Download 8.5 Proving Triangles are Similar

Proving Triangles are Similar
Using Similarity Theorems
• In this lesson, you will study 2
alternate ways of proving that two
triangles are similar: Side-SideSide Similarity Theorem (SSS) and
the Side-Angle-Side Similarity
Theorem (SAS).
Side Side Side(SSS) Similarity
Theorem
• If the corresponding sides of two
triangles are proportional, then the
triangles are similar.
P
A
Q
B
C
AB
PQ
=
BC
QR
=
CA
RP
R
THEN ∆ABC ~ ∆PQR
Side Angle Side (SAS)Similarity
Thm.
• 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.
M
X
P
N
If X  M and
Z
Y
THEN ∆XYZ ~ ∆MNP
ZX
PM
=
XY
MN
Ex. 1: Proof of SSS Theorem
•Given:
RS
LM
=
•Prove
ST
MN
=
TR
∆RST ~ ∆LMN
NL
Locate P on RS so that PS = LM. Draw PQ so that PQ ║ RT.
Then ∆RST ~ ∆PSQ, by the AA Similarity Postulate, and
RS
LM
=
ST
MN
=
TR
NL
Because PS = LM, you can substitute in the
given proportion and find that SQ = MN and
QP = NL. By the SSS Congruence Theorem,
it follows that ∆PSQ  ∆LMN Finally, use the
definition of congruent triangles and the AA
Similarity Postulate to conclude that ∆RST ~
∆LMN.
Ex. 2: Using the SSS Similarity Thm.
• Which of the three triangles are
similar?
E
12
A
C
9
6
6
F
4
D
8
B
14
G
J
10
6
H
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 ∆ABC and ∆DEF.
AB
DE
=
6
4
=
3
CA
2
FD
=
12
8
=
3
BC
2
EF
=
9
6
=
3
2
Because all of the ratios are equal, ∆ABC ~
∆DEF.
Ratios of Side Lengths of ∆ABC ~
∆GHJ
AB
GH
=
6
6
=1
CA
JG
=
12
14
=
6
BC
7
HJ
=
9
10
Because the ratios are not equal, ∆ABC and
∆GHJ are not similar.
Since ∆ABC is similar to ∆DEF and ∆ABC is not
similar to ∆GHJ, ∆DEF is not similar to ∆GHJ.
Ex. 3: Using the SAS Similarity Thm.
• Use the given lengths to prove that
∆RST ~ ∆PSQ.
S
4
P
12
Given: SP=4, PR = 12, SQ
= 5, and QT = 15;
5
Q
Prove: ∆RST ~ ∆PSQ
Use the SAS Similarity
15
R
SR
SP + PR
4 + 12
16
=
=
=
SP
SP
4
4
T
Theorem. Begin by finding
the ratios of the lengths of
the corresponding sides.
= 4
ST
SQ + QT 5 + 15
20
=
=
=
SQ
SQ
5
5
= 4
So, the side lengths SR and ST are
proportional to the corresponding side lengths
of ∆PSQ. Because S is the included angle in
both triangles, use the SAS Similarity Theorem
to conclude that ∆RST ~ ∆PSQ.