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2.2 Special Angles and Postulates 171
171
LEARNING GOALS
A compliment is an expression of praise, admiration, or congratulations. Often
when someone does something noteworthy, you may “pay them a compliment”
to recognize the person’s accomplishments.
Even though they are spelled similarly, the word “complement” means something
very different. To complement something means to complete or to make whole. This
phrase is used in mathematics, linguistics, music, and art. For example,
complementary angles have measures that sum to 180 degrees—making the straight
angle “whole.” In music, a complement is an interval that when added to another
spans an octave—makes it “whole.”
The film Jerry McGuire features the famous line “You complete me,” meaning that the
other person complements them or that together they form a whole. So, a complement
can be quite a compliment indeed!
In this lesson, you will:
Calculate the complement and
supplement of an angle.
Classify adjacent angles, linear pairs,
and vertical angles.
Differentiate between postulates and
theorems.
Differentiate between Euclidean and
non-Euclidean geometries.
And Now From a New AngleSpecial Angles and Postulates
2.2
KEY TERMS
supplementary angles
complementary angles
adjacent angles
linear pair
vertical angles
postulate
theorem
Euclidean geometry
Linear Pair Postulate
Segment Addition Postulate
Angle Addition Postulate
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What are the measures of your angles in the complementary pair?
How did you calculate the measure of the complement of the 622° angle?
Is it possible for two angles to be both congruent and supplementary?
Is it possible for two angles to be both congruent and complementary?
How do you know when to set the sum of the measures of the two angles
equal to 90°?
How do you know when to set the sum of the measures of the two angles
equal to 180°?
Problem 1
Supplementary and
complementary angles are
de"ned. Students use a compass
to draw angles of speci"ed
measures. Then they use algebra
to calculate the complement and
supplement of different angles.
PROBLEM 1 Supplements and Complements
Two angles are supplementary angles if the sum of their angle measures is equal
to 180º.
1. Use a protractor to draw a pair of supplementary angles that
share a common side, and then measure each angle.
125° 55°
2. Use a protractor to draw a pair of supplementary angles that do not share a common
side, and then measure each angle.
70° 110°
3. Calculate the measure of an angle that is supplementary to /KJL.
J
K
22°
L
An angle that is supplementary to /KJL has a measure of 158º.
Supplementary angles that share a
side form a straight line, or a straight angle.
Angle pairs appear
frequently throughout
geometry. Have students
go through Problems
1 and 2, and make a
table organizing the
different types of angle
pairs. The table should
include a diagram to
associate the terms with
a visual image. Students
should understand that
some angle pairs hold
properties of other angle
pairs. For example, a
linear pair contains both
supplementary and
adjacent angles.
Grouping
Have students complete
Questions 1 through 7 with a
partner. Then have students
share their responses as a class.
Guiding Questions for Share Phase, Questions 1 through 7
What are the measures
of your angles in the
supplementary pair?
How did you calculate the
measure of the supplement
of the 22° angle?
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2.2 Special Angles and Postulates 173
Two angles are complementary angles if the sum of their angle measures is equal to 90º.
4. Use a protractor to draw a pair of complementary angles that
share a common side, and then measure each angle.
33°
57°
5. Use a protractor to draw a pair of complementary angles that do not share a common
side, and then measure each angle.
25°
65°
6. Calculate the measure of an angle that is complementary to /J.
62°
J
An angle that is complementary to /J has a measure of 28º.
Complementary angles that share a side form a right
angle.
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7. Determine the measure of each angle. Show your work and explain your reasoning.
a. Two angles are congruent and supplementary.
x 1 x 5 180
2x 5 180
x 5 90
The measure of each angle is 90º.
The sum of the angle measures is 180º.
b. Two angles are congruent and complementary.
x 1 x 5 90
2x 5 90
x 5 45
The measure of each angle is 45º.
The sum of the angle measures is 90º.
c. The complement of an angle is twice the measure of the angle.
x 1 2x 5 90
3x 5 90
x 5 30
The measure of the angle is 30º.
The measure of the complement of the angle is 2 3 30º, or 60º.
The sum of the angle measures is 90º.
d. The supplement of an angle is half the measure of the angle.
x 1 0.5x 5 180
1.5x 5 180
x 5 120
The measure of the angle is 120º.
The measure of the supplement of the angle is 1 __ 2 (120º), or 60º.
The sum of the angle measures is 180º.
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2.2 Special Angles and Postulates 175
Grouping
Have students complete
Question 8 with a partner. Then
share the responses as a class.
Guiding Questions for Share Phase, Question 8
What do parts (a) and (b)
have in common?
What is different parts (a)
and (b)?
Share a general description
for the solution methods of
parts (a) and (b).
8. Determine the angle measures in each diagram.
a.
(2x 1 20)°
(x 2 14)°
(x 2 14) 1 (2x 1 20) 5 90
3x 1 6 5 90
3x 5 84
x 5 28
Measure of 1st angle 5 x 2 14
5 28 2 14
5 14
Measure of 2nd angle 5 2x 1 20
5 2(28) 1 20
5 56 1 20
5 76
The measures of the two angles are 14º and 76º.
b.
(x 2
30)°
(3x 1 10)°
(3x 1 10) 1 (x 2 30) 5 180
4x 2 20 5 180
4x 5 200
x 5 50
Measure of 1st angle 5 x 2 30
5 50 2 30
5 20
Measure of 2nd angle 5 3x 1 10
5 3(50) 1 10
5 150 1 10
5 160
The measures of the two angles are 20º and 160º.
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Problem 2
Students explore adjacent
angles, linear pairs, and vertical
angles.
Grouping
Have students complete
Question 1 with a partner.
Then have students share their
responses as a class.
Guiding Questions for Share Phase, Question 1
Do adjacent angles share a
common vertex?
Do adjacent angles share a
common side?
Is there more than one
way to correctly draw angle 2
so that it is adjacent to
angle 1? Explain.
PROBLEM 2 Angle Relationships
You have learned that angles can be supplementary or complementary. Let’s explore other
angle relationships.
/1 and /2 are adjacent angles. /5 and /6 are not adjacent angles.
1 2
5 6
/3 and /4 are adjacent angles. /7 and /8 are not adjacent angles.
3 4
8
7
1. Analyze the worked example. Then answer each question.
a. Describe adjacent angles.
Two angles are adjacent if they share a common vertex and share
a common side.
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2.2 Special Angles and Postulates 177
b. Draw /2 so that it is adjacent to /1.
1
2
c. Is it possible to draw two angles that share a common vertex but do not share a
common side? If so, draw an example. If not, explain why not.
1 2
d. Is it possible to draw two angles that share a common side, but do not
share a common vertex? If so, draw an example. If not, explain why not.
It is not possible. If two angles share a side, then they must be adjacent on the
same line, and therefore share a common vertex.
Adjacent angles are two angles that share a common vertex and share a common side.
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Grouping
Have students complete
Question 2 with a partner.
Then share their responses
as a class.
Guiding Questions for Share Phase, Question 2
Does a linear pair of angles
always form a line?
Does a line have a
degree measure?
What is the sum of the
measures of the two angles
that form a linear pair?
Are the two angles forming a
linear pair of angles always
adjacent angles?
Are the two non-common
sides of a linear pair always
opposite rays?
/1 and /2 form a linear pair. /5 and /6 do not form a linear pair.
1
2
5 6
/3 and /4 form a linear pair. /7 and /8 do not form a linear pair.
3 4
7
8
2. Analyze the worked example. Then answer each question.
a. Describe a linear pair of angles.
Two angles are a linear pair if they are adjacent and their noncommon sides form
a line.
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2.2 Special Angles and Postulates 179
b. Draw /2 so that it forms a linear pair with /1.
2 1
c. Name all linear pairs in the !gure shown.
1
2
34
The linear pairs are /1 and /4, /1 and /3, /2 and /3, /2 and /4.
d. If the angles that form a linear pair are congruent, what can you conclude?
If the angles that form a linear pair are congruent, then the intersecting lines, line
segments, or rays forming the linear pair are perpendicular.
A linear pair of angles are two adjacent angles that have noncommon sides that form a line.
So, are the angles in a
linear pair always supplementary?
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Guiding Questions for Share Phase, Question 3
Are vertical angles adjacent
or non-adjacent angles?
Do intersecting lines always
form vertical angles?
Two intersecting lines form
how many different pairs of
vertical angles?
Three intersecting lines form
how many different pairs of
vertical angles?
Why do you suppose vertical
angles always congruent?
/1 and /2 are vertical angles. /5 and /6 are not vertical angles.
1
2
5
6
/3 and /4 are vertical angles. /7 and /8 are not vertical angles.
3
4
8 7
3. Analyze the worked example. Then answer each question.
a. Describe vertical angles.
Vertical angles are nonadjacent angles formed by two intersecting lines.
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2.2 Special Angles and Postulates 181
b. Draw /2 so that it forms a vertical angle with /1.
2
1
c. Name all vertical angle pairs in the diagram shown.
1
2
34
/1 and /2 are vertical angles. /3 and /4 are also vertical angles.
d. Measure each angle in part (c). What do you notice?
Vertical angles are always congruent.
Vertical angles are two nonadjacent angles that are formed by two intersecting lines,
line segments, or rays.
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Grouping
Have students complete
Questions 4 and 5 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Questions 4 and 5
Is there more than one way
to draw a correct diagram
to represent the conditional
statements in Question 5?
Identify the hypothesis of
the conditional statements in
Question 5.
Identify the conclusion of
the conditional statements in
Question 5.
4. Determine m/AED. Explain how you determined the angle measure.
(4x 1 25)°
DE
A
C
B
(x 1 94)°
Angles DEB and AEC are vertical angles, so they are congruent.
(x 1 94) 5 (4x 1 25)
94 5 3x 1 25
69 5 3x
23 5 x
m/AEC 5 23 1 94 5 117°
m/DEB 5 117°
Angles AEC and AED are supplementary.
m/AEC 1 m/AED 5 180
117 1 m/AED 5 180
m/AED 5 63
So, the measure of angle AED is 63°.
5. For each conditional statement, draw a diagram and then write the hypothesis as the
“Given” and the conclusion as the “Prove.”
a. m/DEG 1 m/GEF 5 180º, if /DEG and /GEF are a linear pair.
Given: /DEG and /GEF are a linear pair
Prove: m/DEG 1 m/GEF 5 180º
D E F
G
Make sure to carefully read the name of the angle whose measure
you want to know.
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2.2 Special Angles and Postulates 183
b. If /ABD and /DBC are complementary, then
___ › BA '
___
› BC .
Given: /ABD and /DBC are complementary
Prove:
___
› BA '
___
› BC
B
A
C
D
c. If /2 and /3 are vertical angles, then /2 > /3.
Given: /2 and /3 are vertical angles
Prove: /2 > /3
3
2
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Problem 3
The terms postulate and theorem
are de"ned. Euclid’s "rst "ve
postulates and Euclid’s Elements
are mentioned as the foundation
of Euclidean Geometry. Non-
Euclidean Geometry is also
mentioned. Students answer
questions related to parallel lines
and the interior angle measures
of triangles in spherical
geometry. The Linear Pair
Postulate, the Segment Addition
Postulate, and the Angle
Addition Postulate are stated.
Students use each postulate
to complete statements and
answer questions.
Guiding Questions for Discuss Phase
Who was Euclid and what is
he known for?
What is the key difference
between Euclidean and non-
Euclidean geometry?
PROBLEM 3 Postulates and Theorems
A postulate is a statement that is accepted
without proof.
A theorem is a statement that can
be proven.
The Elements is a book written by the Greek
mathematician Euclid. He used a small number of
unde!ned terms and postulates to systematically
prove many theorems. As a result, Euclid was able
to develop a complete system we now know as
Euclidean geometry.
Euclid’s !rst !ve postulates are:
1. A straight line segment can be drawn joining
any two points.
2. Any straight line segment can be extended
inde!nitely in a straight line.
3. Given any straight line segment, a circle can be
drawn that has the segment as its radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn that intersect a third line in such a way that the sum of the inner
angles on one side is less than two right angles, then the two lines inevitably must
intersect each other on that side if extended far enough. (This postulate is equivalent to
what is known as the parallel postulate.)
Euclid used only the !rst four postulates to prove the !rst 28 propositions or theorems of
The Elements, but was forced to use the !fth postulate, the parallel postulate, to prove the
29th theorem.
The Elements also includes !ve “common notions”:
1. Things that equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things that coincide with one another equal one another.
5. The whole is greater than the part.
It is important to note that Euclidean geometry is not the only system of geometry.
Examples of non-Euclidian geometries include hyperbolic and elliptic geometry. The
essential difference between Euclidean and non-Euclidean geometry is the nature of
parallel lines.
Greek mathematician Euclid is sometimes
referred to as the Father of Geometry.
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2.2 Special Angles and Postulates 185
Grouping
Have students complete
Questions 1 through 4 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Questions 1 through 4
Would the shortest distance
between 2 points be an arc in
spherical geometry?
Are lines in"nite in
spherical geometry?
Another way to describe the differences between these geometries is to consider two lines
in a plane that are both perpendicular to a third line.
In Euclidean geometry, the lines remain at a constant distance from each other and are
known as parallels.
In hyperbolic geometry, the lines “curve away” from each other.
In elliptic geometry, the lines “curve toward” each other and eventually intersect.
Spherical geometry, like its name implies, is a geometry that
substitutes a sphere for a plane, which makes it different from
plane geometry in signi!cant ways. In spherical geometry, some
topics that you have learned about, such as parallel lines and the
sum of the interior angles of a triangle are very different.
In spherical geometry, lines are de!ned as great circles
of a sphere, which divide the sphere into two congruent
hemispheres.
1. Which of the !gures in the diagram shown—p, q, or r—appear to be spherical lines?
Explain your reasoning.
Figures q and r are spherical lines, because they are great circles. Figure p is not a line.
2. Describe some similarities and differences between lines in Euclidean geometry and
lines in spherical geometry, given this diagram.
In Euclidean geometry, lines are straight and extend infinitely in two directions.
In spherical geometry, lines are great circles of a sphere. These lines appear to extend
infinitely in two directions also, but a line in spherical geometry overlaps itself.
p
r
q
To better understand
spherical geometry, have
students relate to their
knowledge of using a
globe. On a globe, the
equator is close to an
example of a great circle.
Show that any line of
longitude intersects
the equator at a
90-degree angle.
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Grouping
Have students complete
Questions 5 through 7 with a
partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Questions 5 through 7
What other geometric "gures
can you think of that would
have different properties in
spherical geometry?
What properties do you think
they would have?
3. Do you think that parallel lines can exist in spherical geometry? Explain your reasoning.
Parallel lines do not exist, because lines are defined as great circles. Any two
different great circles of a sphere must intersect.
4. At how many points do 2 different spherical lines intersect? Explain your reasoning.
Because spherical lines are great circles, two different spherical lines intersect
at 2 point.
In spherical geometry, triangles have different properties as well.
Consider the spherical triangle shown. It has 3 right angles, so the
sum of its interior angle measures is 2708.
The sides of a spherical triangle are always arcs of great circles.
5. Describe what you think are some similarities and differences
between triangles in Euclidean geometry and spherical triangles,
given this diagram.
Answers will vary.
The sum of the interior angle measures of a triangle in Euclidean geometry is 1808.
Spherical triangles do not necessarily have interior angle measures whose sum is 1808.
The sides of a Euclidean triangle are all straight line segments, and the sides of a
spherical triangle are curved line segments.
6. What do you think is the greatest sum of the interior angle measures of a spherical
triangle? What is the least sum? Explain your reasoning.
Answers will vary.
The sum of the interior angle measures of a spherical triangle is greater than 1808
and less than 5408.
7. What do you think would be the properties of a right
spherical triangle? Explain your reasoning.
A right spherical triangle could be a triangle with just
one 90-degree angle.
Using this textbook as a guide, you will develop your own
system of geometry, just like Euclid. You already used the
three unde!ned terms point, line, and plane to de!ne
related terms such as line segment and angle.
Your journey continues with the introduction of three
fundamental postulates:
The Linear Pair Postulate
The Segment Addition Postulate
The Angle Addition Postulate
Investigate the properties of lines on a
sphere and spherical triangles by drawing on a ball, such as a
basketball or soccer ball.
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2.2 Special Angles and Postulates 187
Grouping
Have students complete
Questions 8 through 10 with
a partner. Then have students
share their responses
as a class.
Guiding Questions for Share Phase, Questions 8 through 10
What is the hypothesis in the
Linear Pair Postulate?
What is the conclusion in the
Linear Pair Postulate?
Does the Linear Pair
Postulate tell you the sum
of the measures of the two
angles forming the linear pair
is equal to 180°?
What is the de"nition of
supplementary angles?
Does the de"nition of
supplementary angles tell
you the sum of the measures
of the two angles forming the
linear pair is equal to 180°?
What is the hypothesis in the
Segment Addition Postulate?
What is the conclusion in the
Segment Addition Postulate?
Would the Segment Addition
Postulate hold true if point
B was not located between
points A and C?
What is the hypothesis in the
Angle Addition Postulate?
What is the conclusion in the
Angle Addition Postulate?
Would the Angle Addition
Postulate hold true if point
D did not lie in the interior of
angle ABC?
You will use these postulates to make various conjectures. If you are able to prove your
conjectures, then the conjectures will become theorems. These theorems can then be used
to make even more conjectures, which may also become theorems. Mathematicians use this
process to create new mathematical ideas.
The Linear Pair Postulate states: “If two angles form a linear pair, then the angles
are supplementary.”
8. Use the Linear Pair Postulate to complete each representation.
a. Sketch and label a linear pair.
D E F
G
b. Use your sketch and the Linear Pair Postulate to write the hypothesis.
/DEG and /GEF form a linear pair.
c. Use your sketch and the Linear Pair Postulate to write the conclusion.
/DEG and /GEF are supplementary angles.
d. Use your conclusion and the de!nition of supplementary angles to write a statement
about the angles in your !gure.
m/DEG 1 m/GEF 5 180º
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The Segment Addition Postulate states: “If point B is on ___
AC and between points A and C,
then AB 1 BC 5 AC.”
9. Use the Segment Addition Postulate to complete each representation.
a. Sketch and label collinear points D, E, and F with point E between points D
and F.
D E F
b. Use your sketch and the Segment Addition Postulate to write the hypothesis.
Point E is on ___
DF and between points D and F.
c. Use your sketch and the Segment Addition Postulate to write the conclusion.
DE 1 EF 5 DF
d. Write your conclusion using measure notation.
m ___
DE 1 m ___
EF 5 m ___
DF
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The Angle Addition Postulate states: “If point D lies in the interior of /ABC, then
m/ABD 1 m/DBC 5 m/ABC.”
10. Use the Angle Addition Postulate to complete each representation.
a. Sketch and label /DEF with
___
› EG drawn in the interior of /DEF.
E F
G
D
b. Use your sketch and the Angle Addition Postulate to write the hypothesis.
Point D lies in the interior of /ABC.
c. Use your sketch and the Angle Addition Postulate to write the conclusion.
m/DEG 1 m/GEF 5 m/DEF
Be prepared to share your solutions and methods.
2.2 Special Angles and Postulates 189
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Check for Students’ Understanding
Given:
___ › AB '
___
› AC , m/ABC 5 44º, and
‹
___
› CD bisects /ACB
1. Determine the measure of each angle in the diagram.
908
1348
1348
238
238238
2386781138
678 1138
448
4481368
1368
A
CD
B
2. List all of the de"nitions, theorems, or reasons you used to determine the measures of the angles in
the diagram.
1. The sum of the measures of the interior angles of a triangle is equal to 180°
2. Vertical angles are congruent
3. Definition of linear pair
4. Definition of bisect
5. Definition of perpendicular
6. Definition of right angle
7. Definition of supplementary angles