Identify similar polygons Prove certain triangles are similar by using AA~, SSS~, and SAS~ Use triangle similarity to solve problems.

1. Obj. 35 Triangle Similarity
The student is able to (I can):
• Identify similar polygons
• Prove certain triangles are similar by using AA~, SSS~,
and SAS~
• Use triangle similarity to solve problems.

2. similar
polygons
Two polygons are similar if and only if their
corresponding angles are congruent and
their corresponding side lengths are
proportional.
Example:
6
N
5
3
M
O
4
L
12
X
8
10
E
∠N ≅ ∠X
∠L ≅ ∠S
∠E ≅ ∠A
∠O ≅ ∠M
S
6
A
3 4 5 6
= =
=
6 8 10 12
NOEL ~ XMAS

3. Note: A similarity statement describes
two similar polygons by listing their
corresponding vertices.
Example: NOEL ~ XMAS
Note: To check whether two ratios are
equal, cross-multiply them–the
products should be equal.
Example:
3 4
=
6 8
24 = 24

4. Example
Determine whether the rectangles are
similar. If so, write the similarity ratio and
a similarity statement.
Q
15
U
6
D
A
R
25
E
10
T
C
All of the angles are right angles, so all the
angles are congruent.
QUAD ~ RECT
6 15
=
?
sim. ratio: 3
10 25
5
150 = 150

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

6. 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.
N
18
W
12
30
24
H
40
O
16
Y
T
WH HY WY
=
=
NO OT NT
Therefore, ∆WHY ~ ∆NOT by SSS~

7. 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
U
5
2.5
L
2
V
LU LV
=
TE TX
T
4
X
∠L ≅ ∠T
Therefore, ∆LUV ~ ∆TEX by SAS~

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

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