The document discusses properties of similar triangles that can be used to find unknown side lengths or angle measures. These include: the triangle proportionality theorem, triangle angle bisector theorem, angle-angle similarity, side-side-side similarity, and side-angle-side similarity. Examples are provided to demonstrate applying these properties to solve problems involving similar triangles.

1. Similar Triangle Properties
The student is able to (I can):
• Use properties of similar triangles to find segment
lengths.
• Apply proportionality and triangle angle bisector
theorems.
• Apply triangle angle bisector theorems
• Use triangle similarity to solve problems.

2. Triangle Proportionality Theorem
If a line parallel to a side of a triangle
intersects the other two sides then it
divides those sides proportionally.
S
P
A
C
E
>
>
PC SE
AP AC
PS CE
=
Note: This ratio is not the same as the
ratio between the third sides!
≠
AP PC
PS SE

3. Triangle Proportionality Theorem Converse
If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
S
P
A
C
E
>
>
PC SE
AP AC
PS CE
=

4. Two Transversal Proportionality
If three or more parallel lines intersect
two transversals, then they divide the
transversals proportionally.
G
O
D
T
A
C
>
>
>
CA DO
AT OG
=

5. Examples Find PE
10x = (4)(14)
10x = 56
S
C
O
P
E
10101010 14141414
4444
10 14
4 x
=
xxxx
28 3
x 5 5.6
5 5
= = =
>
>

6. Example Verify that
(15)(8) = (10)(12)?
120 = 120 Therefore,
H
O
R
SE
HE OS
15
10
12 8
=
15 10
?
12 8
HE OS

7. Example Solve for x.
6x = (10)(9)
6x = 90
x = 15
>
>
>
x
96
10
10 x
6 9
=

8. Triangle Angle Bisector Theorem
An angle bisector of an angle of a triangle
divides the opposite side in two segments
that are proportional to the other two
sides of the triangle.
=
CD CA
DB AB

9. Example: Solve for x.
=
AD AB
DC BC
=
=
= =
3.5 5
x 12
5x 42
42
x 8.4
5

10. Angle-Angle Similarity (AA~)
If two angles of one triangle are
congruent to two angles of another
triangle, then the triangles are similar.
∠M ≅ ∠P
∠A ≅ ∠O
Therefore, ∆MAC ~ ∆POD by AA~
M
A C
P
O
D

11. Side-Side-Side Similarity (SSS~)
If the three sides of one triangle are
proportional to the three corresponding
sides of another triangle, then the
triangles are similar.
W H
Y
N
O
T
= =
WH HY WY
NO OT NT
Therefore, ∆WHY ~ ∆NOT by SSS~
1230
18
16
40
24

12. Side-Angle-Side Similarity (SAS~)
If two sides of one triangle are
proportional to two sides of another
triangle, and the included angles are
congruent, then the triangles are similar.
E
T X
U
L V
=
LU LV
TE TX ∠L ≅ ∠T
Therefore, ∆LUV ~ ∆TEX by SAS~
4
5
2
2.5

13. Example Explain why the triangles are similar and
write a similarity statement.
90 — 56 = 34º
Therefore m∠V = m∠X, thus ∠V ≅ ∠X.
Since m∠U = m∠E = 90º, ∠U ≅ ∠E
Therefore, ∆LUV ~ ∆TEX by AA~
56º
34º
L
U V
T
E
X

14. Example Verify that ∆SAT ~ ∆ORT
A
S
T
R
O
12
15
20
16
∠ATS ≅ ∠RTO (Vertical angles ≅)
12 15
?
16 20
=
240 = 240
Therefore, ∆SAT ~ ∆ORT by SAS~