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PRE-QUIZ#4: ANGLES AND ANGLE
MEASURES
1. The common endpoint shared by two
rays that form an angle.
2. Instrument to measure an angle.
3. An angle whose measure is greater than
90 but less than 180.
4. Two angles whose sum of measures is
90.
5. Two non-adjacent angles formed by
intersecting lines
Cassiopeia
BA and BC are the sides of the angle.
B is called the vertex.
Definition: ANGLE
A figure is an angle if and only
if it is the union of two noncollinear
rays (the sides) with a common
endpoint (the vertex).
X
Y
Z
WAYS TO NAME AN ANGLE
a. Use the vertex and a point from
each side. The vertex letter is
always in the middle.
Ex. ABC or CBA
A
D
C
B
WAYS TO NAME AN ANGLE
b. Use the vertex only. If there is
only one angle at a vertex, then
the angle can be named with
that vertex.
Ex. B
A
D
C
B
WAYS TO NAME AN ANGLE
c. Use a number
Ex. 1
A
D
2 1
B
C
A
W
S
Y
B
T
C
Y and W are in the interior of the angle.
S and T are in the exterior of the angle.
A, B, and C are on the angle.
Definition: INTERIOR OF AN ANGLE
The interior of ABC is the set of
all points P in the plane of ABC
such that:
(1) P and C are on the same side
of BA, and
(2) P and A are on the same side
of BC.
A
B
P
INTERIOR of the ANGLE
C
Definition: EXTERIOR OF AN ANGLE
The exterior of ABC is the set of
all points of the plane of ABC that
lie neither on the angle nor in its
interior.
A
W
S
Y
B
T
C
Exercises:
a) Name each angle in two ways.
Exercises:
a)Name the angle in three ways.
b)Identify the vertex and its sides.
c)Is the umbra in the interior, exterior,
or on the angle?
The circle graph shows the enrollment in Math
courses at Hayes High School.
a)
Use a protractor
to find the
measure of each
angle of the circle
graph.
b) Classify each
angle as acute,
obtuse, or right.
c) What is the greatest percentage that an acute
angle could represent in a circle graph?
Postulate:
Given any angle, there is a unique
real number between 0 and 180
known as its degree measure.
0 < mY < 180
X
Y
mY = r
r
Z
measure of Y
Definition: ACUTE ANGLE
A is an acute angle if and only if
0 < mA <90.
B
A
C
Definition: RIGHT ANGLE
A is a right angle if and only if
mA = 90.
B
A
C
Definition: OBTUSE ANGLE
A is an obtuse angle if and only
if 90 < mA < 180.
B
A
C
THE PROTRACTOR POSTULATE
In a half-plane with edge AB and any point
S between A and B,
a) there exists a one-to-one correspondence
between the rays that originate at S in that
half-plane and the real numbers between 0
and 180, and
b) to measure an angle formed by two of
these rays, find the absolute value of the
difference of the corresponding real
numbers.
The graph shows the sailing positions.
Suppose a sailboat is in the run position.
How many degrees must the sailboat be
turned so that it is in the close reach
position?
ANGLE ADDITION POSTULATE
If D is in the interior of ABC,
then mABC = m ABD + m DBC.
A
D
B
C
Example #1
If m1 = 35 and m2 = 46, find
mEFG.
E
H
2
F
1
G
Example #2
Find mABC and mCBD if
mABD = 120.
C
D
(5x – 6)
2x
A
B
Definition: CONGRUENT ANGLES
Two angles are congruent if and
only if they have equal measures.
X  Y if and only if mX = mY.
A
C
Y
X
B
D
If mX = 40 and mY = 40,
then X  Y.
Theorem: RIGHT ANGLE THEOREM
All right angles are congruent.
Definiton:
BISECTOR OF AN ANGLE
A ray is a bisector of an angle if and
only if it divides the angle into two
congruent angles.
If OX bisects AOB, then AOX  XOB.
A
X
O
B
ANGLE BISECTOR THEOREM
If OX is a bisector of AOB, then
2 mAOX  mAOB
1
mAOX  mAOB
2
and
2 mXOB  mAOB
1
mXOB  mAOB.
2
If an acute angle is bisected, what
type of angles are formed?
What type of angles are formed
when an obtuse angle is bisected?
What type of angles are formed
when a right angle is bisected?
What are their measures?
If OE bisects LOV and m LOE = 53, find
mVOL.
E
L
O
V
If AE bisects CAR, m RAE = 3x – 5 and
m RAC = 2(2x + 3), find x and m CAE.
C
E
A
R
Find the measure of the indicated
angle.
m1 = 56
m2 = 38
E
B
1
L
mBLU = ?
2
U
Find the measure of the indicated
angle.
E
U
mBLU = 124
m2 = 70
B
m1 = ?
L
Find the value of x and the measure
of the indicated angle.
E
U
B
L
mBLU = 133
m1 = x + 19
m2 = 2(x – 3)
x=?
m1 = ?
LE bisects BLU
E
U
mBLU = 142
m1 = ?
B
L
LE bisects BLU
E
U
B
L
m1 = 51
m2 = ?
mBLU = ?
LE bisects BLU
m1 = 4x + 1
m2 = 5x – 16
E
B
1
L
2
U
x=?
m1 = ?
mBLU = ?
LE bisects BLU
E
U
m1 = 3(x – 2)
mBLU = 5x + 4
B
L
x=?
m2 = ?
LE bisects BLU LD bisects ELU
mBLU = 120
E
B
D
1 2
L
3
U
m1 = ?
m2 = ?
m3 = ?