Download Notes 8.5 - NOHS Teachers

Mtgnt_08 trans wk bk 11/3/03 3:43 PM Page 176
8.5
Proving Triangles are Similar
Goals p Use similarity theorems to prove two triangles are similar.
p Use similar triangles to solve real-life problems.
THEOREM 8.2: SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM
If the lengths of the corresponding sides
of two triangles are proportional, then
the triangles are similar.
AB
BC
CA
If ,
PQ
QR
RP
A
P
Q
B
R
C
then T ABC S T PQR .
THEOREM 8.3: SIDE-ANGLE-SIDE (SAS) SIMILARITY
THEOREM
If an angle of one triangle is congruent
to an angle of a second triangle and
the lengths of the sides including these
angles are proportional, then the
triangles are similar.
ZX
XY
If aX c aM and ,
PM
MN
then T XYZ S T MNP .
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Geometry Notetaking Guide • Chapter 8
X
M
P
Z
Y
N
Mtgnt_08 trans wk bk 11/3/03 3:43 PM Page 177
Using the SSS Similarity Theorem
Example 1
Which of the following three triangles are similar?
R
8
Q
W
Y
T
15
4
10
S
6
10
6
12
V
Z
X
12
To decide which, if any, of the triangles are similar, you need to
consider the ratios of the lengths of corresponding sides.
Ratios of Side Lengths of TQRS and TTVW
Shortest sides
Longest sides
Remaining sides
4
2
RS
,
6
3
VW
10
2
QS
,
15
3
TW
QR
2
8
TV
3
12
Answer Because the ratios are equal, TQRS S TTVW .
Ratios of Side Lengths of TQRS and TXYZ
Shortest sides
Longest sides
Remaining sides
RS
4
2
,
YZ
6
3
QS
10
5
,
XZ
12
6
QR
4
8
XY
5
10
Answer Because the ratios are not equal, TQRS and TXYZ are
not similar .
Checkpoint Complete the following exercise.
1. Which of the three triangles
are similar?
L
20
TKLM and TNPQ
16
12
G
14
22
N
15
Q
K
M
J
18
20
25
P
H
Lesson 8.5 • Geometry Notetaking Guide
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Example 2
Using the SAS Similarity Theorem
Use the given lengths to prove that
TDFR S TMNR.
10
D 2M
Solution
Given: DF 15, MN 12
DM 2, DR 10
R
12
15
Prove: TDFR S TMNR
Paragraph Proof Use the SAS Similarity
Theorem. Begin by finding the ratios of the
lengths of the corresponding sides.
N
F
15
5
DF
12
4
MN
10
5
DR
10
8
4
MR
10 2
So, the lengths of sides DF
*& and DR
*& are proportional to the
lengths of the corresponding sides of TMNR. Because aFDR and
aNMR are right angles , use the SAS Similarity Theorem to
conclude that TDFR S TMNR.
Checkpoint Complete the following exercise.
2. Describe how to prove that TRSV is
similar to TYXV.
X
R
24
15
V
S
20
18
Y
Show that corresponding sides are proportional. Then use
the SAS Similarity Theorem with the vertical angles aRVS
and aYVX.
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Geometry Notetaking Guide • Chapter 8