324 Chapter 5 Relationships Within TrianglesObjective To.docx

324 Chapter 5 Relationships Within Triangles
Objective To use inequalities involving angles and sides of
triangles
In the Solve It, you explored triangles formed by various
lengths of board. You may have
noticed that changing the angle formed by two sides of the
sandbox changes the length
of the third side.
Essential Understanding Th e angles and sides of a triangle have
special
relationships that involve inequalities.
Property Comparison Property of Inequality
If a 5 b 1 c and c . 0, then a . b.
For a neighborhood improvement project, you
volunteer to help build a new sandbox at
the town playground. You have two boards
that will make up two sides of the
triangular sandbox. One is 5 ft long and the
other is 8 ft long. Boards come in the
lengths shown. Which boards can you use
for the third side of the sandbox? Explain.
Inequalities in
One Triangle

5-6
t
t
tt
o
lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
T
A
C T I V I T I
E S T
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC

Proof of the Comparison Property of Inequality
Given: a 5 b 1 c, c . 0
Prove: a . b
Statements Reasons
1) c . 0 1) Given
2) b 1 c . b 1 0 2) Addition Property of Inequality
3) b 1 c . b 3) Identity Property of Addition
4) a 5 b 1 c 4) Given
5) a . b 5) Substitution
Proof
hsm11gmse_NA_0506.indd 324 3/6/09 11:56:15 AM
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tml
Problem 1
Got It?
Lesson 5-6 Inequalities in One Triangle 325
Th e Comparison Property of Inequality allows you to prove the
following corollary to
the Triangle Exterior Angle Th eorem (Th eorem 3-11).

Proof of the Corollary
Given: /1 is an exterior angle of the triangle.
Prove: m/1 . m/2 and m/1 . m/3.
Proof: By the Triangle Exterior Angle Th eorem, m/1 5 m/2 1
m/3. Since
m/2 . 0 and m/3 . 0, you can apply the Comparison Property of
Inequality and conclude that m/1 . m/2 and m/1 . m/3.
Applying the Corollary
Use the fi gure at the right. Why is ml2 S ml3?
In nACD, CB > CD, so by the Isosceles Triangle Th eorem,
m/1 5 m/2. /1 is an exterior angle of nABD, so by the
Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3.
Th en m/2 . m/3 by substitution.
1. Why is m/5 . m/C?
You can use the corollary to Th eorem 3-11 to prove the
following theorem.
Corollary Corollary to the Triangle Exterior Angle Theorem
Corollary
Th e measure of an exterior
angle of a triangle is greater
than the measure of each of
its remote interior angles.
If . . .
/1 is an exterior angle

Then . . .
m/1 . m/2 and
m/1 . m/3
2 1
3
Proof
3
4
1
25
A
CD
B
Theorem 5-10
Theorem
If two sides of a triangle are
not congruent, then the
larger angle lies opposite
the longer side.
If . . .
XZ . XY

Then . . .
m/Y . m/Z
You will prove Theorem 5-10 in Exercise 40.
X
Y
Z
G
U
I
m
C
Th
G
How do you identify
an exterior angle?
An exterior angle
must be formed by the
extension of a side of the
triangle. Here, /1 is an
exterior angle of nABD,
but /2 is not.
tml

Problem 2
Got It?
Using Theorem 5-10
A town park is triangular. A landscape architect
wants to place a bench at the corner with the
largest angle. Which two streets form the corner
with the largest angle?
Hollingsworth Road is the longest street, so it is
opposite the largest angle. MLK Boulevard and
Valley Road form the largest angle.
2. Suppose the landscape architect
wants to place a drinking fountain
at the corner with the second largest
angle. Which two streets form the
corner with the second-largest angle?
Th eorem 5-11 below is the converse of Th eorem 5-10. Th e
proof of Th eorem 5-11 relies
on indirect reasoning.
Indirect Proof of Theorem 5-11
Given: m/A . m/B
Prove: BC . AC
Step 1 Assume temporarily that BC 6 AC . Th at is, assume
temporarily that either

BC , AC or BC 5 AC .
Step 2 If BC , AC , then m/A , m/B (Th eorem 5-10). Th is
contradicts the given
fact that m/A . m/B. Th erefore, BC , AC must be false.
If BC 5 AC , then m/A 5 m/B (Isosceles Triangle Th eorem). Th
is also
contradicts m/A . m/B. Th erefore, BC 5 AC must be false.
Step 3 Th e temporary assumption BC 6 AC is false, so BC .
AC .
120 yd
175 yd
105 yd
Hol
ling
swo
rth
Roa
d
Va
lle
y
Ro
ad

MLK Boulevard
Theorem 5-11
Theorem
If two angles of a triangle
are not congruent, then the
longer side lies opposite
the larger angle.
If . . .
m/A . m/B
Then . . .
BC . AC
B
A
C
Proof
G
A
w
l
w
H
o
V
How do you fi nd the

side opposite an
angle?
Choose an angle of
a triangle. The side
opposite it is the only
side that does not have
an endpoint at the vertex
of the angle.
tml
Problem 3
Got It?
Problem 4
Got It?
Using Theorem 5-11
Multiple Choice Which choice shows the sides of kTUV in
order from
shortest to longest?
TV, UV, UT UV, UT, TV
UT, UV, TV TV, UT, UV
By the Triangle Angle-Sum Th eorem, m/T 5 60. 58 , 60 , 62, so

m/U , m/T , m/V. By Th eorem 5-11, TV , UV , UT. Choice A is
correct.
3. Reasoning In the fi gure at the right, m/S 5 24
and m/O 5 130. Which side of nSOX is the
shortest side? Explain your reasoning.
For three segments to form a triangle, their lengths must be
related in a certain way.
Notice that only one of the sets of segments below can form a
triangle. Th e sum of the
smallest two lengths must be greater than the greatest length.
Using the Triangle Inequality Theorem
Can a triangle have sides with the given lengths? Explain.
A 3 ft, 7 ft, 8 ft B 5 ft, 10 ft, 15 ft
3 1 7 . 8 7 1 8 . 3 8 1 3 . 7 5 1 10 6 15
10 . 8 15 . 3 11 . 7 15 6 15
Yes. Th e sum of the lengths of any two No. Th e sum of 5
and 10 is not
sides is greater than the length of the greater than 15. Th is
contradicts
third side. Th
eorem 5-12.
4. Can a triangle have sides with the given lengths? Explain.
a. 2 m, 6 m, and 9 m b. 4 yd, 6 yd, and 9 yd
T
U V

58 62
S
O
X
3 cm 2 cm2 cm3 cm
6 cm5 cm
Theorem 5-12 Triangle Inequality Theorem
Th e sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
XY 1 YZ . XZ YZ 1 XZ . XY XZ 1 XY . YZ
You will prove Theorem 5-12 in Exercise 45.
X
Y
Z
G
M
s
B
m
G

How do you use
the angle measures
to order the
side lengths?
List the angle measures
in order from smallest to
largest. Then replace the
measure of each angle
with the length of the
side opposite.
C
A
How do you use the
Triangle Inequality
Theorem?
Test each pair of side
lengths. The sum of each
pair must be greater than
the third length.
tml
Problem 5
Got It?

Finding Possible Side Lengths
Algebra In the Solve It, you explored the possible dimensions of
a triangular
sandbox. Two of the sides are 5 ft and 8 ft long. What is the
range of possible
lengths for the third side?
Let x represent the length of the third side. Use the Triangle
Inequality Th eorem
to write three inequalities. Th en solve each inequality for x.
x 1 5 . 8 x 1 8 . 5 5 1 8 . x
x . 3 x . 23 x , 13
Numbers that satisfy x . 3 and x . 23 must be greater than 3. So,
the third
side must be greater than 3 ft and less than 13 ft.
5. A triangle has side lengths of 4 in. and 7 in. What is the
range of
possible lengths for the third side?
The lengths of two sides of
the triangle are 5 ft and 8 ft.
The range of possible
lengths of the third
side
Use the Triangle Inequality Theorem to
write three inequalities. Use the solutions
of the inequalities to determine the
greatest and least possible lengths.

Practice and Problem-Solving Exercises
Explain why ml1 S ml2.
6. 7. 8.
PracticeA See Problem 1.
2
3
1 4
2
1 34
3
1 4
2
Lesson Check
Do you know HOW?
Use kABC for Exercises 1 and 2.
1. Which side is the longest?
2. Which angle is the smallest?
3. Can a triangle have sides of
lengths 4, 5, and 10? Explain.

Do you UNDERSTAND?
4. Error Analysis A friend tells you that she drew a
triangle with perimeter 16 and one side of length 8.
How do you know she made an error in her drawing?
5. Reasoning Is it possible to draw a right triangle
with an exterior angle measuring 88? Explain your
reasoning.
A C
B
85
5
4
tml
For Exercises 9–14, list the angles of each triangle in order
from smallest
to largest.
9. 10. 11.
12. nABC, where AB 5 8, 13. nDEF , where DE 5 15, 14. nXYZ
, where XY 5 12,

BC 5 5, and CA 5 7 EF 5 18, and DF 5 5 YZ 5 24, and ZX 5 30
For Exercises 15–20, list the sides of each triangle in order
from shortest
to longest.
15. 16. 17.
18. nABC , with 19. nDEF , with 20. nXYZ , with
m/A 5 90, m/D 5 20, m/X 5 51,
m/B 5 40, and m/E 5 120, and m/Y 5 59, and
m/C 5 50 m/F 5 40 m/Z 5 70
Can a triangle have sides with the given lengths? Explain.
21. 2 in., 3 in., 6 in. 22. 11 cm, 12 cm, 15 cm 23. 8 m, 10 m, 19
m
24. 1 cm, 15 cm, 15 cm 25. 2 yd, 9 yd, 10 yd 26. 4 m, 5 m, 9 m
Algebra Th e lengths of two sides of a triangle are given. Find
the range of
possible lengths for the third side.
27. 8 ft, 12 ft 28. 5 in., 16 in. 29. 6 cm, 6 cm
30. 18 m, 23 m 31. 4 yd, 7 yd 32. 20 km, 35 km
33. Think About a Plan You are setting up a study area where
you will
do your homework each evening. It is triangular with an
entrance on
one side. You want to put your computer in the corner with the
largest
angle and a bookshelf on the longest side. Where should you
place your

computer? On which side should you place the bookshelf?
Explain.
• What type of triangle is shown in the fi gure?
• Once you fi nd the largest angle of a triangle, how do you fi
nd the
longest side?
34. Algebra Find the longest side of nABC , with m/A 5 70,
m/B 5 2x 2 10,
and m/C 5 3x 1 20.
See Problem 2.
5.8
2.7
4.3
L
M
K
C
E
D
105
3x
x JG

4
6
H
See Problem 3.
O
M
N
75
45
G
F
H
28
110
T
U V
30
See Problem 4.
See Problem 5.
7 ft

EntranceApplyB
tml
35. Writing You and a friend compete in a scavenger hunt
at a museum. Th e two of you walk from the Picasso exhibit
to the Native American gallery along the dashed red line. When
he sees that another team is ahead of you, your friend says,
“Th ey must have cut through the courtyard.” Explain what your
friend means.
36. Error Analysis Your family drives across Kansas on
Interstate 70. A sign reads, “Wichita 90 mi, Topeka 110 mi.”
Your little brother says, “I didn’t know that it was only 20
miles
from Wichita to Topeka.” Explain why the distance between the
two cities does not have to be 20 mi.
Reasoning Determine which segment is shortest in each
diagram.
37. 38. 39.
40. Developing Proof Fill in the blanks for a proof of Th eorem
5-10: If two sides of a
triangle are not congruent, then the larger angle lies opposite
the longer side.
Given: nTOY, with YO . YT

Prove: a. 9 . b. 9
Mark P on YO so that YP > YT. Draw TP.
Statements Reasons
1) YP > YT 1) Ruler Postulate
2) m/1 5 m/2 2) c. 9
3) m/OTY 5 m/4 1 m/2 3) d. 9
4) m/OTY . m/2 4) e. 9
5) m/OTY . m/1 5) f. 9
6) m/1 . m/3 6) g. 9
7) m/OTY . m/3 7) h. 9
41. Prove this corollary to Th eorem 5-11: Th e perpendicular
segment from
a point to a line is the shortest segment from the point to the
line.
Given: PT ' TA
Prove: PA . PT
42. Probability A student has two straws. One is 6 cm long and
the other is 9 cm long.
She picks a third straw at random from a group of four straws
whose lengths are
3 cm, 5 cm, 11 cm, and 15 cm. What is the probability that the
straw she picks will

allow her to form a triangle? Justify your answer.
Native American Gallery
Picasso
Exhibit
SR
P
Q
30
40
C D
B A
114
110
32
30
X
W
Y
Z
48
47 95

40
3
4
1
2
O
YT
P
Proof
P
T A
ChallengeC
tml
For Exercises 43 and 44, x and y are integers such that 1 R x R
5 and 2 R y R 9.
43. Th e sides of a triangle are 5 cm, x cm, and y cm. List all

possible (x, y) pairs.
44. Probability What is the probability that you can draw an
isosceles triangle that has
sides 5 cm, x cm, and y cm, with x and y chosen at random?
45. Prove the Triangle Inequality Th eorem: Th e sum of the
lengths of any two sides of
a triangle is greater than the length of the third side.
Given: nABC
Prove: AC 1 CB . AB
(Hint: On BC
)
, mark a point D not on BC , so that DC 5 AC .
Draw DA and use Th eorem 5-11 with nABD.)
A
C
B
Proof
Mixed Review
Write the fi rst step of an indirect proof of the given statement.
50. Th e side is at least 2 ft long. 51. nPQR has two congruent
angles.
52. You know that AB > XY, BC > YZ, and CA > ZX . By what
theorem or

postulate can you conclude that nABC > nXYZ?
Get Ready! To prepare for Lesson 5-7, do Exercises 53–55.
Use the fi gure at the right.
53. What is m/P?
54. What is m/D?
55. Is it possible for AW to equal OG?
See Lesson 5-5.
See Lesson 4-2.
See Lesson 3-5.A O
D
G
P W
35 30
105
125
Standardized Test Prep
46. Th e fi gure shows the walkways connecting four
dormitories on
a college campus. What is the greatest possible whole-number
length, in yards, for the walkway between South dorm and
East dorm?

47. What is the length of a segment with endpoints A(213, 216)
and B(29, 213)?
48. How many sides does a convex quadrilateral have?
49. /1 and /2 are corresponding angles formed by two parallel
lines and a
transversal. If m/1 5 33x 1 2 and m/2 5 68, what is the value of
x?
SAT/ACT
West
South
42 yd
31 yd
57 yd
North
East
tml
copyright:
Question 1 of 25
1.0 Points
Certain guidelines should be kept in mind when coding. Among
these is(are) the following:

A.the coding scheme should be appropriate to the analysis
intended in the study.
B.the coding scheme should be appropriate to the theoretical
concepts being examined.
C.the reliability of the coder(s) should always be verified.
D.code categories should be both mutually exclusive and
exhaustive
E.all of these choices are correct
Question 2 of 25
1.0 Points
The end product of the coding process in quantitative analysis
refers to
A.the conversion of data items into numerical codes
B.the conversions of an attribute to a variable
C.the assignment of a number or numeral to each questionnaire
item
D.the transfer of variables to the computer
E.the assignment of a variable to a computer column
Question 3 of 25
1.0 Points

An example of multivariate analysis would be
A.an examination of the ages of all women who are corporate
executives
B.an analysis of the relationship between age, sex, and type of
nightspot frequented in a given city in the Midwest
C.an analysis of the relationship that exists between types of
undergraduate major and positions held in business
D.an analysis of the relationship between type of offense and
length of prison sentence for those who had a jury trial
E.all of these choices illustrate multivariate analyses
Question 4 of 25
1.0 Points
In reading a table that someone else has constructed, the rule of
thumb is
A.if the table is percentaged down, read across, and if the table
is percentaged across, read down
B.if the table is percentaged down, read down
C.if the table is percentaged across, read across
D.if the table is percentaged down, read down and if the table is
percentaged across, read across
E.all of these choices are good rules of thumb for reading a
table that someone else constructed

Question 5 of 25
1.0 Points
The College of Arts and Science at Delta University has nine
departments. The number of faculty in each department is
shown below. What is the median number of faculty in the
College of Arts and Science? 8, 12, 9, 15, 17, 11, 13, 14, 7
A.12
B.17
C.4.5
D.5
E.there is no median in these data
Question 6 of 25
1.0 Points
The College of Arts and Science at Delta University has nine
departments. The number of faculty in each department is
shown below. What is the mean number of faculty in the
College of Arts and Science at Delta University? 8, 12, 9, 15,
17, 11, 13, 14, 7
A.12
B.17
C.10.7

D.11.8
E.There is no mean for these data
Question 7 of 25
1.0 Points
The most frequent attribute, in either grouped or ungrouped
data, is the
A.mean
B.median
C.mode
D.range
E.marginal
Question 8 of 25
1.0 Points
A measure of dispersion describes
A.where the data are clustered
B.which data are the most important
C.how spread the data are around some central value
D.which data are appropriate for analysis
E.discrete data only

Question 9 of 25
1.0 Points
Subgroup comparisons are done by
A.dividing cases into the appropriate subgroups, describing
each subgroup in terms of a given variable, and comparing those
descriptions across subgroups
B.dividing, describing, and comparing subgroups on
independent and dependent variables
C.analyzing the simultaneous relationships among several
variables among subgroups
D.dividing, describing and comparing univariate independent
variables on subgroups
E.all of these choices reflect how subgroup comparisons are
done
Question 10 of 25
1.0 Points
Which of the following measures of central tendency can be
used at any level of measurement?
A.mean
B.mode
C.median

D.standard deviation
E.the mean, the mode, and the median
Question 11 of 25
1.0 Points
A description of the number of times that the various attributes
of a variable are observed is called a
A.frequency distribution
B.mean
C.measure of dispersion
D.contingency table
E.multivariate table
Question 12 of 25
1.0 Points
You are helping code one of your professor's data. You are
coding occupation and notice that there is no code for a
respondent's indicated occupation. Which of the following
statements would not be incorrect in relating the problem to the
professor?
A.The coding scheme for occupation is not mutually exclusive
B.The lack of a code for this particular occupation signals a
mismatch between the data and the coding scheme

C.The coding scheme for occupation is not exhaustive
D.The coding scheme is not a good scheme
E.All of these choices would be correct statements
Question 13 of 25
1.0 Points
Which of the following statements about data entry is FALSE?
A.There are many ways to enter data on the computer
B.Optical scan sheets can be fed into machines that convert the
marks into data
C.Data entry can sometimes occur in the process of data
collection
D.Data entry specialists can enter data from a questionnaire into
a data matrix or spreadsheet
E.All of these choices are TRUE
Question 14 of 25
1.0 Points
While evaluating a research report, Dr. Childs focused on the
issue of research design. Which of the following questions
would be the least useful for evaluating the design?
A.What was the unit of analysis?
B.Is the study cross-sectional or longitudinal?

C.What population does the researcher want to draw
conclusions about
D.Was the purpose of the study exploration, description, or a
combination of these?
E.All of the above would be helpful questions for evaluating the
design
Question 15 of 25
1.0 Points
If a reader asks "Who originally collected the data being
reanalyzed?" or "When were the data collected?" that reader is
most clearly examining issues that focus on
A.research design
B.survey research
C.theoretical orientation
D.field research
E.analysis of existing statistics
Question 16 of 25
1.0 Points
Which of the following is NOT a suggestion used for evaluating
a web site?
A.Who is the author of the web site?

B.Does the site advocate a particular viewpoint?
C.Does the site give accurate and complete references?
D.Are the data up-to-date?
E.All of these choices are suggestions for evaluating a web site.
Question 17 of 25
1.0 Points
Which of the following guidelines for reporting on the analysis
of qualitative data is FALSE?
A.Provide enough detail so that your reader has a sense of
having made the observations with you
B.Provide only those data that support your interpretations
C.Provide enough information so that the reader might reach a
conclusion that differs from yours
D.Provide enough detail so that the reader could replicate your
study
E.All of the above are TRUE
Question 18 of 25
1.0 Points
Sam wants to prepare a short report on his research for an
academic journal. He simply wants to tell the audience about his
findings. The most appropriate form for his report is

A.a working paper
B.an article
C.a book
D.a research note
E.all of these choices are equally appropriate forms for Sam's
work
Question 19 of 25
1.0 Points
Which of the following is NOT plagiarism?
A.Using another writer's exact words without using quotation
marks
B.Taking a passage of eight or more words without a citation
C.Editing another's words and presenting those as your own
D.Paraphrasing another's words and presenting those as your
own
E.All of these choices illustrate plagiarism
Question 20 of 25
1.0 Points
Which of the following statements is FALSE about the purpose
of a literature review? Literature reviews should:

A.point out general agreements and minimize the disagreements
among previous researchers
B.show where your research fits into the general body of
scientific knowledge
C.lay the groundwork for your study
D.show why your research may have value in the larger scheme
of things
E.all of these choices about literature reviews are TRUE
Question 21 of 25
1.0 Points
A typical report begins with a(n)
A.literature review
B.purpose and overview statement
C.analysis and description statement
D.summary statement
E.conclusions statement
Question 22 of 25
1.0 Points
A good abstract DOES NOT include a statement about

A.the purpose of the research
B.the findings of each table and graph
C.the major findings
D.the methods used
E.all of these choices are included in a good abstract
Question 23 of 25
1.0 Points
Which of the following statements is FALSE concerning data on
the web?
A.Information on the web is available to a large proportion of
the population
B.Information that is placed on the web is typically screened for
accuracy
C.Out-of-date data may be reported on the web
D.University research centers are usually committed to peer
review
E.All of these statements are TRUE
Question 24 of 25
1.0 Points
A good literature review will make use of a process that is akin
to which type of sampling?

A.Simple random sampling
B.Stratified sampling
C.Quota sampling
D.Cluster sampling
E.Snowball sampling
Question 25 of 25
1.0 Points
In reading a research report Laura focused on whether indexes
or scales were used and whether there were multiple dimensions
to concepts and variables. In focusing on these issues Laura was
focusing on:
A.theory
B.sampling
C.data analysis
D.measurement
E.research design

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