Section 8.6 –Proportions and Similar Triangles
Theorem 8.4 – Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects
the other two sides then it divides the two sides
proportionally
Q
T
R
U
S
RT RU
If TU QS then TQ  US
Theorem 8.5 – Converse of the Triangle Proportionality
Theorem
 If a line divides two sides of a triangle proportionally,
then it is parallel to the third side
RT RU
If TQ  US then TU QS
C
Example #1:
4
D
8
B
AB DE , what is the length of EC ?
Solution:
DC EC

BD AE (Triangle Proportionality Theorem)
4 EC

8 12
4(12)
8 = EC, EC = 6
E
12
A
Example #2: Determine whether or not MN GH .
G
21
M
56
L
48
N
16
H
Solution:
Begin by finding and simplifying the ratios of the two sides
divided by MN .
LM 56 8


MG 21 3
LN 48 3


NH 16 1
8 3

Because 3 1 , MN is not parallel to GH
Theorem 8.6– If three parallel lines intersect two
transversals, then they divide the transversals
proportionally
r s and s t, and l and m intersect r, s, and t, then….
UW VX

WY
XZ
Example: 1  2  3, and PQ = 9, QR = 15, and ST = 11.
What is the length of
TU ?
P
S
1
9
Q
11
T
2
15
R
U
3
Solution:
PQ ST

QR TU
9
11

15 TU
9(TU) = 15(11)
Theorem 8.7 – Triangle Bisector Theorem
 If a ray bisects an angle of a triangle, then it divides
the opposite side into segments whose lengths are
proportional to the lengths of the other two sides
AD CA

CD
If
bisects ACB, then DB
CB
Example: CAD  DAB. Use the given side lengths to
find the length of DC .
9
B
A
15
D
14
C
Solution:
Let x = DC. Then BD = 14 – x
AB BD

AC DC
9(x) = 15(14-x)
9(x) = 210 – 15x
9 14  x

15
x
24x = 210
x = 8.75
so….. DC = 8.75 units
Example #3: 1  2, find AD and DB.
Solution:
A
x
D
15
5
15 - x
1
2
C
Solution:
x/15-x = 5/12 (Triangle Angle Bisector Theorem)
12x = 75-5x
17x = 75
x = 75/17 = AD
BD = 15-x = 15 - 75/17 = 180/17  10.59
B
12