8-5 PROPORTIONS IN TRIANGLES (p. 446-452)
Theorem 8-4 Side-splitter Theorem
If a line is parallel to one side of a triangle and intersects the other two sides, then
it divides those two sides proportionally.
In other words, the parallel line will split the two sides of the triangle in the same ratio.
Examine the proof for Theorem 8-4 on p. 446 and ask these questions.
1. In #2, did you have to use the same theorem twice? Why is Q  Q?
2. What is our abbreviation for reason 4?
RQ SQ
XR YS

?
3. If

, is it also true that
XR YS
RQ SQ
A quick way to remember the Side-splitter Theorem is
Top
Top

(along the sides joined by the parallel segment)
Bottom Bottom
Example: In the following diagram, MN
proportion.
B
z
C
12
M
10
N
6
A
BA . Find z by setting up and solving a
Example: In the following figure, LK  JV and WV  JV. Find x by setting up and
solving a proportion. Round to the nearest hundredth. Also, find the fraction which is
equal to this decimal.
J
12
x+2.5
L
K
7.5
x
V
W
The following statement is a corollary to the Side-splitter Theorem. It basically states
that three or more parallel lines will split transversals in the same ratio. A corollary is
really a theorem that uses a closely related theorem to prove it.
You will prove that this corollary is true by doing homework problem #34.
Corollary Corollary to Theorem 8-4
If three parallel lines intersect two transversals, then the segments intercepted on
the transversals are proportional.
E
G
a
x
b
y
J
F
x a

y b
Again,
top
top

.
bottom bottom
Example: In the diagram of the following trapezoid, the segments joining the legs of
trapezoid RSTU are parallel to its bases. Set up and solve proportions to find x and y.
R
S
5
12.5
6
x
y
9
U
T
Do 2 on p. 447.
We will not cover the details of the proof for the next theorem. Perhaps the teacher will
provide the statements for the proof of this theorem on a future quiz and students can
provide the reasons? In any event, the Side-splitter theorem is an important reason in its
proof.
Optional Activity: Using the four lengths in the following figure, have students use their
NP
MN
and
. How do these ratios compare?
calculators and find
PO
MO
Theorem 8-5 Triangle-Angle-Bisector Theorem
If a ray bisects an angle of a triangle, then it divides the opposite side into two
segments that are proportional to the other two sides of the triangle.
NP = 2 .3 3 cm
MN = 2 .8 7 cm
PO = 4 .5 8 cm
MO = 5 .6 4 cm
M
N
P
O
NP MN

PO MO
A quick way to remember this theorem is
Left
Left

Right
Right
Example: In the following diagram, KG is bisecting JGH. Find x by setting up and
solving a proportion.
G
40
24
J
x
K
H
30
Example: In the following diagram,
solving a proportion.
T
12.2
W
x
m
TUW  m VUW. Find x by setting up and
V
5
8
U
Homework p. 448-452: 3,6,9,14,15,19,24,26,27,29,33,34,36,52,53,56,58,62
27. Solve
5
x

12 13 - x
29. Solve
3
x

because x + y = 30 and y = 30 – x
2 30 - x
36. Solve
5 7 .5
5
x

and 
3
x
3 7.5