Angles

1
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

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Angles
What You'll Learn
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
3
Angles
Opposite rays
___________ are two rays that are part of a the
same line and have only theirendpoints in common.
opposite rays
straight angle
The figure formed by opposite rays is also
referred to as a ____________.
Straight Angle (Video)
4
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.
The common endpoint is called the ______,
vertex
and the two rays that make up the sides ofthe
angle are called the sides of the angle.
side
R
side
5
Angles
There are several ways to name this angle.
1) Use the vertex and a point from each side.
or
The vertex letter is always in the middle.
side
2) Use the vertex only.
1
If there is only one angle at a vertex, then
theangle can be named with that vertex.
R
side
3) Use a number.
6
Angles
Definitionof Angle An angle is a figure formed by two noncollinear rays that have a common endpoint.
Symbols
Naming Angles (Video)
7
Angles
1) Name the angle in four ways.
2) Identify the vertex and sides of this angle.
vertex
Point B
sides
8
Angles
1) Name all angles having W as their vertex.
X
W
1
2
Y
Z
No!
9
Angles
An angle separates a plane into three parts
interior
1) the ______
exterior
2) the ______
angle itself
3) the _________
In the figure shown, point B and all other
points in the blue region are in the interiorof
the angle.
Point A and all other points in the greenregion
are in the exterior of the angle.
Points Y, W, and Z are on the angle.
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Angles
Is point B in the interior of the angle,
exterior of the angle,
or on the angle?
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
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3.2 Angle Measure
What You'll Learn
You will learn to measure, draw, and classify
angles.
1) Degrees 2) Protractor 3) Right Angle 4) Acute
Angle 5) Obtuse Angle
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3.2 Angle Measure
degrees
In geometry, angles are measured in units called
_______.
The symbol for degree is .
In the figure to the right, the angle is 75
degrees.
In notation, there is no degree symbol with
75 because the measure of an angle is a real
number with no unit of measure.
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3.2 Angle Measure
Postulate 3-1 Angles Measure Postulate For every angle, there is a unique positive number between __ and ____ called the degree measure of the angle.
0
180
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3.2 Angle Measure
protractor
You can use a _________ to measure angles and
sketch angles of givenmeasure.
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3.2 Angle Measure
70
Find the measurement of
180 150 30
180
45
150 45 105
150
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3.2 Angle Measure
Use a protractor to draw an angle having a
measure of 135.
2) Place the center point of the protractor
on A. Align the mark labeled 0 with the
ray.
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3.2 Angle Measure
Once the measure of an angle is known, the angle
can be classified as oneof three types of
angles. These types are defined in relation to a
right angle.
Types of Angles

Angle Classification (Video)
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3.2 Angle Measure
Classify each angle as acute, obtuse, or right.
Obtuse
Acute
Right
Obtuse
Acute
Acute
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3.2 Angle Measure
Given (What do you know?)
Given (What do you know?)
9y 4 67
5x 7 138
Check!
Check!
9y 63
5x 145
9(7) 4 ?
5(29) -7 ?
y 7
x 29
63 4 ?
145 -7 ?
67 67
138 138
20
? ? ?
60
60
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End of Lesson
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3.3 The Angle Addition Postulate
What You'll Learn
You will learn to find the measure of an angle
and the bisectorof an angle.
NOTHING NEW!
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3.3 The Angle Addition Postulate
1) Draw an acute, an obtuse, or a
right angle. Label the angle RST.
45
75
30
3) For each angle, find m?RSX, m?XST, and
?RST.
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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
2) Make a conjecture about the
relationship between the two smaller angles
and the larger angle.
45
The sum of the measures of the twosmaller angles
is equal to the measureof the larger angle.
The Angle Addition Postulate (Video)
75
30
25
3.3 The Angle Addition Postulate
Postulate 3-3 Angle Addition Postulate For any angle PQR, if A is in the interior of ?PQR, thenm?PQA m?AQR m?PQR.
m?1 m?2 m?PQR.
There are two equations that can be derived using
Postulate 3 3.
m?1 m?PQR m?2
These equations are true no matter where A is
locatedin the interior of ?PQR.
m?2 m?PQR m?1
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3.3 The Angle Addition Postulate
Find m?2 if m?XYZ 86 and m?1 22.
m?2 m?1 m?XYZ
Postulate 3 3.
m?2 m?XYZ m?1
m?2 86 22
m?2 64
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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
m?ABC 2x
m?CBD 5x 6
m?CBD 5(18) 6
m?ABC 2(18)
m?ABC 36
m?CBD 90 6
m?CBD 84
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3.3 The Angle Addition Postulate
Just as every segment has a midpoint that bisects
the segment, every angle has a ___ that bisects
the angle.
ray
angle bisector
This ray is called an ____________ .
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3.3 The Angle Addition Postulate
Definition of an Angle Bisector The bisector of an angle is the ray with its endpoint at thevertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equalmeasure.
m?1 m?2
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3.3 The Angle Addition Postulate
If bisects ?CAN and m?CAN 130,
find ?1 and ?2.
?1 ?2 ?CAN
Postulate 3 - 3
?1 ?2 130
Replace ?CAN with 130
?1 ?1 130
Replace ?2 with ?1
2(?1) 130
Combine like terms
(?1) 65
Divide each side by 2
Since ?1 ?2, ?2 65
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End of Lesson
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Adjacent Angles and Linear Pairs of Angles
What You'll Learn
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
2
1
adjacent next to, joining.
?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 and Linear Pairs of 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 and Linear Pairs of 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 ____
35
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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Adjacent Angles and 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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Adjacent Angles and 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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End of Lesson
39
3.5 Complementary and Supplementary Angles
What You'll Learn
You will learn to identify and use Complementary
and Supplementary angles
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3.5 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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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.
Complementary angles DO NOT need to have a common
side or even the same vertex.
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3.5 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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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.
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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3.5 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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End of Lesson
46
3.6 Congruent Angles
What You'll Learn
You will learn to identify and use congruent
and vertical angles.
Recall that congruent segments have the same
________.
measure
Congruent angles
_______________ also have the same measure.
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3.6 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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3.6 Congruent Angles
arcs
To show that ?1 is congruent to ?2, we use
____.
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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3.6 Congruent 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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3.6 Congruent 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
51
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
Hands-On
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
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3.6 Congruent Angles
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
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3.6 Congruent 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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3.6 Congruent 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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3.6 Congruent 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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3.6 Congruent Angles
Suppose two angles are congruent. What do you
think is true about their complements?
?1 ? ?2
?2 y 90
?1 x 90
x is the complement of ?1
y is the complement of ?2
y 90 - ?2
x 90 - ?1
Because ?1 ? ?2, a substitution is made.
y 90 - ?1
x 90 - ?1
x y
?x ? ?y
If two angles are congruent, their complements
are congruent.
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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.
?A ? ?B
Theorem 3-3
If two angles are congruent, then their
supplements are _________.
congruent
The measure of angles supplementary to ?1 and
?4 is 110.
110
110
70
70
?4 ? ?1
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3.6 Congruent Angles
Theorem 3-4
If two angles are complementary to the same
angle, then they are _________.
congruent
?3 is complementary to ?4
?5 is complementary to ?4
4
3
5
?5 ? ?3
Theorem 3-5
If two angles are supplementary to the same
angle, then they are _________.
congruent
?1 is supplementary to ?2
?3 is supplementary to ?2
?1 ? ?3
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3.6 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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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.
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3.6 Congruent Angles
1) If m?1 2x 3 and the m?3 3x - 14,
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
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Suppose you draw two angles that are congruent
and supplementary. What is true about the angles?
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3.6 Congruent Angles
Theorem 3-6
If two angles are congruent and supplementary
then each is a __________.
right angle
?1 is supplementary to ?2
?1 and ?2 90
Theorem 3-7
All right angles are _________.
congruent
?A ? ?B ? ?C
64
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?
They are vertical angles.
?1 ? ?3
m ?1 m?3
m ?3 64
?3 is supplementary to ?4
Given
Definition of supplementary.
m?3 m?4 180
64 m?4 180
m?4 180 64
m?4 116
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End of Lesson
66
3.7 Perpendicular Lines
What You'll Learn
You will learn to identify, use properties of,
and construct perpendicular lines and segments.
67
3.7 Perpendicular Lines
perpendicular lines
Lines that intersect at an angle of 90 degrees
are _________________.
68
3.7 Perpendicular Lines
Definition of Perpendicular Lines
Perpendicular lines are lines that intersect to
form a right angle.
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3.7 Perpendicular Lines
In the figure below, l ? m. The following
statements are true.
m
l
1) ?1 is a right angle.
Definition of Perpendicular Lines
Vertical angles are congruent
2) ?1 ? ?3.
Definition of Linear Pair
3) ?1 and ?4 form a linear pair.
Linear pairs are supplementary
4) ?1 and ?4 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
70
3.7 Perpendicular Lines
Theorem 3-8
If two lines are perpendicular, then they form
four rightangles.
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3.7 Perpendicular Lines
1) ?PRN is an acute angle.
False.
2) ?4 ? ?8
True
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3.7 Perpendicular Lines
Theorem 3-9
If a line m is in a plane and T is a point in
m, then there exists exactly ___ line in that
plane that is perpendicular to m at T.
one
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End of Lesson