.
• Ex.

WE SAY m?k 30
6
If two angles have the same measure, we say
?ABC ? ?CBJ CONGRUENT ANGLES REFERS TO
ANGLES WITH THE SAME MEASURE.
7
NAME THE ANGLES

• If the angles both measure 65 then write a
  statement using proper notation.

8
CLASSIFYING ANGLES


25
Angles whose measures are less than 90 are
called _________________?
ACUTE ANGLES
Angles whose measures are 90 are called
_________________?
RIGHT ANGLES
115
Angles whose measures are greater than 90 but
less than 180 are called _________________?
OBTUSE ANGLES
9
Angles whose measures equal 180 are called
_________________?
STRAIGHT ANGLES
A straight angle is a line, Formed by two
opposite rays.
Name the opposite rays that form the above angle
10
Think about this situation
m ?TPV 110 m ?TPC 60 Find m ?CPV
50
This leads to the next postulate
11
ANGLE ADDITION POSTULATE (1.8)

• IF R IS in THE INTERIOR OF
• ?PQS, THEN
• m ?PQR m ?RQS m ?PQS

12
EXAMPLES
C
G
F
D
m ?CDF 115 m ?CDG 3x 5 m ?GDF 2x
X 22
13
EXAMPLES
m ?AOB 4X 2 m ?BOC 5X 10 m ?COD 2X
14 Find m ?AOD
110
14
ANGLE BISECTOR

• A Ray or Segment that DIVIDES AN ANGLE INTO
  CONGRUENT ANGLES
• CONGRUENT ANGLES HAVE THE SAME MEASURE

A
B
D
?ABD ? ?DBC
C
IS AN ANGLE BISECTOR OF ?ABC
15
EXAMPLES
45
16
EXAMPLES
X 6 m ?EFG 72
17
EXAMPLES
18
ANGLE RELATIONSHIPS
ADJACENT ANGLES ANGLES THAT SHARE A COMMON SIDE
and VERTEX, BUT NO COMMON INTERIOR POINTS
Name pairs of adjacent angles
19
EXAMPLES
G
C
F
D
Are thes angles adjacent angles?
m ?GDF 65 Find m ?CDG
115
20
ANGLE RELATIONSHIPS

• LINEAR PAIR
• ADJACENT ANGLES WHOSE NON-COMMON SIDES ARE
  OPPOSITE RAYS. That is the two angles form a
  straight angle.

1
2
21
ANGLE RELATIONSHIPS

• Make observations about the angles in the
  worksheet.
• THE SUM OF THE MEASURES OF THE ANGLES IN A LINEAR
  PAIR IS 180?

22
EXAMPLES
G
C
F
D
Are thes angles a linear pair?
m ?GDF 7x 2 and m ?CDG 3x 8 Find x and
each angle.
X 17 m ?GDF 121 m ?CDG 59
23
EXAMPLE
The sum of the measures of the angles in a linear
pair is 180
FIND x FIND Y FIND Z
24
ANGLE RELATIONSHIPS

• VERTICAL ANGLES
• TWO NONADJACENT ANGLES FORMED BY TWO
  INTERSECTING LINES.

Name a pair of vertical angles

25
ANGLE RELATIONSHIPS
Vertical angles are congruent
C
B
100
4Y
80
A
2X
K
H
FIND M?KAH FIND M?KAB FIND X Find Y
2x 100 x 50
2y 80 y 20
26
EXAMPLES
m ?BIK 10X 5 m ?HIC 2X 21 Find X Find m
?BIK Find m ?BIC
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ANGLE RELATIONSHIPS

• PERPENDICULAR LINES
• INTERSECTING LINES THAT FORM 4 RIGHT ANGLES

28
a
d
b
c
29
ANGLE RELATIONSHIPS
COMPLEMENTARY ANGLES TWO ANGLES WHOSE MEASURES
ADD UP TO 90
SUPPLEMENTARY ANGLES TWO ANGLES WHOSE MEASURES
ADD UP TO 180
30
EXAMPLES
Two angles are complementary and one angle has 3
times the measure of the other. Find the two
angles. Hint write an equation and solve.
x 3x 90 4x 90 X 22.50
So one angle is 22.5 and its complement is 90
- 22.5 67.50
31
EXAMPLES
Two angles are supplementary and one angle
measures 40 less than three times the other.
Find both angles. Hint write an equation and
solve.
x 3x - 40 180 4x 220 X 55
So one angle is 55 and its supplement is 180
- 55 125