Proving Triangles Similar through
SSS and SAS
CH 6.5
Side Side Side Similarity Theorem
• If the corresponding side lengths of 2 triangles
are proportional, then the triangles are similar
To prove 2 triangles similar using SSS
• In order to prove similarity using SSS, you
must check each possible proportion of the
side lengths of a triangle.
Not similar
Use SSS to find the Scale Factor and
determine whether the triangles are
similar…if they are similar name the
triangles correctly
∆ ABC ~∆DEF
AC 9 3
 
DF 6 2
BC 15 3


EF 10 2
AB 12 3


DE 8 2
Use SSS to find the Scale Factor and
determine whether the triangles are
similar
Not Similar
AB
DE
5

4
AC
DF
8

7
BC
EF
7

6
Assuming that ∆ ABC~ ∆ DEF find x.
Each proportion will equal the scale factor
AB AC BC


DE DF EF
8
4
= 3( x  1)
12
4(3x+3) = 8(12)
12x + 12 = 96
12x = 84 x = 7
Assuming that ∆ XYZ~ ∆ PQR find x.
Each proportion will equal the scale factor
XY XZ YZ


PQ PR QR
20 2

30 3
3(12) = 2(3x -6)
2
12
36 = 6x -12

3 3( x  2) 48 = 6x x = 8
Side Angle Side Similarity Theorem
• If 2 triangles have a corresponding congruent
angle and the sides including that angle are
proportional, then the 2 triangles are similar.
Are the Triangles similar?
How?
yes
SAS
Name the corresponding Side, Angle, and Side
for each triangle
BC 9 3
AC 18 3




ACB  DCE
CE 15 5
CD 30 5
Are the Triangles similar?
How?
yes
SAS
Name the corresponding Side, Angle, and Side
for each triangle
Find the scale factor to back it up
RT
28
4
SR 24 4




R  N
NQ
21
3
PN 18 3
Are the Triangles similar?
How?
yes
SAS
or
SSS
Name the corresponding Side, Angle, and Side and
Side, Side, Side for each triangle.
Find the scale factor to back it up
WX 20 4 WZX  XZY XZ 12 4 WZ 16 4


 
 
XY 15 3
ZY 9 3 XZ 12 3
Find the Scale Factor and determine
whether the triangles are similar using SAS
∆ RST ~ ∆ XYZ
S  Y
RS 4 2
 
XY 6 3
ST 6 2
 
YZ 9 3
Is there enough information to determine whether
the triangles are similar?
no
Why?
The sides are not proportional and it
does not follow SAS.
Is there enough information to determine whether
the triangles are similar?
yes
Which Similarity Postulate
allows us to say yes?
SAS
CD CG C  C

CE CF
15 20

27 36
5 5

9 9
Are the triangles similar? Which similarity
postulate allows us to say it is similar?
yes
SAS
The sides are
proportional and
the included angles
are congruent.
Are the triangles similar? Which similarity
postulate allows us to say it is similar?
yes
SAS
2 sides are proportional
and the included angle
is congruent.
Assuming that these triangles are similar.
Let’s solve for the missing variables.
3x + 8
13y - 38
12
4x - 5
15
6y + 11
Page 391
• #3- 9, 15 - 23