Download 1.6 Angle Pair Relationships

Angles – sides and vertex
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
vertex
The common endpoint is called the ______,
and the two rays that make up the sides of
the angle are called the sides of the angle.
R
vertex
side
T
Naming Angles
There are several ways to name this angle.
S
1) Use the vertex and a point from each side.
SRT
or
TRS
The vertex letter is always in the middle.
2) Use the vertex only.
R
1
side
R
vertex
If there is only one angle at a vertex, then the
angle can be named with that vertex.
3) Use a number.
1
T
Angles
An angle is a figure formed by two noncollinear rays that
have a common endpoint.
Symbols:
D
Definition
of Angle
DEF
FED
E
2
E
F
2
Angles
1) Name the angle in four ways.
C
ABC
A
CBA
1
B
1
B
2) Identify the vertex and sides of this angle.
vertex: Point B
sides:
BA
and BC
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 Measure
Classify each angle as acute, obtuse, or right.
110°
40°
90°
Obtuse
Right
Acute
50°
130°
Acute
Obtuse
75°
Acute
Straight 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 ____________.
The figure formed by opposite rays is also referred to as a
straight angle A straight angle measures 180 degrees.
____________.
Congruent Angles
measure
Recall that congruent segments have the same ________.
Congruent
angles
_______________
also have the same measure.
Congruent Angles
Two angles are congruent if they have the same
degree measure
______________.
Definition of
Congruent
Angles
B  V iff
50°
50°
B
V
mB = mV
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
X
mX = mZ
Z
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?
Angle Pair
Relationships
Angle Pair Relationship
Essential Questions
How are special angle pairs
identified?
Adjacent Angles
When you “split” an angle, you create two angles.
The two angles are called
_____________
adjacent angles
adjacent = next to, joining.
A
B
2
1
1 and 2 are examples of adjacent angles.
They share a common ray.
Name the ray that 1 and 2 have in common.
C
BD
____
Adjacent 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
R
1 and 2 are adjacent
with the same vertex R and
2
1
common side RM
N
Adjacent 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 ____
Vertical Angles
When two lines intersect, four
____ angles are formed.
There are two pair of nonadjacent angles.
vertical angles
These pairs are called _____________.
4
1
3
2
Vertical Angles
Two angles are vertical if 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
Vertical Angles
Vertical angles are congruent.
Theorem 3-1
Vertical
Angle
Theorem
n
2
m
1  3
3
1
4
2  4
Vertical Angles
Find the value of x in the figure:
130°
x°
The angles are vertical angles.
So, the value of x is 130°.
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 _________
Linear Pairs of Angles
Two angles form a linear pair if:
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
Linear Pairs of Angles
In the figure, CM and CE are opposite rays.
1) Name the angle that forms a
linear pair with 1.
ACE
H
T
A
2
1
ACE and 1 have a common side CA
the same vertex C, and opposite rays
3 4
C
M
CM and CE
2) Do 3 and TCM form a linear pair? Justify your answer.
No. Their noncommon sides are not opposite rays.
E
Complementary Angles
Two angles are complementary if
the sum of their degree measure is 90.
E
D
A
Definition of
Complementary
Angles
B
30°
60°
F
C
mABC + mDEF = 30 + 60 = 90
Complementary 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
A
B
D
30°
C
60°
F
Complementary angles DO NOT need to have a common side
or even the same vertex.
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
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 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
Supplementary Angles
Some examples of supplementary angles are shown below.
H
75°
105°
I
mH + mI = 180
Q
130°
50°
H
P
S
U
V
60°
120°
60°
Z
T
mPHQ + mQHS = 180
mTZU + mUZV = 180
and
W
mTZU + mVZW = 180
Vertical Angles
Find the value of x in the figure:
(x – 10)°
125°
The angles are vertical angles.
(x – 10) = 125.
x – 10 = 125.
x = 135.
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
Congruent Angles
G
D
1
1) If m1 = 2x + 3 and the
m3 = 3x + 2, then find the
m3
x = 17; 3 = 37°
A
4
3
2
B
C
E
H
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°