Transcript MEASURING SEGMENTS

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
NAMING ANGLES
EXTERIOR
POINTS
E A
The MIDDLE
side
LETTER OF the
J
NAME ALWAYS
INTERIOR
POINTS
VERTEX
B
vertex
ABC
side
C
USE 3 LETTERS
ONE FOR THE VERTEX
ONE FOR A POINT ON
EACH SIDE OF THE
ANGLE
ALTERNATIVE ANGLE
NAMES
A
side
vertex
B
4
side
C
USE 1 LETTER (THE VERTEX)
OR
1 NUMBER (THE VERTEX)
FOR THE NAME.
 WE SAY
4 or B
Name the angles in this the
diagram…
T
E
1
M
2
V
How does your answer
change if I ask for the
distinct angles in the
diagram?
ANGLE NOTATION
- ANGLES ARE MEASURED
IN DEGREES.
- WHEN WE WANT TO STATE
THE MEASURE OF AN
ANGLE WE USE LOWER
m IN FRONT OF THE
CASE
NAME.
Ex.
k
30°
WE SAY mk = 30°
If two angles have the
same measure, we say
ABC  CBJ
A
B
C
20°
20°
J
CONGRUENT ANGLES
REFERS TO ANGLES WITH
THE SAME MEASURE.
NAME THE ANGLES
C
T
3
K
M
5
P
L
• If the angles both measure 65°
then write a statement using
proper notation.
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
K
P
C
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…
PK and PC
Think about this situation…
V
P
T
C
m TPV = 110°
m TPC = 60 °
Find m CPV
= 50°
This leads to the next
postulate…
ANGLE ADDITION
POSTULATE (1.8)
 IF R IS in THE INTERIOR OF
PQS, THEN
m PQR + m RQS = m PQS
P
Q
R
S
EXAMPLES
C
G
F
D
m CDF = 115°
m CDG = 3x + 5
m GDF = 2x
X = 22°
EXAMPLES
A
B
C
O
D
m AOB = 4X – 2
m BOC = 5X + 10
m COD = 2X + 14
Find m AOD = 110°
ANGLE BISECTOR
A Ray or Segment that
DIVIDES AN ANGLE INTO
CONGRUENT ANGLES
CONGRUENT ANGLES
HAVE THE SAME MEASURE
A
B
D
ABD  DBC
BD
IS AN ANGLE BISECTOR OF
C
ABC
EXAMPLES
A
C
B
D
IF BC IS AN ANGLE
BISECTOR OF ACD,
FIND m ACB. = 45°
EXAMPLES
F
E
G
X=6
m EFG = 72
T
IF FT IS AN ANGLE
BISECTOR OF EFG, and
mEFT = 4x + 12 and
mTFG = 6x
Find x:
FIND m
EFG.
EXAMPLES
H
G
K
J
IF HJ IS AN ANGLE
BISECTOR OF GHK, and
mGHJ = 6x + 10 and
mJHK = 3x + 28
Find x:
FIND m
GHK.
ANGLE RELATIONSHIPS
ADJACENT ANGLES:
ANGLES THAT
SHARE A COMMON
SIDE and VERTEX,
BUT NO COMMON
INTERIOR POINTS
A
B
C
Name pairs of adjacent
angles
D
O
EXAMPLES
G
C
D
F
Are thes angles adjacent
angles?
m GDF = 65°
Find m CDG = 115°
ANGLE RELATIONSHIPS
• LINEAR PAIR:
ADJACENT ANGLES
WHOSE NON-COMMON
SIDES ARE OPPOSITE
RAYS. That is the two
angles form a straight
angle.
1
2
ANGLE RELATIONSHIPS
• Make observations about the
angles in the worksheet.
• THE SUM OF THE
MEASURES OF THE
ANGLES IN A LINEAR PAIR
IS 180
EXAMPLES
G
C
D
F
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°
EXAMPLE
The sum of the measures of
the angles in a linear pair is
180°
C
B
Y
Z
60°
K
H
FIND x
FIND Y
FIND Z
ANGLE RELATIONSHIPS
• VERTICAL ANGLES:
TWO NONADJACENT
ANGLES FORMED BY TWO
INTERSECTING LINES.
Name a pair of vertical
angles
2
3
4
5
ANGLE RELATIONSHIPS
Vertical angles are congruent
C
B
100°
4Y
A
2X
80°
K
H
FIND MKAH
FIND MKAB
FIND X
2x = 100 x = 50
Find Y
2y = 80 y = 20
EXAMPLES
m BIK = 10X + 5
m HIC = 2X + 21
Find X
Find m BIK
Find m BIC
C
B
10x + 5
2x + 21
I
H
K
ANGLE RELATIONSHIPS
• PERPENDICULAR LINES:
INTERSECTING LINES
THAT FORM 4 RIGHT
ANGLES
a
b
c
d
ANGLE RELATIONSHIPS
COMPLEMENTARY ANGLES…
TWO ANGLES WHOSE
MEASURES ADD UP TO
90°
SUPPLEMENTARY ANGLES…
TWO ANGLES WHOSE
MEASURES ADD UP TO
180°
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 °
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°