This document discusses proportional lengths in triangles using the Triangle Proportionality Theorem and Triangle Angle-Bisector Theorem. It provides examples of using these theorems to set up proportions to solve problems such as finding unknown segment lengths. Several examples are worked through, applying the theorems to determine lengths, whether lines are parallel, and using proportionality.

1. 7-6 Proportional Lengths
Chapter 7 Ratio, Proportion and
Similarity
Objective: I can use the Triangle Proportionality
Theorem and the Triangle Angle-Bisector Theorem
to set up proportions to solve problems.

3. Theorem 7-3 Triangle Proportionality
Theorem
 If a line parallel to one side of a triangle
intersects the other two sides, then it
divides those sides proportionally.
In the figure to the right, name some of the
proportions that may be justified with this
theorem.
Answers will be given in class.

4. Corollary
 If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
Some proportions that you can
get from the given figure:
𝑎
𝑐
=
𝑏
𝑑
,
𝑎
𝑗
=
𝑐
𝑘
,
𝑏
𝑗
=
𝑑
𝑘
See if you can find any
more.

5. Theorem 7-4 Triangle Angle-Bisector
Theorem
If a ray bisects an angle of a triangle, then it divides
the opposite side into segments proportional to the
other two sides.
𝑎
𝑐
=
𝑏
𝑑

6. Ex. 1: Finding the length of a
segment
 In the diagram 𝐴𝐵 ∥ 𝐸𝐷, BD = 8, DC = 4, and AE =
12. What is the length of EC?
12
8
4
C
B A
D E
Answer: 6

7. Ex. 2: Determining Parallels
 Given the diagram, determine whether 𝑀𝑁 ∥ 𝐺𝐻.
21
16
48
56
L
G
H
M
N
LM
MG
56
21
=
8
3
=
LN
NH
48
16
=
3
1
=
8
3
3
1
≠
MN is not parallel to GH.

8. Ex. 3: Using Proportionality
 In the diagram 1  2  3, and
PQ = 9, QR = 15, and ST = 11. What
is the length of TU?
11
15
9
3
2
1
S
T
UR
Q
P
Answer:
55
3

9. Ex. 4: Using the Angle-Bisector
Theorem
 In the diagram, CAD 
DAB. Use the given side
lengths to find the length of
DC.
15
9
14
D
A
C
B
Answer:
35
4
Since AD is an angle bisector of
CAB, you can apply the Angle-
Bisector Theorem.
Let x = DC. Then BD = 14 – x.