This document discusses properties of similar triangles. It defines three ways triangles can be similar: angle-angle similarity (AA~) if two angles of one triangle are congruent to two angles of another; side-side-side similarity (SSS~) if the three sides of one triangle are proportional to the three sides of another; and side-angle-side similarity (SAS~) if two sides of one triangle are proportional to two sides of another and the included angles are congruent. Examples are provided to demonstrate applying these properties to determine if triangles are similar and write similarity statements.

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

2. 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, ΔMAC ~ ΔPOD by AA~
M
A C
P
O
D

3. 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, ΔWHY ~ ΔNOT by SSS~
1230
18
16
40
24

4. 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, ΔLUV ~ ΔTEX by SAS~
4
5
2
2.5

5. Example:
Explain why the triangles are similar and write a similarity
statement.
56°
34°
L
U V
T
E
X

6. Example:
Explain why the triangles are similar and write a similarity
statement.
90 – 56 = 34°
Therefore m∠V = m∠X, thus ∠V ≅ ∠X.
Since m∠U = m∠E = 90°, ∠U ≅ ∠E
Therefore, ΔLUV ~ ΔTEX by AA~
56°
34°
L
U V
T
E
X

7. Example:
Verify that ΔSAT ~ ΔORT
S
T
R
O
12
15
20
16

8. Example:
Verify that ΔSAT ~ ΔORT
A
S
T
R
O
12
15
20
16
∠ATS ≅ ∠RTO (Vertical angles ≅)
12 15
?
16 20
=
240 = 240
Therefore, ΔSAT ~ ΔORT by SAS~