Download Answers for the lesson “Use Proportionality Theorems”

LESSON
6.5
Answers for the lesson “Use Proportionality
Theorems”
12. The length of }
CD is not 20;
Skill Practice
1. If a line parallel to one side of a
triangle intersects the other two
sides, then it divides the two
sides proportionally.
B
20 2 x
x
10
16
}5}
13. C
14. 27
15. 9
16. a 5 22.8125, b 5 15.625, c 5 15,
d 5 5, e 5 4, f 5 8
E
17. a 5 9, b 5 4, c 5 3, d 5 2
C
]›
18. AD must bisect ŽA to use
Theorem 6.7.
D
A
CE CD
EB DA
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2. In the Midsegment Theorem
the segment connecting the
midpoints of two sides of a
triangle is parallel to the third
side which is a special case of
the Converse of the Triangle
Proportionality Theorem.
3. 9
4. 21
12
8
5. Parallel; } 5 }, so the Converse
7.5
5
of the Triangle Proportionality
Theorem applies.
18
24
6. not parallel; } Þ }
15
10
25
20
7. Parallel; } 5 }, so the
22.5
18
19. a–b. See figure in part (c)
c.
C
G
F
E
D
A
J
K
L
B
Theorem 6.6 guarantees
that parallel lines divide
transversals proportionally.
AD
DE
EF
5}
5}
51
Since }
EF
FG
DE
AJ
JK
KL
implies }
5}
5}
51
KL
LB
JK
which means AJ 5 JK 5
KL 5 LB.
Converse of the Triangle
Proportionality Theorem applies.
8. C
10. 12
9. 10
11. 1
Geometry
Answer Transparencies for Checking Homework
183
‹]›
23. Draw AD . (Through any two
20.
points, there is exactly one line.)
Let G be the point of intersection
‹]›
‹]›
of AD
and BE
. Since k1 i k2 and
k2 i k3, by the Triangle
Proportionality Theorem
t
CB
BA
DG
GA
DG
GA
DE
EF
} 5 } and } 5 }. Using
x
the Transitive Property of
r
CB
s
A
B
C
DE
Equality, }
5}
.
EF
BA
24. a. Lot A 5 50.9 yd,
Lot B 5 58.4 yd,
Lot C 5 64.7 yd
Problem Solving
21. 350 yd
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
22. Since }
QS i }
TU ŽS > ŽTUR and
ŽQ > ŽUTR using the
Corresponding Angles Postulate.
nSRQ , nURT using the AA
Similarity Postulate.
QR
TR
SR
UR
} 5 } using the
definition of similarity,
QR 5 QT 1 TR and
SR 5 SU 1 UR by the Segment
Addition Postulate. Substituting
you get
QT 1 TR
TR
b. Lot C
c. About $114,735; about
$127,112. Sample answer:
Solve
100,000
50.9
x
58.4
100,000
50.9
} 5 }.
x
64.7
} 5 } and
SU 1 UR
UR
} 5 } which
QT
SU
simplifies to }
5}
.
UR
TR
Geometry
Answer Transparencies for Checking Homework
184
27. Since }
XW i }
AZ, ŽXZA > ŽWXZ
25.
In an isosceles triangle, the legs
are congruent, so the ratio of their
lengths is 1 : 1 . By Theorem 6.7,
this ratio is equal to the ratio of
the lengths of the segments
created by the ray, so it is
also 1 : 1 .
RU 1 US
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5}
and
showing }
US
TQ
RQ
YW
XY
YW
XY
get }
5}
. Substituting you
AX
WZ
get }
5}
.
XZ
WZ
28. a. about 4.3 cm
26. Sample answer: Begin by
RT 1 TQ
using the Alternate Interior
Angles Congruence Theorem.
This makes nAXZ isosceles
because it is shown that
ŽA > ŽWXZ and by the
Converse of the Base Angles
Theorem, }
AX > }
XZ. Since
}
}
XW i AZ using the Triangle
Proportionality Theorem you
RS
simplifying this to }
5}
.
US
TQ
Use the proportions to solve for
TQ
US
} and use the Transitive
Property of Equality. Show
nRTU , nRQS using the SAS
Similarity Theorem and show
ŽRTU > ŽRQS by definition
of similar triangles. Then use the
Corresponding Angles Converse
QS i }
TU.
to show }
b. Sample answer: The line
connecting the top left to
the bottom right of Car 1
is parallel to the line
connecting the top left to the
bottom right of Car 2; the
triangle with vertices
consisting of the vanishing
point, the top left of Car 1,
and the bottom right of Car 1
is similar to the triangle with
vertices consisting of the
vanishing point, the top left of
Car 2, and the bottom right
of Car 2.
c. about 4.7 cm
Geometry
Answer Transparencies for Checking Homework
185
} and CM
} so they are both
29. Draw AN
}. nAPN , nMPC,
parallel to BY
nCXM , nBXP, and nBZP ,
nAZN using the AA Similarity
Postulate. From nAPN , nMPC
AN
AP
you get }
5}
using the
MC
MP
definition of similarity. Similarly
from nCXM ,nBXP and
nBZP , nAZN you get
CX
BX
MC
PB
BZ
AZ
BP
AN
} 5 } and } 5 },
respectively. Using Theorem 6.4
AY
AP
and nACM, you get }
5}
.
PM
YC
AY
CX
BZ
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
Now }
+}+}5
YC BX AZ
AN
MC
MC
PB
BP
AN
} + } + } 5 1.
Geometry
Answer Transparencies for Checking Homework
186