Mrs. Hayden
Name______________________________
Date_______
202 Notes 6.4 and 6.5
6.4 Parallel lines and proportional parts.
Theorem 6.4—Triangle Proportionality Theorem—If a line is parallel to one
side of a triangle and intersects the other two sides in two distinct points, then it
separates these sides into segments of proportional lengths.
Ex:
A
c
h
a
g
E
e
B
d
D
b
f
C
Example problems:
9
x
8
y
12
12
3
6
x
5
Theorem 6.5—Converse of the triangle proportionality Theorem—If a line
intersects two sides of a triangle and separates these sides into segments of
proportional length then the line is parallel to the third side.
Ex:
T
U
V
X
W
Mrs. Hayden
Midsegment—of a triangle is a segment whose endpoints are the midpoints of
two sides of a triangle.
A
Ex:
N
M
C
B
Theorem 6.6—Triangle Midsegment theorem—A midsegment of a triangle is
parallel to one side of the triangle, and its length is ½ the length of that side.
A
Ex:
N
M
C
B
Ex problems:
9
12
x
9
12
20
Corollary 6.1—If three or more parallel lines intersect two transversals, then they
cut the transversals proportionally
Corollary 6.2—If three or more parallel lines cut off congruent segments on one
transversal, then they cut off congruent segments on every transversal.
Ex:
a
b
c
d
e
f
g
h
Mrs. Hayden
2
Ex problems: 5
2x +2
x
3x-4
3
5y
8
6.5 Parts of Similar Triangles
Ex:
3
A
Scale factor
ABC P=______
DEF P=______
Ratio of perimeters
5
D
3
B
4
y+7
C
10
6
E
8
F
Theorem 6.7—Proportional Perimeters Theorem—If two triangles are similar,
then the perimeters are proportional to the measures of the corresponding sides.
Ex:
The s are ~.
Find the
perimeter of
the larger .
16 5
16
24
32
Special segments of similar triangles
Theorem 6.8—If two triangles are similar, then the measures of the
corresponding altitudes are proportional to the measures of the corresponding
sides.
Theorem 6.9—If two triangles are similar, then the measures of the
corresponding angle bisectors are proportional to the measures of the
corresponding sides.
Theorem 6.10—If two triangles are similar, then the measures of the
corresponding medians are proportional to the measures of the corresponding
sides.
Examples of:
Thm. 6.8
Thm. 6.9
Thm 6.10
Mrs. Hayden
B
M
P
A
B
A
D
C
O
I
G
H
J
G
K
L
E
Ex problems:
ABC ~ GFE What is EH?
G
A
EH = x
CD = 3
D
H
C
E
5
8
B
F
Theorem 6.11—Angle Bisector Proportion Theorem—an angle bisector in a
triangle separates the opposite side into segments that have the same ratio as
the other two sides.
Ex:
S
SUN
UV bisects SUN
U
1
2
V
N
C
E
N
F
D
H
F
Mrs. Hayden
Ex problems:
r
k
b
t
s
m
h
j
x
Find x.
y
w
25 - x
22
z
x
8
x+3
x
3
x+6
2x + 1
9
15
3x - 6
Find AB.
B
x
4
1.2
A
4.9
5
3
16
50
E
y
3
C
50
D