GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
Skill Check
✓
2. Any two of the points;
as shown in Example 3
on p. 481, any two
points on a line can be
used to calculate the
slope of the line.
3 APPLY
1. If ¤ABC ~ ¤XYZ, AB = 6, and XY = 4, what is the scale factor of the
triangles? 3:2
ASSIGNMENT GUIDE
BASIC
2. The points A(2, 3), B(º1, 6), C(4, 1), and D(0, 5) lie on a line. Which two
points could be used to calculate the slope of the line? Explain. See margin.
3. Can you assume that corresponding sides and corresponding angles of any
two similar triangles are congruent? No; corresp. sides of ~ ◊ are not £ unless
the scale factor of the ◊ is 1:1. Corresp. ™s are £, however.
Determine whether ¤CDE ~ ¤FGH.
4. no
G
D
39ⴗ
C
72ⴗ
E
41ⴗ
F
72ⴗ
5. yes
D
G
60ⴗ
C
60ⴗ
E
60ⴗ
F
H
9. ™J and ™F, ™K and ™G,
JK
KL
™L and ™H, }} = }} = In the diagram shown ¤JKL ~ ¤MNP.
GH
FG
JL
}}
FH
N
6. Find m™J, m™N, and m™P. 37°, 90°, 53°
10. ™V and ™S, ™T and ™T,
VT
ST
TW
TU
LM
QP
MN
PN
™N and ™N, }} = }} =
LN
}}
QN
8
M
L
P
EXERCISE LEVELS
Level A: Easier
9–17
Level B: More Difficult
18–47, 50–54, 56, 57
Level C: Most Difficult
48, 49, 55, 58–60
B
do you know that ¤ABC ~ ¤BDC?
Explain your answer.
By the Reflexive Prop. of Cong., ™C £ ™C, so A
¤ABC ~ ¤BDC by the AA Similarity Post.
D
C
PRACTICE AND APPLICATIONS
STUDENT HELP
Extra Practice
to help you master
skills is on p. 818.
USING SIMILARITY STATEMENTS The triangles shown are similar. List all
the pairs of congruent angles and write the statement of proportionality.
9–11. See margin.
P
9.
10. V
11.
G
K
S
F
J
N
L
H
L
W
U
q
M
T
LOGICAL REASONING Use the diagram to complete the following.
STUDENT HELP
HOMEWORK HELP
Example 1: Exs. 9–17,
33–38
Example 2: Exs. 18–26
Example 3: Exs. 27–32
Example 4: Exs. 39–44,
53, 55, 56
Example 5: Exs. 45–47
?㛭 ¤LMN
12. ¤PQR ~ 㛭㛭㛭㛭㛭㛭
Day 1: pp. 483–485 Exs. 9–38
Day 2: pp. 485–487 Exs. 39–53,
55–71
pp. 483–487 Exs. 9–38 (with 8.3)
pp. 485–487 Exs. 39–53, 55–57,
61–71 (with 8.5)
3
53ⴗ
37ⴗ
Day 1: pp. 483–485 Exs. 9–38
Day 2: pp. 485–487 Exs. 39–53,
55–57, 61–71
BLOCK SCHEDULE
K
5
8. Given that ™CAB £ ™CBD, how
11. ™L and ™Q, ™M and ™P,
4
7. Find MP and PN. 10, 6
™W and ™U, }} = }} =
VW
}}
SU
J
AVERAGE
ADVANCED
60ⴗ
H
Day 1: pp. 483–485 Exs. 9–38
Day 2: pp. 485–487 Exs. 39–52,
57, 61–71
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 10,
14, 20, 28, 30, 36, 42, 46, 48. See
also the Daily Homework Quiz:
• Blackline Master (Chapter 8
Resource Book, p. 65)
•
Transparency (p. 59)
P
PQ
QR
RP
13. ᎏᎏ = ᎏᎏ = ᎏᎏ LM; MN; NL
?
?
?
?
20
y
14. ᎏᎏ = ᎏᎏ 15; y
12
?
?
18
15. ᎏᎏ = ᎏᎏ 15; x
20
?
q
?㛭
16
16. y = 㛭㛭㛭㛭㛭㛭
L
x
18
12
20
R
M
15
N
?㛭 24
17. x = 㛭㛭㛭㛭㛭㛭
8.4 Similar Triangles
483
483
COMMON ERROR
EXERCISES 23 AND 24
In problems such as these, where
the triangles do not have the corresponding parts aligned, it is common
for students to have trouble identifying the corresponding parts. To help
students avoid this difficulty, have
them copy the diagram and mark the
corresponding angles in the triangles. Some students may also need
to trace one of the triangles and
rotate it so that the corresponding
parts are more evident.
COMMON ERROR
EXERCISE 26 In problems such
as this, where the triangles are
nested and/or rotated, it is common
for student to have trouble identifying corresponding parts. To help
students avoid this difficulty, have
them trace both triangles on separate sheets of paper. They can then
rotate one of the triangles until the
corresponding parts of both triangles align and mark congruent any
angles that are shared by both
triangles in the original figure.
DETERMINING SIMILARITY Determine whether the triangles can be proved
similar. If they are similar, write a similarity statement. If they are not
similar, explain why.
18. No; m™C = 31° and
m™D = 47°; the
corresponding Å are
not £.
18.
R
19.
D
20.
V
41ⴗ
92ⴗ
A
20
16
33ⴗ
E
P
S
F
26
20
R 12 T
M 15 q
RS
ST
MP
PQ
57ⴗ
no; }} ≠ }}, so the
75ⴗ
72ⴗ W
92ⴗ
B
q
C
P
lengths of corresponding
sides are not proportional.
yes; ¤PQR ~ ¤WPV
22. No; m™E = 94°,
m™B = 94°, and
m™A = 54°; the
corresponding Å are
not £.
21.
H
F
22.
9
B
D
M
65ⴗ
32ⴗ
6
X
N
23.
A
55ⴗ
J
C
53ⴗ
48ⴗ
48ⴗ
Y
E
Z
F
18
K
yes; ¤JMN ~¤JLK
V
25.
A
50ⴗ
L
See margin.
yes; ¤XYZ ~ ¤GFH
24.
65ⴗ
33ⴗ
G
77ⴗ
26.
P
50ⴗ
S
C
D
Y
50ⴗ
B
E
yes; ¤ABC ~ ¤EDC
Z
W
T
q
R
X
yes; ¤PQR ~ ¤PST
yes; ¤VWX ~ ¤VYZ
xy USING ALGEBRA Using the labeled points, find the slope of the line. To
verify your answer, choose another pair of points and find the slope using
the new points. Compare the results.
27.
2
º}}
5
28.
y
y
1
}}
3
(⫺8, 3)
(⫺3, 1)
(5, 0)
x
(⫺1, ⫺2)
x
(2, ⫺1)
(2, ⫺1)
(⫺4, ⫺3)
(7, ⫺3)
xy USING ALGEBRA Find coordinates for point E so that ¤OBC ~ ¤ODE.
y
29. O(0, 0), B(0, 3), C(6, 0), D(0, 5) (10, 0)
30. O(0, 0), B(0, 4), C(3, 0), D(0, 7)
冉5 }14}, 0冊 D
31. O(0, 0), B(0, 1), C(5, 0), D(0, 6) (30, 0)
32. O(0, 0), B(0, 8), C(4, 0), D(0, 9)
冉4 }12}, 0冊
B
O
484
484
Chapter 8 Similarity
C
E(?, 0)
x
xy USING ALGEBRA You are given that ABCD is a trapezoid, AB = 8,
AE = 6, EC = 15, and DE = 10.
STUDENT HELP NOTES
A 8 B
AB
AE
BE
? CDE 34. ᎏ
ᎏ = ᎏᎏ = ᎏᎏ
33. ¤ABE ~ ¤
?
?
?
y
CD; CE; DE
6
E
6
8 6
15
10 15 10
8
35. ᎏᎏ = ᎏᎏ }15} = }x}
36. ᎏᎏ = ᎏᎏ }} = }}
y
?
?
?
? 6
10
15
? 20
? 4
37. x = 38. y = Homework Help Students
can find help for Exs. 39–44
at www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
D
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with problem
solving in Exs. 39–44.
x
C
SIMILAR TRIANGLES The triangles are similar. Find the value of the
variable.
39. 14
6
11
2}}
40.
8
7
CONCEPT QUESTION
EXERCISES 45–47 Ask students
to identify each type of special
segment shown in these exercises.
4
r
Ex. 45: altitude; Ex. 46: median;
Ex. 47: angle bisector
11
p
7
16
yⴚ3
41. 27
42. 36
18
z
24
44. 13}1}
43. 100
35ⴗ
6
5
xⴗ
45
4
2
45ⴗ
55
44
32
s
SIMILAR TRIANGLES The segments in blue are special segments in the
similar triangles. Find the value of the variable.
45.
12
12
15
46.
80
8
47. 25
48
x
y ⴚ 20
18
36
14
18
y
27
Æ
zⴚ4
Æ
48. Since
KM fi JL and
Æ Æ
JK fi KL , ™JMK and
™JKL are right Å.
Since all right Å are
£, ™JMK £ ™JKL.
By the Reflexive Prop.
of Cong., ™J £ ™J,
so ¤JKL ~ ¤JMK by
the AA Similarity Post.
48.
PROOF Write a paragraph or two-column proof.
GIVEN
Æ
Æ Æ
K
Æ
䉴 KM fi JL , JK fi KL
PROVE 䉴 ¤JKL ~ ¤JMK
J
M
8.4 Similar Triangles
L
485
485
49.
MATHEMATICAL REASONING
EXERCISE 51 Ask students what
they can conclude by showing that
this statement is false. All equilateral triangles are similar. Explain
that this statement is the negation of
the statement given in the exercise.
STUDENT HELP NOTES
Software Help Instructions for
several software packages are
available in blackline format in the
Chapter 8 Resource Book, p. 54
and at www.mcdougallittell.com.
GIVEN 䉴 ™ECD is a right angle,
49, 55. See Additional Answers
beginning on page AA1.
E
See margin.
51. False; all Å of any 2
LOGICAL REASONING In Exercises 50–52, decide whether the
equilateral ◊ are £, so statement is true or false. Explain your reasoning.
the ◊ are ~ by the AA
Similarity Post. (Note also, 50. If an acute angle of a right triangle is congruent to an acute angle of another
that if one ¤ has sides of
right triangle, then the triangles are similar.
length x and the other has
True; all right Å are £, so the ◊ are ~ by the AA Similarity Post.
sides of length y, then the 51. Some equilateral triangles are not similar.
ratio of any two side
52. All isosceles triangles with a 40° vertex angle are similar. See margin.
x
lengths is }}. Then all
y
ICE HOCKEY A hockey player passes
corresp. side lengths are in 53.
the puck to a teammate by bouncing the
puck off the wall of the rink as shown.
From physics, the angles that the path of
the puck makes with the wall are
congruent. How far from the wall will the
pass be picked up by his teammate? 1.5 m
proportion, so the def. of ~
◊ can also be used to
show that any 2 equilateral
◊ are ~.)
STUDENT HELP
NE
ER T
54.
SOFTWARE HELP
Visit our Web site
www.mcdougallittell.com
to see instructions for
several software
applications.
puck
1m
wall
2.4 m
6m
Æ
Æ Æ
Æ
P
56.
For Lesson 8.4:
•
Search the Test and Practice
Generator for key words or
specific lessons.
486
ESTIMATING HEIGHT On a
sunny day, use a rod or pole to
estimate the height of your school
building. Use the method that Thales
used to estimate the height of the
Great Pyramid in Exercise 55.
Answers will vary.
486
Chapter 8 Similarity
Not drawn to scale
S
q
780 ft
In the figure, PQ fi QT, SR fi QT, and
PR ∞ ST. Write a paragraph proof to
show that the height of the pyramid is 480 feet. See margin.
ADDITIONAL PRACTICE
AND RETEACHING
d
TECHNOLOGY Use geometry software to verify that any two points on a
line can be used to calculate the slope of the line. Draw a line k with a
negative slope in a coordinate plane. Draw two right triangles of different size
whose hypotenuses lie along line k and whose other sides are parallel to the xand y-axes. Calculate the slope of each triangle by finding the ratio of the vertical
side length to the horizontal side length. Are the slopes equal? yes
Æ Æ
For more Mixed Review:
B
A
PROVE 䉴 ¤ABE ~ ¤CDE
52. True; since the vertex ™
55.
THE GREAT PYRAMID The
of each isosceles ¤ has
Greek mathematician Thales (640–546
measure 40°, the measure
B.C.) calculated the height of the Great
of each base ™ is
1
Pyramid in Egypt by placing a rod at
}} (180° º 40°) = 70°. Then
2
the tip of the pyramid’s shadow and
the ◊ are ~ by the AA
using similar triangles.
Similarity Post.
• Practice Levels A, B, and C
(Chapter 8 Resource Book, p. 55)
• Reteaching with Practice
(Chapter 8 Resource Book, p. 58)
•
See Lesson 8.4 of the
Personal Student Tutor
D
C
™EAB is a right angle.
INT
TEACHING TIPS
EXERCISE 56 Students will need
some time at school to do this
exercise. You might turn this exercise into a class activity. Divide
the class into small groups and
give each group a pole of different
length. They can then compare
their results to see that the height
of the building is independent of the
length of the pole.
PROOF Write a paragraph proof or a two-column
proof. The National Humanities Center is located in
Research Triangle Park in North Carolina. Some of
its windows consist of nested right triangles, as
shown in the diagram. Prove that ¤ABE ~ ¤CDE.
4 ft
R
T
6.5 ft
Test
Preparation
57. MULTI-STEP PROBLEM Use the following information.
57. a. Right Å ABX and DCX are
£ and vertical Å AXB
and DXC are £. Then
¤ABX ~ ¤DCX by the
AA Similarity Post.
Going from his own house to Raul’s house, Mark
drives due south one mile, due east three miles,
and due south again three miles. What is the
distance between the two houses as the crow flies?
Mark‘s house
DAILY HOMEWORK QUIZ
1 mi
B
a. Explain how to prove that ¤ABX ~ ¤DCX.
X
C
N
b. Use corresponding side lengths of the triangles
W
to calculate BX. }3} mi
4
E
3 mi
The diagram shows how similar triangles relate to human vision. An image
similar to a viewed object appears on the retina. The actual height of the object h
is proportional to the size of the image as it appears on the retina r. In the same
manner, the distances from the object to the lens of the eye d and from the lens to
the retina, 25 mm in the diagram, are also proportional.
www.mcdougallittell.com
appears on the retina as 1 mm
tall. How far away is the object?
25 mm
lens
h
r
Not drawn to scale
retina
25 m
MIXED REVIEW
61. USING THE DISTANCE FORMULA Find the distance between the points
57.d. Sample answer: ConÆ
A(º17, 12) and B(14, º21). (Review 1.3) 5兹82
苶
struct a line ∞ to BC
through A and extend
K
N
L
Æ
TRIANGLE MIDSEGMENTS M, N, and P
DC up to this line to
are
the
midpoints
of
the
sides
of
¤JKL.
intersect it at E. Since
Æ Æ Æ Æ
Complete the statement.
M
P
AE ∞ BC , CE ∞ AB ,
(Review 5.4 for 8.5)
and ™B is a right ™,
J
ABCE is a rectangle.
Æ
Æ
?
? . 46
63. If NP = 23, then KJ = Thus AB = CE = 1 mi 62. NP ∞ KJ
and BC = AE = 3 mi.
? . 16
? . 12
65. If JL = 24, then MN = Since ™E is a right ™, 64. If KN = 16, then MP = ¤AED is a right triangle
PROPORTIONS Solve the proportion. (Review 8.1)
Æ
with hypotenuse AD
x
3
3
12
17
11
and legs of 1 mi and
66. ᎏᎏ = ᎏᎏ }9}
67. ᎏᎏ = ᎏᎏ 8
68. ᎏᎏ = ᎏᎏ 51
3 mi.
12
8 2
y
32
x
33
34
x+6
69. ᎏᎏ = ᎏᎏ }3}6
11
11
3
x
23
70. ᎏᎏ = ᎏᎏ 69
72
24
Y
6
EXTRA CHALLENGE NOTE
59. An object that is 10 meters away
EXTRA CHALLENGE
X
h
58. Write a proportion that relates r, d, h, and 25 mm. Sample answer: }25} = }r}
400 mm
60. An object that is 1 meter tall
V
3. What is VW ? 10
4. What is XZ ? 9
5. If m⬔U = 50° and m⬔Y = 30°,
what is m⬔Z ? 100°
HUMAN VISION In Exercises 58–60, use the following information.
d
5
1. 䉭UVW ⬃ 䉭
XYZ
2. What is the scale factor of
䉭UVW to 䉭XYZ ? }56}
Writing Using the properties of rectangles,
explain a way that a point E
Æ
appears on the retina as 1 mm
tall. Find the height of the object.
12
7.5
U
d
Z
D
could be added to the diagram so that AD would be the hypotenuse of
Æ
Æ
¤AED, and AE and ED would be its legs of known length. See margin.
★ Challenge
Transparency Available
Use the diagram for
Exercises 1–5.
W
S
Raul‘s house
c. Use the Pythagorean Theorem to calculate AX,
and then DX. Then find AD. 1.25 mi; 3.75 mi; 5 mi
d.
4 ASSESS
3 mi
A
Challenge problems for
Lesson 8.4 are available in
blackline format in the Chapter 8
Resource Book, p. 62 and at
www.mcdougallittell.com.
ADDITIONAL TEST
PREPARATION
1. WRITING Explain how to use
shadows to indirectly measure
the height of a tall object.
Illustrate your explanation with a
sketch. Sample answer: Erect a
pole perpendicular to the ground.
Measure the height of the pole
and the length of its shadow. Then
measure the length of the shadow
of the tall object. Use similar triangles to find the height of the tall
object. Check students’ sketches.
x
8
71. ᎏᎏ = ᎏᎏ º16, 16
32
x
8.4 Similar Triangles
487
487