5-5 Inequalities in Triangles

5-5
5-5
Inequalities in Triangles
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To use inequalities involving
Graph the triangles with the given vertices. List the sides in order from
shortest to longest. 1–4. See back of book.
angles of triangles
• To use inequalities involving
sides of triangles
. . . And Why
To locate the largest corners
on a triangular backyard
deck, as in Example 2
Lessons 1-8 and 5-4
1. A(5, 0), B(0, 8), C(0, 0)
2. P(2, 4), Q(-5, 1), R(0, 0)
3. G(3, 0), H(4, 3), J(8, 0)
4. X(-4, 3), Y(-1, 1), Z(-1, 4)
1
2
To use inequalities involving
angles of triangles
To use inequalities involving
sides of triangles
Examples
1
2
3
4
Recall the steps for indirect proof.
5. You want to prove m&A . m&B. Assume that mlA K mlB.
Write the first step of an indirect proof.
6. In an indirect proof, you deduce that AB $ AC is false. What conclusion
can you make? AB R AC
1
Objectives
5
Applying the Corollary
Real-World Connection
Using Theorem 5-11
Using the Triangle Inequality
Theorem
Finding Possible Side Lengths
Math Background
Inequalities Involving Angles of Triangles
Theorems 5-10 and 5-11 can be
treated as extending the Isosceles
Triangle Theorem and its converse
to the case of inequality. These
theorems enable students to prove
in Exercise 41 that the shortest
segment from a point to a line
is perpendicular to the line.
When you empty a container of juice into two
glasses, it is difficult to be sure that the glasses
get equal amounts. You can be sure, however,
that each glass holds less than the original amount
in the container. This is a simple application
of the Comparison Property of Inequality.
More Math Background: p. 256D
Key Concepts
Property
Comparison Property of Inequality
Lesson Planning and
Resources
If a = b + c and c . 0, then a . b.
Proof
See p. 256E for a list of the
resources that support this lesson.
Proof of the Comparison Property
Given: a = b + c, c . 0
PowerPoint
Prove: a . b
Statements
1.
c.0
2. b + c . b + 0
3. b + c . b
4.
a=b+c
5.
a .b
Bell Ringer Practice
Reasons
1.
2.
3.
4.
5.
Check Skills You’ll Need
Given
Addition Property of Inequality
Simplify.
Given
Substitute a for b + c in Statement 3.
For intervention, direct students to:
Finding Distance
Lesson 1-8: Example 1
Extra Skills, Word Problems, Proof
Practice, Ch. 1
The Comparison Property of Inequality allows you to prove the following corollary
to the Exterior Angle Theorem for triangles (Theorem 3-13).
Lesson 5-5 Inequalities in Triangles
Special Needs
Below Level
L1
To illustrate that Theorem 4-10 and 4-11 only apply to
one triangle, draw TUV along with a similar but
smaller triangle, ABC. Show that TV AB does not
imply that TV AC.
learning style: visual
289
Indirect Proof
Lesson 5-4: Examples 3, 5
Extra Skills, Word Problems, Proof
Practice, Ch. 5
L2
Using geometry software to alter the sides and angles
of triangles (beginning with an isosceles triangle) may
help students understand Theorems 5-10 and 5-11.
learning style: visual
289
2. Teach
Key Concepts
Corollary
Corollary to the Triangle Exterior Angle Theorem
The measure of an exterior angle of a triangle is
greater than the measure of each of its remote
interior angles.
Guided Instruction
3
1
EXAMPLE
Remind students that a corollary
is both a statement that follows
directly from a theorem and a
theorem itself.
2
1
2
m&1 . m&2 and m&1 . m&3
Proof
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 Exterior Angle Theorem, m&1 = m&2 + 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.
Tactile Learners
EXAMPLE
Have students construct a triangle
to model the problem, using sides
18 cm, 21 cm, and 27 cm long to
see that the larger angles are
opposite the longer sides.
1
Applying the Corollary
Additional Examples
4
X
1
3
3
P
1
By the corollary to the Exterior Angle Theorem,
m&1 . m&3. So, m&2 . m&3 by substitution.
1 Explain why m&4 m&5.
B
5
Y
2
Quick Check
4
2
1 Explain why m&OTY . m&3. mlOTY S ml2
T
by the Comparison Prop. of Ineq. Since it was
proven that ml2 S ml3, then by the Trans. Prop. mlOTY S ml3.
You will prove the following inequality theorem in the exercises.
Y
C
Key Concepts
ml4 S ml2 by the Corollary
to the Exterior Angle Theorem,
ml2 ≠ ml5 because l2 and
l5 are congruent corresponding angles, and
ml4 S ml5 by substitution.
Theorem 5-10
Y
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.
2 In RGY, RG = 14, GY = 12,
and RY = 20. List the angles from
largest to smallest. lG, lY, lR
2
Real-World
Connection
Careers Landscape architects
blend structures with
decorative plantings.
Quick Check
290
EXAMPLE
Real-World
Z
X
Connection
Deck Design A landscape architect is
designing a triangular deck. She wants to
place benches in the two larger corners.
Which corners have the larger angles?
Corners B and C have the larger angles.
They are opposite the two longer sides
of 27 ft and 21 ft.
2 List the angles of #ABC in order from
smallest to largest. lA, lC, lB
A
27 ft
C
21 ft
18 ft
B
Chapter 5 Relationships Within Triangles
Advanced Learners
English Language Learners ELL
L4
Have students find the range of possible values for
the length of the third side of a triangle whose other
side lengths are a and b. If bL a, b–a RcRa±b.
290
O
In the diagram, m&2 = m&1 by the Isosceles
Triangle Theorem. Explain why m&2 . m&3.
PowerPoint
A
EXAMPLE
learning style: verbal
Use the proof of the corollary before Example 1 to
review the meaning of the term corollary. A corollary
is a statement that can be proved easily by applying
the theorem.
learning style: verbal
2
1
Guided Instruction
Inequalities Involving Sides of Triangles
Theorem 5-10 on the preceding page states that the larger angle is opposite the
longer side. The converse is also true.
Key Concepts
Theorem 5-11
If two angles of a triangle are not congruent,
then the longer side lies opposite the larger angle.
B
C
A
Indirect Proof of Theorem 5-11
Given: m&A . m&B
Prove: BC . AC
Step 2 If BC , AC, then m&A , m&B (Theorem 5-10). This contradicts the
given fact that m&A . m&B. Therefore, BC , AC must be false.
If BC = AC, then m&A = m&B (Isosceles Triangle Theorem). This also
contradicts m&A . m&B. Therefore, BC = AC must be false.
Step 3 The assumption BC AC is false, so BC . AC.
A
1
4
5
A
A
B
B
E
D
D
C
C
Math Tip
EXAMPLE
Discuss why “x 2 and x -2”
can be written as x 2. Point
out that the possible lengths are
written as a compound inequality.
PowerPoint
3 In ABC, &C is a right angle.
Which is the longest side? AB
Using Theorem 5-11
E
D
C
B
EXAMPLE
EXAMPLE
Additional Examples
E
D
C
B
A
3
C
B
A
2
3
E
D
C
B
Error Prevention
Ask: Why does comparing only
the sum of the two shorter sides
and the longest side tell whether
a triangle can have the given
lengths? If the least sum is
greater than the greatest length,
the other inequalities must be
true also.
5
Step 1 Assume BC AC. That is, assume BC , AC or BC = AC.
EXAMPLE
Emphasize that Theorems 5-10
and 5-11 apply only within
triangles, not between triangles.
4
If m&A . m&B, then BC . AC.
Proof
3
D
E
E
Test-Taking Tip
Don’t be distracted!
Choice B lists the sides
in order, but from
longest to shortest, not
shortest to longest.
Multiple Choice Which choice shows the sides of
#TUV in order from shortest to longest?
TV, UV, UT
UT, UV, TV
UV, UT, TV
T
TV, UT, UV
U
58
V
By the Triangle Angle-Sum Theorem, m&T 60.
58 60 62, so m&U m&T m&V. By Theorem 5-11,
TV UV UT. The correct choice is A.
Quick Check
62
4 Can a triangle have sides with
the given lengths? Explain.
a. 2 cm, 2 cm, 4 cm no; 2 ± 2 4
b. 8 in., 15 in., 12 in. yes;
8 ± 12 S 15
3 List the sides of the #XYZ in order from
shortest to longest. Explain your listing.
YZ R XY R XZ since mlY ≠ 80.
X
5 In FGH, FG = 9 m and
GH = 17 m. Describe the possible
lengths of FH. 8 R FH R 26
Y
40
60
Z
The lengths of three segments must be related in a certain way to form a triangle.
3 cm
3 cm
2 cm
Resources
• Daily Notetaking Guide 5-5 L3
• Daily Notetaking Guide 5-5—
L1
Adapted Instruction
2 cm
5 cm
6 cm
3 cm, 3 cm, 5 cm
2 cm, 2 cm, 6 cm
Closure
Explain why each triangle below
is impossible.
Notice that only one of the sets of three segments above can form a triangle.
The sum of the smallest two lengths must be greater than the greatest length.
This is Theorem 5-12 (see next page). You will prove it in the exercises.
Lesson 5-5 Inequalities in Triangles
12 100° 15
50°
30°
25
291
18
12
32
In the first triangle, the side
opposite the smallest angle
is not the shortest side; the
second triangle violates the
Triangle Inequality Theorem.
291
3. Practice
Key Concepts
Theorem 5-12
Triangle Inequality Theorem
The sum of the lengths of any two sides of a triangle
is greater than the length of the third side.
Assignment Guide
Y
XY + YZ . XZ
1 A B 1-9, 30, 33
YZ + ZX . YX
2 A B
10-29, 31, 32, 34-37
C Challenge
38-41
Test Prep
Mixed Review
42-49
50-61
4
Homework Quick Check
Using the Triangle Inequality Theorem
Can a triangle have sides with the given lengths? Explain.
To check students’ understanding
of key skills and concepts, go over
Exercises 5, 22, 29, 30, 33.
Exercise 5 Ask: How do you know
x 0? Side lengths must be
positive numbers.
EXAMPLE
Z
X
ZX + XY . ZY
a. 3 ft, 7 ft, 8 ft
Quick Check
Visual Learners
Exercises 7–9, 13–15 Have
3 + 6 10
8+7.3
The sum of 3 and 6
is not greater than 10,
contradicting Theorem 5-12.
The sum of any two lengths
is greater than the third length.
No
4 Can a triangle have sides with the given lengths? Explain.
a. 2 m, 7 m, and 9 m
b. 4 yd, 6 yd, and 9 yd
no; 2 ± 7 w 9
yes; 4 ± 6 S 9; 6 ± 9 S 4; and 4 ± 9 S 6
5
students draw and label each
triangle to make sure that they
identify opposite angles correctly.
3+7.8
3 + 8 . 7 Yes
For: Triangle Inequality Activity
Use: Interactive Textbook, 5-5
Exercise 6 Make sure that
students can explain why &I is
the largest angle in GHI.
b. 3 cm, 6 cm, 10 cm
EXAMPLE
Finding Possible Side Lengths
Algebra A triangle has sides of lengths 8 cm and 10 cm. Describe the lengths
possible for the third side.
Let x represent the length of the third side. By the Triangle Inequality Theorem,
x + 8 . 10
x + 10 . 8
x.2
8 + 10 . x
x . -2
x , 18
The third side must be longer than 2 cm and shorter than 18 cm.
Quick Check
GPS Guided Problem Solving
L3
L4
Enrichment
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
Practice 5-5
Inequalities in Triangles
1 ft
N
M
C
2.
3.
25 m
B
Practice and Problem Solving
A
Practice by Example
4.1 cm
1.9 cm
D
6 ft
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Explain why ml1 S ml2. 1–3. See margin.
Q
14 m
18 m
5.5 ft
Example 1
EXERCISES
L3
Date
Determine the two largest angles in each triangle.
1.
5 A triangle has sides of lengths 3 in. and 12 in. Describe the lengths possible for the
third side. 9 R x R 15
S
4.0 cm
Example 1
R
1.
P
5.
13 cm
I
7 cm
A
15 yd
(page 290)
39 cm
K
20 yd
11 cm
R
S
6.
24 cm
T
25 yd
B
Can a triangle have sides with the given lengths? Explain.
7. 4 m, 7 m, and 8 m
8. 6 m, 10 m, and 17 m
9. 4 in., 4 in., and 4 in.
10. 1 yd, 9 yd, and 9 yd
11. 11 m, 12 m, and 13 m
12. 18 ft, 20 ft, and 40 ft
14. 8 12 yd, 9 14 yd, and 18 yd
15. 2.5 m, 3.5 m, and 6 m
13. 1.2 cm, 2.6 cm, and 4.9 cm
3.
3
GO for
Help
55 cm
A
2.
2
L
4.
3 2
2
1
1
4
1
4 3
List the sides of each triangle in order from shortest to longest.
B
16.
C
47ⴗ
17. L
S
18.
75ⴗ
T
© Pearson Education, Inc. All rights reserved.
41ⴗ
56ⴗ
O
107ⴗ
R
A
292
B
List the angles of each triangle in order from largest to smallest.
19. S
1.7
3.4
D
21. P
S
20.
Chapter 5 Relationships Within Triangles
28
R
13
25
2.6
38
26
N
A
J
21
O
The lengths of two sides of a triangle are given. Describe the lengths possible
for the third side.
22. 4 in., 7 in.
23. 9 cm, 17 cm
24. 5 ft, 5 ft
25. 11 m, 20 m
26. 6 km, 8 km
27. 24 in., 37 in.
292
1. l3 O l2 because they
are vertical ' and ml1
S ml3 by Corollary
to the Ext. l Thm. So,
ml1 S ml2 by subst.
2. An ext. l of a k is larger
than either remote int. l.
3. ml1 S ml4 by Corollary
to the Ext. l Thm. and l4
O l2 because if n lines,
then alt. int. ' are O.
16. No; 2 ± 3 w 6.
17. Yes; 11 ± 12 S 15;
12 ± 15 S 11;
11 ± 15 S 12.
4
Example 2
(page 290)
4.
K
5. lD, lC, lE
4.3
2.7
M
C
5.8
L
lM, lL, lK
7. #ABC, where AB = 8,
BC = 5, and CA = 7
lA, lB, lC
Example 3
(page 291)
Careers
List the angles of each triangle in order from smallest to largest.
D
6.
Exercise 28 Traveling sales
H
4
x 105
E
3x
G
6
lG, lH, lI
I
9. #XYZ,where XY = 12,
YZ = 24, and ZX = 30
lZ, lX, lY
List the sides of each triangle in order from shortest to longest.
10.
O
8. #DEF, where DE = 15,
EF = 18, and DF = 5
lE, lF, lD
11. G
12.
28
Exercise 29 Check that students
T
use the Triangle Inequality
Theorem in their explanation.
45
(page 292)
Example 5
(page 292)
H
30
U
N
V
F
TU,
UV,
TV
MN, ON, MO
FH, GF, GH
13. #ABC, with
14. #DEF, with
15. #XYZ, with
m&A = 90,
m&D = 20,
m&X = 51,
m&B = 40, and
m&E = 120, and
m&Y = 59, and
m&C = 50
m&F = 40
m&Z = 70
AC, AB, CB
EF, DE, DF
ZY, XZ, XY
Can a triangle have sides with the given lengths? Explain. 16–21. See margin.
M
Example 4
110
75
16. 2 in., 3 in., 6 in.
17. 11 cm, 12 cm, 15 cm
18. 8 m, 10 m, 19 m
19. 1 cm, 15 cm, 15 cm
20. 2 yd, 9 yd, 10 yd
21. 4 m, 5 m, 9 m
27. 20 km, 35 km 15 R s R 5
28. Error Analysis The Shau family is crossing Kansas on Highway 70. A sign reads
“Wichita 90 miles, Topeka 110 miles.” Avi says, “I didn’t know that it was only
20 miles from Wichita to Topeka.” Explain to Avi why the distance between the
two cities doesn’t have to be 20 miles. See margin.
Suppose two sides of one triangle are congruent to two sides of another
triangle. If the included angle of the first triangle is larger than the included
angle of the second triangle, then 9.
a. Draw a diagram to illustrate the hypothesis.
b. The conclusion of the Hinge Theorem concerns the sides opposite the two
angles mentioned in the hypothesis. Write the conclusion.
c. Draw a diagram to illustrate the converse.
d. Converse of the Hinge Theorem Write the conclusion to this theorem.
Suppose two sides of one triangle are congruent to two sides of another
triangle. If the third side of the first triangle is greater than the third side of
the second triangle, then 9.
Lesson 5-5 Inequalities in Triangles
18. No; 8 ± 10 w 19.
19. Yes; 1 ± 15 S 15;
15 ± 15 S 1.
20. Yes; 2 ± 9 S 10;
9 ± 10 S 2;
2 ± 10 S 9.
geometry software to investigate
the Hinge Theorem and its
converse.
23. 5 in., 16 in.11 R s R 21 24. 6 cm, 6 cm 0 R s R 12
29. Writing Explain why the distance between the two peaks in the photograph is
greater than the difference of the distances from the hiker to each of the peaks.
See margin.
30. The Hinge Theorem The hypothesis of the Hinge Theorem is stated below.
GPS The conclusion is missing. a–d. See margin.
Exercise 29
Exercise 30 Students can use
possible for the third side.
25. 18 m, 23 m 5 R s R 41 26. 4 yd, 7 yd 3 R s R 11
Apply Your Skills
Technology Tip
x 2 Algebra The lengths of two sides of a triangle are given. Describe the lengths
22. 8 ft, 12 ft 4 R s R 20
B
representatives make a great
effort to plan their routes
efficiently. When traveling to
multiple destinations over the
course of a day, week, or month,
a sales representative makes a
schedule of routes that minimizes
travel time and maximizes time
with customers.
21. No; 4 ± 5 w 9.
28. Answers may vary.
Sample: If Y is the
distance between
Wichita and Topeka,
then 20 R Y R 200.
293
29. Let the distance between
the peaks be d and the
distances from the hiker
to each of the peaks be
a and b. Then d ± a S b
and d ± b S a. Thus, d S
b – a and d S a – b.
30. a.
D
A
B
C E
F
b. The third side of the
1st k is longer than
the third side of the
2nd k.
c. See diagram in part
(a).
d. The included l of the
first k is greater than
the included l of the
second k.
31. Answers may vary.
Sample: The shortcut
across the grass is
shorter than the sum of
the two paths.
293
4. Assess & Reteach
31. Shortcuts Explain how the student in the photograph is applying the Triangle
Inequality Theorem. See margin.
x 2 32. Algebra Find the longest side of #ABC, if m&A = 70, m&B = 2x - 10, and
PowerPoint
m&C = 3x + 20. AB
Lesson Quiz
33. Developing Proof Fill in the blanks for a proof of Theorem 5-10: If two sides of
a triangle are not congruent, then the larger angle lies opposite the longer side.
Use the figure for
Exercises 1–3.
Given: #TOY, with YO . YT.
25
2
3 4
12
3
P
1
Mark P on YO so that YP > YT. Draw TP.
1
B
O
Prove: a. 9 . b. 9 mlOTY; ml3
A
C
27°
D
1. Explain why m&4 m&1.
ml4 S ml1 by the Corollary
to the Exterior Angle Theorem.
Real-World
Statements
Connection
2. Explain why m&3 m&1. By
Theorem 5-10, if two sides are
not congruent, the larger
angle lies opposite the longer
side, so ml3 S ml1.
d. 9 l Add. Post.
4. m&OTY . m&2
e. 9 Comparison Prop. of Ineq.
Y
f. 9 Sub. (step 2)
g. 9 An ext. l of a k is greater than
either remote int. l.
7. m&OTY . m&3
h. 9
Trans. Prop. of Ineq.
34. Prove this corollary to Theorem 5-11:
P
The perpendicular segment from a point to a line
is the shortest segment from the point to the line.
See margin
Given: PT ' TA
p. 295.
T
Prove: PA . PT
A
6. m&1 . m&3
Proof
4. Can a triangle have lengths of
2 mm, 3 mm, and 6 mm?
Explain. no; 2 ± 3 6
Critical Thinking Determine which segment is shortest in each diagram.
GO
6. In PQT, m&P = 50 and m&T
= 70. Which side is shortest?
QT
Homework Help
P
30
Visit: PHSchool.com
Web Code: aue-0505
40
R
Challenge
36. C
Q
35. RS
nline
C
110
37.
D CD
32
114
S
B
X
48
30
A
XY
W
Y
95
47
40
Z
38. Probability A student has two straws, one 6 cm long and the other 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? 12
Alternative Assessment
Have each student write a
paragraph explaining the
Comparison Property of
Inequality, the Corollary to the
Exterior Angle Theorem, the
two theorems that relate the
relative positions of the angles
and sides of a triangle, and the
Triangle Inequality Theorem.
3. m&OTY = m&4 + m&2
1. Ruler Post.
5. m&OTY . m&1
3. Use ABC to describe the
possible lengths of AC .
13 R AC R 37
5. In XYZ, XY = 5, YZ = 8, and
XZ = 7. Which angle
is largest? lX
2. m&1 = m&2
2
T
c. 9 Base ' of an isos. k are O.
1. YP > YT
Some recall the Triangle
Inequality Theorem as “The
shortest path between two
points is the straight path.”
4
Reasons
39. (2, 4), (2, 6), (3, 3),
(3, 4), (3, 5), (3, 6),
(3, 7), (4, 3), (4, 4),
(4, 5), (4, 6), (4, 7),
(4, 8)
For Exercises 39 and 40, x and y are whole numbers, 1 R x R 5, and 2 R y R 9.
39. The sides of a triangle are 5 cm, x cm, and y cm. List possible (x, y) pairs. See left.
40. Probability What is the probability that you can draw an isosceles triangle that
5
has sides 5 cm, x cm, and y cm, with x and y chosen at random? 18
Proof
41. Prove Theorem 5-12: The sum of the lengths of any two sides
of a triangle is greater than the length of the third side.
Given: #ABC
C
See margin.
Prove: AC + CB . AB
)
A
(Hint: On BC mark a point D not on BC, so that DC = AC.
Draw DA and use Theorem 5-11 with #ABD.)
34. lT is the largest l in
kPTA. Thus PA S PT
because the longest
side of a k is opp. the
largest l.
294
Chapter 5 Relationships Within Triangles
41. D
C
A
294
B
CD ≠ AC is given so
kACD is isosc. by def.
of isosc. k. This means
mlD ≠ mlCAD. Then
mlDAB S mlCAD by
the Comparison Prop.
of Ineq. So by subst.,
mlDAB S mlD and by
Thm. 5-11 DB S AB.
Since DC ± CB ≠ DB,
by subst.
DC ± CB S AB. Using
subst. again, AC ±
CB S AB.
B
Test Prep
Test Prep
Multiple Choice
Resources
P
42. For #PQR, which is the best estimate for PR? D
A. 137 m
B. 145 m
C. 163 m
D. 187 m
(2a + 12)
184 m
43. Two sides of a triangle measure 13 and 15.
Which length is NOT possible for the third side? F
F. 2
G. 8
H. 14
J. 20
Q
44. Which statement is true for the figure at the right? D
A. JN . JB
B. JN . BN
C. The shortest side is JB.
D. The longest side is BN.
B
4a
114
145 m R
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 301
• Test-Taking Strategies, p. 296
• Test-Taking Strategies with
Transparencies
J
4c
(c - 25)
130
N
not to scale
45. For #ABC, what must be true about an exterior angle at A? J
F. It is larger than &A.
G. It is smaller than &A.
H. It is larger than &B.
J. It is smaller than &C.
46. Which lengths can be lengths for the sides of a triangle? C
A. 1, 2, 5
B. 3, 2, 5
C. 5, 2, 5
D. 7, 2, 5
47. For #JKL, L J , JK , KL. What must be true about angles J, K, and L? J
F. m&L , m&J , m&K
G. m&L . m&J . m&K
H. m&J , m&L , m&K
J. m&J . m&L . m&K
Short Response
48. In #PQR, PQ . PR . QR. One angle measures 170°. List all possible whole
number values for m&P. 1, 2, 3, 4, 5, 6, 7, 8, 9
49. In #ABC, m&A . m&C . m&B. a–b. See margin.
a. Of AB and AC , one measures 5 inches and the other measures 9 inches.
Which measures 9 inches? Explain.
b. Based on your conclusion for part (a), find all possible whole-number
measures for the third side. Explain.
Mixed Review
GO for
Help
Lesson 5-4
Write the negation of each statement.
50. m&A # m&B mlA S mlB
Lesson 2-5
Lesson 1-9
51. m&X . m&B mlX K mlB
52. The angle is a right angle.
53. The triangle is not obtuse.
The triangle is obtuse.
The angle is not a right angle.
Use the diagram. Find the measure of each angle.
E
C
54. &ADH 90
55. &GDH 35
A
35 D
56. &CDH 145
57. &ADG 55
G
H
Find to the nearest tenth of a square unit the area
of each circle with the given radius r or diameter d.
58. r = 1.6 ft
8.0 ft2
lesson quiz, PHSchool.com, Web Code: aua-0505
49. [2] a. Since mlA S
mlC S mlB, the
sides opp. them
are related in the
same way:
BC S AB S AC. Of
AB and AC , AB
59. d = 35 mm
962.1 mm2
60. r = 0.5 m
0.8 m2
61. d = 20 mi
314.2 mi 2
Lesson 5-5 Inequalities in Triangles
is longer than AC .
Since 9 in. S 5 in.,
AB ≠ 9 in. and
AC ≠ 5 in.
b. BC is the longest
side, and 9 R BC R
14. The possible
295
whole number
measures for BC
are 10 in., 11 in.,
12 in., and 13 in.
[1] part (a) OR part (b)
incorrect
295
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