Geom 5.2 Worksheets

NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Lesson Reading Guide
Inequalities and Triangles
Get Ready for the Lesson
Read the introduction to Lesson 5-2 in your textbook.
• Which side of the patio is opposite the largest corner?
• Which side of the patio is opposite the smallest corner?
Read the Lesson
1. Name the property of inequality that is illustrated by each of the following.
a. If x ! 8 and 8 ! y, then x ! y.
b. If x " y, then x # 7.5 " y # 7.5.
c. If x ! y, then #3x " #3y.
d. If x is any real number, x ! 0, x $ 0, or x " 0.
2. Use the definition of inequality to write an equation that shows that each inequality is true.
a. 20 ! 12
b. 101 ! 99
c. 8 ! #2
d. 7 ! #7
e. #11 ! #12
f. #30 ! #45
3. In the figure, m!IJK $ 45 and m!H ! m!I.
a. Arrange the following angles in order from largest to
smallest: !I, !IJK, !H, !IJH
H
J
K
c. Is "HIJ an acute, right, or obtuse triangle? Explain your reasoning.
d. Is "HIJ scalene, isosceles, or equilateral? Explain your reasoning.
Remember What You Learned
4. A good way to remember a new geometric theorem is to relate it to a theorem you
learned earlier. Explain how the Exterior Angle Inequality Theorem is related to the
Exterior Angle Theorem, and why the Exterior Angle Inequality Theorem must be true if
the Exterior Angle Theorem is true.
Chapter 5
12
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
b. Arrange the sides of "HIJ in order from shortest to longest.
I
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Study Guide and Intervention
Inequalities and Triangles
Angle Inequalities Properties of inequalities, including the Transitive, Addition,
Subtraction, Multiplication, and Division Properties of Inequality, can be used with
measures of angles and segments. There is also a Comparison Property of Inequality.
For any real numbers a and b, either a " b, a $ b, or a ! b.
The Exterior Angle Theorem can be used to prove this inequality involving an exterior angle.
Exterior Angle
Inequality Theorem
If an angle is an exterior angle of a
triangle, then its measure is greater than
the measure of either of its corresponding
remote interior angles.
B
A
1
C
D
m!1 ! m!A, m!1 ! m!B
List all angles of !EFG whose measures are
less than m"1.
The measure of an exterior angle is greater than the measure of
either remote interior angle. So m!3 " m!1 and m!4 " m!1.
G
4
1 2
3
E
H
F
Exercises
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
List all angles that satisfy the stated condition.
L
1. all angles whose measures are less than m!1
3
1 2
M
2. all angles whose measures are greater than m!3
U
3. all angles whose measures are less than m!1
4. all angles whose measures are greater than m!1
5
4
J
K
Exercises 1–2
3 5
7
X
1 4
2
T
W
Exercises 3–8
6
V
5. all angles whose measures are less than m!7
6. all angles whose measures are greater than m!2
7. all angles whose measures are greater than m!5
S
8. all angles whose measures are less than m!4
8
Q
9. all angles whose measures are less than m!1
10. all angles whose measures are greater than m!4
Chapter 5
13
N
7
1
R
2
3
6
5
4
O
Exercises 9–10
P
Glencoe Geometry
Lesson 5-2
Example
NAME ______________________________________________ DATE
5-2
Study Guide and Intervention
____________ PERIOD _____
(continued)
Inequalities and Triangles
Angle-Side Relationships
When the sides of triangles are
not congruent, there is a relationship between the sides and
angles of the triangles.
A
B
• If one side of a triangle is longer than another side, then the
angle opposite the longer side has a greater measure than the
angle opposite the shorter side.
• If one angle of a triangle has a greater measure than another
angle, then the side opposite the greater angle is longer than
the side opposite the lesser angle.
Example 1
If AC ! AB, then m!B ! m!C.
If m!A ! m!C, then BC ! AB.
Example 2
List the angles in order
from least to greatest measure.
List the sides in order
from shortest to longest.
S
C
6 cm
R
C
35!
7 cm
9 cm
T
20!
A
125!
B
C
!B
!, !
AB
!, !
AC
!
!T, !R, !S
Exercises
List the angles or sides in order from least to greatest measure.
1.
R
T
35 cm
2.
80!
23.7 cm
S
3.
S
R
60!
B
4.3
3.8
40!
T
A
C
4.0
Determine the relationship between the measures of the
given angles.
22
U
35
24
4. !R, !RUS
R
5. !T, !UST
T
24
21.6
13 V
S
25
6. !UVS, !R
Determine the relationship between the lengths of the
given sides.
C
30!
30!
7. A
!C
!, !
BC
!
A
8. B
!C
!, !
DB
!
30!
D
90!
B
9. A
!C
!, !
DB
!
Chapter 5
14
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
48 cm
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Skills Practice
Inequalities and Triangles
Determine which angle has the greatest measure.
1. !1, !3, !4
2. !4, !5, !7
3. !2, !3, !6
4. !5, !6, !8
3
1 2
5
4 6 7
Use the Exterior Angle Inequality Theorem to list all
angles that satisfy the stated condition.
8
2 4 7
5. all angles whose measures are less than m!1
1
3
5 6
8
9
Lesson 5-2
6. all angles whose measures are less than m!9
7. all angles whose measures are greater than m!5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
8. all angles whose measures are greater than m!8
Determine the relationship between the measures of
the given angles.
9. m!ABD, m!BAD
23
A
10. m!ADB, m!BAD
B
34
35
39
C
41
D
11. m!BCD, m!CDB
12. m!CBD, m!CDB
Determine the relationship between the lengths of the
given sides.
13. !
LM
!, !
LP
!
M
83! 57!
L
14. M
!P
!, !
MN
!
38!
59!
79! N
44!
P
15. !
MN
!, !
NP
!
Chapter 5
16. M
!P
!, !
LP
!
15
Glencoe Geometry
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Practice
Inequalities and Triangles
Determine which angle has the greatest measure.
1. !1, !3, !4
2. !4, !8, !9
10
9
8
3
4
2
3. !2, !3, !7
7
6
5
1
4. !7, !8, !10
Use the Exterior Angle Inequality Theorem to list
all angles that satisfy the stated condition.
1
2
3
5. all angles whose measures are less than m!1
5 6
4
7
8
9
6. all angles whose measures are less than m!3
7. all angles whose measures are greater than m!7
8. all angles whose measures are greater than m!2
9. m!QRW, m!RWQ
R
47
10. m!RTW, m!TWR
Q
11. m!RST, m!TRS
D
14
T
22
S
E
113!
F
14. D
!E
!, !
DG
!
120!
17!
32!
G
16. D
!E
!, !
EG
!
17. SPORTS The figure shows the position of three trees on one
part of a Frisbee™ course. At which tree position is the angle
between the trees the greatest?
2
40 ft
3
Chapter 5
W
45
48!
H
15. !
EG
!, !
FG
!
34
12. m!WQR, m!QRW
Determine the relationship between the lengths
of the given sides.
13. !
DH
!, !
GH
!
44
35
16
37.5 ft
53 ft
1
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Determine the relationship between the measures
of the given angles.
NAME ______________________________________________ DATE
5-2
____________ PERIOD _____
Word Problem Practice
Inequalities and Triangles
1. DISTANCE Carl and Rose live on the
same straight road. From their balconies
they can see a flagpole in the distance.
The angle that each person’s line of
sight to the flagpole makes with the
road is the same. How do their distances
from the flagpole compare?
4. SQUARES Matthew has three different
squares. He arranges the squares to
form a triangle as shown. Based on
the information, list the squares in
order from the one with the smallest
perimeter to the one with the largest
perimeter.
2. OBTUSE TRIANGLES Don notices that
the side opposite the right angle in a
right triangle is always the longest of
the three sides. Is this also true of the
side opposite the obtuse angle in an
obtuse triangle? Explain.
1
2
47˚
54˚
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
3. STRING Jake built a triangular
structure with three black sticks. He
tied one end of a string to vertex M
and the other end to a point on the
stick opposite M, pulling the string
taut. Prove that the length of the
string cannot exceed the longer of the
two sides of the structure.
CITIES For Exercises 5 and
Dallas
6, use the following
64˚
information.
59˚
Stella is going to
Abilene
Texas to visit a friend.
As she was looking at a
Austin
map to see where she
might want to go, she noticed
the cities Austin, Dallas, and Abilene
formed a triangle. She wanted to determine
how the distances between the cities were
related, so she used a protractor to measure
two angles.
M
string
5. Based on the information in the figure,
which of the two cities are nearest to
each other?
6. Based on the information in the figure,
which of the two cities are farthest apart
from each other?
Chapter 5
17
Glencoe Geometry
Lesson 5-2
3

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