NAME
DATE
7-4
PERIOD
Study Guide and Intervention
Parallel Lines and Proportional Parts
Proportional Parts within Triangles
In any triangle, a
line parallel to one side of a triangle separates the other two
sides proportionally. This is the Triangle Proportionality Theorem.
The converse is also true.
R
X
T
RX
SY
RX
SY
, then −
.
RS
RS
=−
. If −
=−
, then XY
If XY
XT
YT
XT
YT
S
Y
−−
−−
−−
If X and Y are the midpoints of RT and ST, then XY is a
midsegment of the triangle. The Triangle Midsegment Theorem
states that a midsegment is parallel to the third side and is half its length.
−−
1
and XY = −
RS
If XY is a midsegment, then XY
RS.
2
−− −−
In ABC, EF CB. Find x.
Example 1
E
6
Example 2
In GHJ, HK = 5,
KG = 10, and JL is one-half the length
−−
−−− −−
of LG. Is HK KL?
C
)
18
A
x + 22
F x+2
+
B
,
-
−− −−− AF
AE
Since EF CB, −
=−
.
FB
EC
(
6x + 132 = 18x + 36
96 = 12x
8=x
Using the converse of the Triangle
Proportionality Theorem, show that
JL
HK
−
=−
.
KG
LG
Let JL = x and LG = 2 x.
JL
x
1
−
=−
=−
2
2
10
2x
LG
1
1
Since − = −, the sides are proportional and
2
2
−−− −−
5
HK
1
−
=−
=−
KG
HJ KL.
Exercises
ALGEBRA Find the value of x.
1.
5
7
7
2.
x
5
x
4.
Chapter 7
5.
17.5
3.
x
9
x
11
x + 12
30
10
18
x
8
24
20
35
12
33
5
6. x
x + 10
30
10
24
10
Glencoe Geometry
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x + 22
18
−=−
x+2
6
NAME
DATE
7-4
PERIOD
Study Guide and Intervention (continued)
Parallel Lines and Proportional Parts
Proportional Parts with Parallel Lines When
three or more parallel lines cut two transversals,
they separate the transversals into proportional
parts. If the ratio of the parts is 1, then the parallel
lines separate the transversals into congruent parts.
s
t
1
2
c
a
d
b
v
w
n
x
4 5 6
If 1 2 3,
a
c
then −
=−
.
b
Example
u
3
d
m
If 4 5 6 and
u
w
−
v = 1, then −
x = 1.
Refer to lines 1, 2, and 3 above. If a = 3, b = 8, and c = 5, find d.
3
5
1
1 2 3 so −
=−
. Then 3d = 40 and d = 13 −
.
8
3
d
Exercises
ALGEBRA Find x and y.
1.
5x
2.
3x
12
2x - 6
x+3
x=8
x=9
3.
4.
2x + 4
3x - 1
3
x
4
1
y = 3−
3
6.
32
16
x+4
y
x = 12
Chapter 7
y+2
8
2y + 2
x = 5; y = 2
5.
y
5
3y
Lesson 7-4
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
x + 12 12
y+3
y=3
25
Glencoe Geometry