3.8.3 Similar Triangle Properties

Use properties of similar triangles to find segment lengths
Apply proportionality and triangle angle bisector theorems
Use ratios to make indirect measurements
Use scale drawings to solve problems

1. 3.8.3 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 ratios to make indirect measurements
• Use scale drawings 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
T
AP AC PS CE
=
Note: This ratio is notthe 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
T
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
101010 10141414 14
4444 10 14 4 x
=
xxxx
28 3
x 5 5.6
5 5
= = =
>
>

6. Example
Verify that
(15)(8) = (10)(12)?
120 = 120 TTherefore,
H
O
R
S E
HE OS
T
15
10
12 8
=
15 10
?
12 8
HE OS
T

7. Example
Solve for x.
6x = (10)(9)
6x = 90
x = 15
>
>
>
x
9 6
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. indirect
measurement
Any method that uses formulas, similar
figures, and/or proportions to measure an
object.
Example: An 8 foot tall stick casts a
6 foot shadow. At the same time, a tall
flagpole casts an 18 foot shadow. How tall
is the flagpole?
6
8
18
x
The triangles are similar by AA~.
8 x 6 18
=
6x = 144 →x = 24 feet

11. Example
Miriam saw a mirror on the ground and
noticed that she could see the top of
Reunion Tower in the mirror. Her line of
sight was 5’ above the ground, and the
mirror was 2’ away from her. She measured
the distance from that position to the
base of Reunion Tower, and it was 224 feet.
How high is Reunion Tower?
The reflection creates congruent angles, so
the triangles are similar by AA~.

12. Example
5’
2’ 224’
x
5 x 2 224
2x 1120
x 560 ft.
=
=
=