special angles and postulates 2 - ihs math · a new angle special angles and postulates 2.2 key...

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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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172 Chapter 2 Introduction to Proof

2

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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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.


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.


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.


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Grouping


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


Question 1 with a partner.

Then have students share their

responses as a class.


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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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


Question 2 with a partner.

Then share their responses

as a class.


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


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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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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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


a. Describe vertical angles.

Vertical angles are nonadjacent angles formed by two intersecting lines.

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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


Questions 4 and 5 with a


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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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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Grouping





as a class.


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





as a class.


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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Grouping


Questions 8 through 10 with

a partner. Then have students


as a class.


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°?


Segment Addition Postulate?


Segment Addition Postulate?

Would the Segment Addition

Postulate hold true if point

B was not located between

points A and C?


Angle Addition Postulate?


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.


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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

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