Angle Pair Relationships

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Angle Pair Relationships
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Angle Pair Relationship Essential Questions

• How are special angle pairs identified?

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Straight Angles
Opposite rays
___________ are two rays that are part of a the
same line and have only their endpoints in common.
opposite rays
The figure formed by opposite rays is also
referred to as a ____________. A straight angle
measures 180 degrees.
straight angle
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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.
side
vertex
The common endpoint is called the ______,
and the two rays that make up the sides ofthe
angle are called the sides of the angle.
side
R
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Naming Angles
There are several ways to name this angle.
1) Use the vertex and a point from each side.
or
side
The vertex letter is always in the middle.
2) Use the vertex only.
1
R
side
If there is only one angle at a vertex, then
theangle can be named with that vertex.
3) Use a number.
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Angles
Definitionof Angle An angle is a figure formed by two noncollinear rays that have a common endpoint.
Symbols
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Angles
1) Name the angle in four ways.
2) Identify the vertex and sides of this angle.
vertex
Point B
sides
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Angles
1) Name all angles having W as their vertex.
X
W
1
2
Y
Z
No!
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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

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Angle Measure
Classify each angle as acute, obtuse, or right.
Obtuse
Acute
Right
Obtuse
Acute
Acute
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Adjacent Angles
When you split an angle, you create two angles.
The two angles are called _____________
adjacent angles
adjacent next to, joining.
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.
____
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Adjacent Angles
Definition of 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
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Adjacent Angles
Determine whether ?1 and ?2 are adjacent
angles.
No. They have a common vertex B, but
_____________
no common side
Yes. They have the same vertex G and a
common side with no interior points in
common.
No. They do not have a common vertex or
____________
a common side
The side of ?1 is ____
The side of ?2 is ____
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Adjacent Angles and Linear Pairs of Angles
Determine whether ?1 and ?2 are adjacent
angles.
No.
Yes.
In this example, the noncommon sides of the
adjacent angles form a ___________.
straight line
linear pair
These angles are called a _________
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Linear Pairs of Angles
Definition of Linear Pairs
Two angles form a linear pair if and only if
(iff)
A) they are adjacent and
B) their noncommon sides are opposite rays
?1 and ?2 are a linear pair.
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Linear Pairs of Angles
1) Name the angle that forms a linear pair
with ?1.
?ACE
2) Do ?3 and ?TCM form a linear pair?
Justify your answer.
No. Their noncommon sides are not opposite rays.
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Complementary and Supplementary Angles
Definition of Complementary Angles
Two angles are complementary if and only if (iff)
The sum of their degree measure is 90.
m?ABC m?DEF 30 60 90
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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.
Complementary angles DO NOT need to have a common
side or even the same vertex.
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Complementary and Supplementary Angles
Some examples of complementary angles are shown
below.
m?H m?I 90
m?PHQ m?QHS 90
m?TZU m?VZW 90
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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.
Definition of Supplementary Angles
Two angles are supplementary if and only if (iff)
the sum of their degree measure is 180.
m?ABC m?DEF 50 130 180
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Complementary and Supplementary Angles
Some examples of supplementary angles are shown
below.
m?H m?I 180
m?PHQ m?QHS 180
m?TZU m?UZV 180
and
m?TZU m?VZW 180
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Congruent Angles
measure
Recall that congruent segments have the same
________.
Congruent angles
_______________ also have the same measure.
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Congruent Angles
Definition of Congruent Angles
Two angles are congruent iff, they have the
same ______________.
degree measure
?B ? ?V iff
m?B m?V
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Congruent Angles
To show that ?1 is congruent to ?2, we use
____.
arcs
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
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Vertical Angles
When two lines intersect, ____ angles are formed.
four
There are two pair of nonadjacent angles.
vertical angles
These pairs are called _____________.
1
4
2
3
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Vertical Angles
Definition of Vertical Angles
Two angles are vertical iff they are two
nonadjacent angles formed by a pair of
intersecting lines.
Vertical angles
?1 and ?3
1
4
2
?2 and ?4
3
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Vertical Angles
Theorem 3-1 Vertical Angle Theorem
Vertical angles are congruent.
n
m
2
?1 ? ?3
3
1
?2 ? ?4
4
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Vertical Angles
Find the value of x in the figure
The angles are vertical angles.
So, the value of x is 130.
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Vertical Angles
Find the value of x in the figure
The angles are vertical angles.
(x 10) 125.
(x 10)
x 10 125.
125
x 135.
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Congruent Angles
Suppose ?A ? ?B and m?A 52.
Find the measure of an angle that is
supplementary to ?B.
1
?B ?1 180
?1 180 ?B
?1 180 52
?1 128
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Congruent Angles
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