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Angles

§ 3.1 Angles

§ 3.2 Angle Measure

§ 3.3 The Angle Addition Postulate

§ 3.4 Adjacent Angles and Linear Pairs of Angles

§ 3.5 Complementary and Supplementary Angles

§ 3.6 Congruent Angles

§ 3.7 Perpendicular Lines
Angles
You will learn to name and identify parts of an angle.
1) Opposite Rays
2) Straight Angle
3) Angle
4) Vertex
5) Sides
6) Interior
7) Exterior
Angles
Opposite
rays are two rays that are part of a the same line and have only their
___________
endpoints in common.
Y
X
Z
opposite rays
XY and XZ are ____________.
straight angle
The figure formed by opposite rays is also referred to as a ____________.
Straight Angle (Video)
Angles
There is another case where two rays can have a common endpoint.
angle
This figure is called an _____.
Some parts of angles have special names.
S
The common endpoint is called the ______,
vertex
and the two rays that make up the sides of
the angle are called the sides of the angle.
R
side
vertex
T
Angles
There are several ways to name this angle.
1) Use the vertex and a point from each side.
SRT
or
TRS
S
The vertex letter is always in the middle.
2) Use the vertex only.
R
If there is only one angle at a vertex, then the
angle can be named with that vertex.
R
side
vertex
3) Use a number.
1
1
T
Angles
An angle is a figure formed by two noncollinear rays that
have a common endpoint.
D
Symbols:
Definition
of Angle
E
DEF
FED
2
E
F
2
Naming Angles (Video)
Angles
1) Name the angle in four ways.
ABC
C
A
CBA
1
B
1
B
2) Identify the vertex and sides of this angle.
vertex:
sides:
Point B
BA
and
BC
Angles
1) Name all angles having W as their vertex.
X
1
2
W
1
2
XWZ
Y
2) What are other names for
XWY
or
1?
YWX
3) Is there an angle that can be named
No!
Z
W?
Angles
An angle separates a plane into three parts:
interior
1) the ______
exterior
exterior
2) the ______
angle itself
3) the _________
In the figure shown, point B and all other
points in the blue region are in the interior
of the angle.
Point A and all other points in the green
region are in the exterior of the angle.
Points Y, W, and Z are on the angle.
W
A
Y
Z
interior
B
Angles
Is point B in the interior of the angle,
exterior of the angle,
or on the angle?
P
G
Exterior
B
Is point G in the interior of the angle,
exterior of the angle,
or on the angle?
On the angle
Is point P in the interior of the angle,
exterior of the angle,
or on the angle?
Interior
§3.2 Angle Measure
You will learn to measure, draw, and classify angles.
1) Degrees
2) Protractor
3) Right Angle
4) Acute Angle
5) Obtuse Angle
§3.2 Angle Measure
degrees
In geometry, angles are measured in units called _______.
The symbol for degree is °.
P
In the figure to the right,
the angle is 75 degrees.
75°
In notation, there is no degree symbol with 75
because the measure of an angle is a real
number with no unit of measure.
Q
R
m
PQR = 75
§3.2 Angle Measure
For every angle, there is a unique positive number
0 and ____
between __
180 called the degree measure of the
angle.
Postulate 3-1
Angles
Measure
Postulate
A
B
m
ABC = n
and 0 < n < 180
n°
C
§3.2 Angle Measure
protractor to measure angles and sketch angles of given
You can use a _________
measure.
Use a protractor to measure
SRQ.
1) Place the center point of the protractor
on vertex R.
Align the straightedge with side RS.
2) Use the scale that begins with 0
at RS.
Read where the other side
of the angle, RQ, crosses
this scale.
Q
R
S
Using a Protractor (Video)
§3.2 Angle Measure
Find the measurement of:
m
SRQ = 180
m
SRJ = 45
m

m
SRH
m
QRG = 180 – 150
= 30
m
GRJ = 150 – 45
= 105
SRG = 150
70
H
J
G
Q
R
S
§3.2 Angle Measure
Use a protractor to draw an angle having a measure of 135.
1) Draw AB
3) Locate and draw point C at the
mark labeled 135. Draw AC.
2) Place the center point of the
protractor on A. Align the mark
labeled 0 with the ray.
C
A
B
§3.2 Angle Measure
Once the measure of an angle is known, the angle can be classified as one
of three types of angles. These types are defined in relation to a right angle.
Types of Angles
A
obtuse angle
90 < m
A < 180
A
A
right angle
m
A = 90
acute angle
0<m
A < 90
Angle Classification (Video)
§3.2 Angle Measure
Classify each angle as acute, obtuse, or right.
110°
40°
90°
Obtuse
Right
Acute
50°
130°
Acute
Obtuse
75°
Acute
???
§3.2 Angle Measure
The measure of
Solve for x.
B is 138.
The measure of
Solve for y.
H is 67.
H
9y + 4
5x - 7
B
Given:
Given:
(What do you know?)
B = 5x – 7 and
5x – 7 = 138
5x = 145
x = 29
B = 138
Check!
5(29) -7 = ?
145 -7 = ?
138 = 138
(What do you know?)
H = 9y + 4 and
9y + 4 = 67
9y = 63
y=7
H = 67
Check!
9(7) + 4 = ?
63 + 4 = ?
67 = 67
???
Is m
a
a larger than m
60°
b?
b
60°
§3.3 The Angle Addition Postulate
You will learn to find the measure of an angle and the bisector
of an angle.
NOTHING NEW!
§3.3 The Angle Addition Postulate
1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
45°
2) Draw and label
a point X in the
interior of the
angle. Then
draw SX.
X
75°
S
3) For each angle, find mRSX, mXST, and RST.
30°
T
§3.3 The Angle Addition Postulate
1) How does the sum of mRSX and mXST compare to mRST ?
Their sum is equal to the measure of RST .
mXST = 30
+ mRSX = 45
= mRST = 75
R
2) Make a conjecture about the
relationship between the two
smaller angles and the larger angle.
45°
X
The sum of the measures of the two
smaller
angles
is equal
to the (Video)
measure
The
Angle
Addition
Postulate
of the larger angle.
75°
S
30°
T
§3.3 The Angle Addition Postulate
For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
P
1
Q
2
A
m1 + m2 = mPQR.
R
There are two equations that can be derived using Postulate 3 – 3.
m1 = mPQR – m2
m2 = mPQR – m1
These equations are true no matter where A is located
in the interior of PQR.
§3.3 The Angle Addition Postulate
Find m2 if mXYZ = 86 and m1 = 22.
m2 + m1 = mXYZ
m2 = mXYZ – m1
Postulate 3 – 3.
X
1
W
2
m2 = 86 – 22
m2 = 64
Z
Y
§3.3 The Angle Addition Postulate
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD
Postulate 3 – 3.
2x + (5x – 6) = 120
Substitution
7x – 6 = 120
Combine like terms
7x = 126
Add 6 to both sides
x = 18
Divide each side by 7
36 + 84 = 120
C
mABC = 2x
mCBD = 5x – 6
mABC = 2(18)
mCBD = 5(18) – 6
mABC = 36
mCBD = 90 – 6
mCBD = 84
D
(5x – 6)°
2x°
A
B
§3.3 The Angle Addition Postulate
Just as every segment has a midpoint that bisects the segment, every angle
ray that bisects the angle.
has a ___
angle bisector .
This ray is called an ____________
§3.3 The Angle Addition Postulate
The bisector of an angle is the ray with its endpoint at the
vertex of the angle, extending into the interior of the
angle.
The bisector separates the angle into two angles of equal
measure.
Definition of
an Angle
Bisector
P
PW
1
Q
2
is the bisector of Q.
A
m1 = m2
R
§3.3 The Angle Addition Postulate
If
AT
Since
bisects CAN and mCAN = 130, find 1 and 2.
AT bisects CAN,
1 = 2.
N
1 + 2 = CAN
Postulate 3 - 3
1 + 2 = 130
Replace CAN with 130
1 + 1 = 130
Replace 2 with 1
2(1) = 130
(1) = 65
T
Combine like terms
2
Divide each side by 2
1
Since 1 = 2,
2 = 65
C
A
Adjacent Angles and Linear Pairs of Angles
You will learn to identify and use adjacent angles and
linear pairs of angles.
When you “split” an angle, you create two angles.
The two angles are called
adjacent
angles
_____________
B
A
2
1
adjacent = next to, joining.
C
1 and 2 are examples of adjacent angles. They share a common ray.
Name the ray that 1 and 2 have in common.
BD
____
Adjacent Angles and Linear Pairs of Angles
Adjacent angles are angles that:
A) share a common side
B) have the same vertex, and
C) have no interior points in common
Definition of
Adjacent
Angles
J
1 and 2 are adjacent
R
2
with the same vertex R and
1
common side
N
RM
Adjacent Angles and Linear Pairs of Angles
Determine whether 1 and 2 are adjacent angles.
No. They have a common vertex B, but
_____________
no common side
2
1
B
1
Yes. They have the same vertex G and a
common side with no interior points in
common.
2
G
N
L
J
2
1
No. They do not have a common vertex or
a common side
____________
LN
The side of 1 is ____
JN
The side of 2 is ____
Adjacent Angles and Linear Pairs of Angles
Determine whether 1 and 2 are adjacent angles.
No.
1
2
Yes.
1
X
2
D
Z
In this example, the noncommon sides of the adjacent angles form a
straight line
___________.
linear pair
These angles are called a _________
Adjacent Angles and Linear Pairs of Angles
Two angles form a linear pair if and only if (iff):
A) they are adjacent and
B) their noncommon sides are opposite rays
A
Definition of
Linear Pairs
D
B
1
2
1 and 2 are a linear pair.
BA and BD form AD
1  2  180
Adjacent Angles and Linear Pairs of Angles
In the figure, CM and CE are opposite rays.
H
1) Name the angle that forms a
linear pair with 1.
ACE
ACE and 1 have a common side CA ,
the same vertex C, and opposite rays
CM
and
T
A
2
3 4
1
M
CE
2) Do 3 and TCM form a linear pair? Justify your answer.
No. Their noncommon sides are not opposite rays.
C
E
§3.5 Complementary and Supplementary Angles
You will learn to identify and use Complementary and
Supplementary angles
§3.5 Complementary and Supplementary Angles
Two angles are complementary if and only if (iff) the
sum of their degree measure is 90.
E
D
A
Definition of
Complementary
B
30°
60°
F
C
Angles
mABC + mDEF = 30 + 60 = 90
§3.5 Complementary and Supplementary Angles
If two angles are complementary, each angle is a complement of the other.
ABC is the complement of DEF and DEF is the complement of ABC.
E
D
A
B
30°
60°
F
C
Complementary angles DO NOT need to have a common side or even the
same vertex.
§3.5 Complementary and Supplementary Angles
Some examples of complementary angles are shown below.
75°
15°
H
P
mH + mI = 90
Q
40°
mPHQ + mQHS = 90
50°
H
S
U
T
I
60°
V
mTZU + mVZW = 90
30°
Z
W
§3.5 Complementary and Supplementary Angles
If the sum of the measure of two angles is 180, they form a special pair of
angles called supplementary angles.
Two angles are supplementary if and only if (iff) the
sum of their degree measure is 180.
D
C
Definition of
Supplementary
Angles
50°
A
130°
B
E
mABC + mDEF = 50 + 130 = 180
F
§3.5 Complementary and Supplementary Angles
Some examples of supplementary angles are shown below.
H
75°
105°
I
mH + mI = 180
Q
130°
50°
mPHQ + mQHS = 180
H
P
S
U
V
60°
120°
60°
Z
T
mTZU + mUZV = 180
and
mTZU + mVZW = 180
W
§3.6 Congruent Angles
You will learn to identify and use congruent and
vertical angles.
measure
Recall that congruent segments have the same ________.
Congruent angles also have the same measure.
_______________
§3.6 Congruent Angles
Two angles are congruent iff, they have the same
degree measure
______________.
Definition of
B  V iff
Congruent
Angles
50°
50°
B
V
mB = mV
§3.6 Congruent Angles
arcs
To show that 1 is congruent to 2, we use ____.
1
2
To show that there is a second set of congruent angles, X and Z,
we use double arcs.
This “arc” notation states that:
X  Z
mX = mZ
X
Z
§3.6 Congruent Angles
four angles are formed.
When two lines intersect, ____
There are two pair of nonadjacent angles.
vertical angles
These pairs are called _____________.
4
1
3
2
§3.6 Congruent Angles
Two angles are vertical iff they are two nonadjacent
angles formed by a pair of intersecting lines.
Vertical angles:
Definition of
Vertical
Angles
4
1
3
1 and 3
2
2 and 4
§3.6 Congruent Angles
1) On a sheet of paper, construct two intersecting lines
that are not perpendicular.
2) With a protractor, measure each angle formed.
3) Make a conjecture about vertical angles.
Consider:
A.
1 is supplementary to 4.
m1 + m4 = 180
B.
3 is supplementary to 4.
m3 + m4 = 180
Therefore, it can be shown that 1  3
Likewise, it can be shown that 2  4
4
1
3
2
§3.6 Congruent Angles
4
1) If m1 = 4x + 3 and the m3 = 2x + 11, then find the m3
x = 4; 3 = 19°
2) If m2 = x + 9 and the m3 = 2x + 3, then find the m4
x = 56; 4 = 65°
3) If m2 = 6x - 1 and the m4 = 4x + 17, then find the m3
x = 9; 3 = 127°
4) If m1 = 9x - 7 and the m3 = 6x + 23, then find the m4
x = 10; 4 = 97°
1
3
2
§3.6 Congruent Angles
Vertical angles are congruent.
Theorem 3-1
Vertical
Angle
Theorem
n
2
m
1  3
3
1
4
2  4
§3.6 Congruent Angles
Find the value of x in the figure:
The angles are vertical angles.
130°
x°
So, the value of x is 130°.
§3.6 Congruent Angles
Find the value of x in the figure:
The angles are vertical angles.
(x – 10)°
(x – 10) = 125.
x – 10 = 125.
125°
x = 135.
§3.6 Congruent Angles
Suppose two angles are congruent.
What do you think is true about their complements?
1  2
1 + x = 90
x is the complement
of 1
x = 90 - 1
2 + y = 90
y is the complement
of 2
y = 90 - 2
Because 1  2, a “substitution” is made.
x = 90 - 1
x=y
y = 90 - 1
x  y
If two angles are congruent, their complements are congruent.
§3.6 Congruent Angles
Theorem 3-2
If two angles are congruent, then their complements are
congruent
_________.
The measure of angles complementary to A and
B
is 30.
60°
60°
A
B
A  B
If two angles are congruent, then their supplements are
congruent
_________.
The measure of angles supplementary to 1 and 4
is 110.
Theorem 3-3
70°
4
110°
110°
2
3
4  1
70°
1
§3.6 Congruent Angles
If two angles are complementary to the same angle,
congruent
then they are _________.
Theorem 3-4
3 is complementary to 4
5 is complementary to 4
3
5  3
4
5
If two angles are supplementary to the same angle,
congruent
then they are _________.
Theorem 3-5
1 is supplementary to 2
3
1
3 is supplementary to 2
1  3
2
§3.6 Congruent Angles
Suppose A  B and mA = 52.
Find the measure of an angle that is supplementary to B.
A
B
52°
B + 1 = 180
1 = 180 – B
1 = 180 – 52
1 = 128°
1
§3.6 Congruent Angles
If 1 is complementary to 3,
2 is complementary to 3,
and m3 = 25,
What are m1 and m2 ?
m1 + m3 = 90
Definition of complementary angles.
m1 = 90 - m3
Subtract m3 from both sides.
m1 = 90 - 25
Substitute 25 in for m3.
m1 = 65
Simplify the right side.
You solve for m2
m2 + m3 = 90
Definition of complementary angles.
m2 = 90 - m3
Subtract m3 from both sides.
m2 = 90 - 25
Substitute 25 in for m3.
m2 = 65
Simplify the right side.
§3.6 Congruent Angles
G
D
1
A
4
3
2
B
E
H
1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3
x = 17; 3 = 37°
2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC
x = 29; EBC = 121°
3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4
x = 16; 4 = 39°
4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1
x = 18; 1 = 43°
C
Suppose you draw two angles that are congruent and supplementary.
What is true about the angles?
§3.6 Congruent Angles
If two angles are congruent and supplementary
right angle
then each is a __________.
Theorem 3-6
1 is supplementary to 2
1 and 2 = 90
1
2
congruent
All right angles are _________.
Theorem 3-7
C
A
B
A  B  C
§3.6 Congruent Angles
If 1 is supplementary to 4, 3 is supplementary to 4, and
m 1 = 64, what are m 3 and m 4?
1  3
They are vertical angles.
A
2
E
1 4
m 1 = m3
D
3 is supplementary to 4
Given
m3 + m4 = 180
Definition of supplementary.
m4 = 180 – 64
m4 = 116
3
C
m 3 = 64
64 + m4 = 180
B
§3.7 Perpendicular Lines
You will learn to identify, use properties of, and consrtuct
perpendicular lines and segments.
§3.7 Perpendicular Lines
perpendicular lines
Lines that intersect at an angle of 90 degrees are _________________.
In the figure below, lines AB and CD are perpendicular.
A
C
1
2
3
4
B
D
§3.7 Perpendicular Lines
Perpendicular lines are lines that intersect to form a
right angle.
m
Definition of
Perpendicular
Lines
n
mn
§3.7 Perpendicular Lines
In the figure below, l  m. The following statements are true.
m
2
1
3
4
l
1) 1 is a right angle.
Definition of Perpendicular Lines
2) 1  3.
Vertical angles are congruent
3) 1 and 4 form a linear pair.
Definition of Linear Pair
4) 1 and 4 are supplementary. Linear pairs are supplementary
5) 4 is a right angle.
m4 + 90 = 180, m4 = 90
6) 2 is a right angle.
Vertical angles are congruent
§3.7 Perpendicular Lines
If two lines are perpendicular, then they form four right
angles.
a
Theorem 3-8
2
1
3
4
b
ab
m1  90
m1  90
m1  90
m1  90
§3.7 Perpendicular Lines
In the figure, OP  MN and NP  QS. Determine whether each of
the following is true or false.
1) PRN is an acute angle.
False. Since OP  MN,
PRN is a right angle.
7
P
M
1
Q
2
2) 4  8
True 4 and 8 are vertical angles,
and vertical angles are
congruent.
O
5
R
8
3
4
6
N
§3.7 Perpendicular Lines
If a line m is in a plane and T is a point in m, then there
exists exactly ___
one line in that plane that is perpendicular to
m at T.
m
Theorem 3-9
T
Section Title
SRT
AB
Section Title
A
0
1
2
3
4
5
6
7
8
9
10