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6.5 Prove Triangles Similar by SSS and SAS
Goal  Use the SSS and SAS Similarity Theorems.
Your Notes
THEOREM 6.2: SIDE-SIDE-SIDE (SSS) SIMILARITY THEOREM
If the corresponding side lengths of two triangles are _proportional_, then the triangles
are similar.
BC
CA
If AB =
=
then ABC ~ RST.
ST
TR
RS
Example 1
Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
Solution
Compare ABC and DEF by finding ratios of corresponding side lengths.
Shortest sides
AB = 8 = _2_
DE 4
Longest sides
CA = 12 = 3
8
2
FD
Remaining sides
BC = 9 = 3
A
6
2
EF
When using the SSS
Similarity Theorem,
compare the shortest sides,
the longest sides, and then
the remaining sides.
All the ratios are _not equal_ , so ABC and DEF are _not similar_.
Compare ABC and GHJ by finding ratios of corresponding side lengths.
Shortest sides
1
AB 8
= =
GH 16 2
Longest sides
CA 12 1
=
=
JG
24 2
D
Remaining sides
BC = 9 = 1
18 2
A
HJ
All the ratios are _equal_ , so _ABC_ ~ _GHJ_.
Your Notes
Example 2
Use the SSS Similarity Theorem
Find the value of x that makes ABC ~ DEF.
Solution
Step 1
Find the value of x that makes corresponding side lengths
proportional.
4
10
=x  3
20
Write proportion.
4. _20_ = _10_(x  3)
Cross Products Property
_80_ = _10_x  _30_
Simplify.
_11_ = x
Solve for x.
Check that the side lengths are proportional when x = _11_.
DF = 2x + 3 = _25_
AB = x  3 = _8_
Step 2
BC =? AB
DE
EF
8
4
=
20
10
10
BC ? AC 4
=
=
EF
DF 10 25
When x = _11_, the triangles are similar by the
_SSS Similarity Theorem_.
Checkpoint Complete the following exercises.
1. Which of the three triangles are similar?
PQR  ZXY
2. Suppose AB is not given in ABC. What length for AB would make ABC similar
to QRP?
AB = 36
Your Notes
THEOREM 6.3: 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.
If X  M , and
ZX
XY
=
, then XYZ  MNP.
PM MN
Example 3
Use the SAS Similarity Theorem
Birdfeeder You are drawing a design for a birdfeeder. Can you construct the top so it is
similar to the bottom using the angle measure and lengths shown?
Solution
Both m_B_ and m_E_ equal 87°, so _B_  _E_. Next, compare the ratios of the
lengths of the sides that include B and E.
BC 32 8
AB



EF
20 5
DE
The lengths of the sides that include B and E are _proportional_.
So, by the _SAS Similarity Theorem_, ABC ~ DEF. Yes, you can make the top similar
to the bottom.
Checkpoint Complete the following exercise.
3. In Example 3, suppose you use equilateral triangles on the top and bottom. Are the
top and bottom similar? Explain.
Yes, the top and bottom are similar. If the side length of the top is a and the side
a
length of the bottom is b, the ratios of the side lengths are
and the angles are all
b
60°. The triangles are similar by SAS or SSS.
Your Notes
TRIANGLE SIMILARITY POSTULATE AND THEOREMS
AA Similarity Postulate If A  D and B  E, then ABC ~ DEF.
SSS Similarity Theorem If AB = BC = AC , then ABC ~ DEF.
DE EF
DF
AB
AC
SAS Similarity Theorem If A  D and
=
, then ABC  DEF.
DE DF
Example 4
Choose a method
Tell what method you would use to show that the triangles are similar.
Solution
Find the ratios of the lengths of the corresponding sides.
To identify
corresponding
parts, redraw the
triangles so that
the corresponding
parts have the
same orientation.
Shorter sides
QR 9 3
 
RS 12 4
Longer sides PR  18  3
RT 24 4
The corresponding side lengths are _proportional_.
The included angles PRQ and TRS are _congruent_ because they are _vertical_
angles. So, PQR ~ TSR by the _SAS Similarity Theorem_.
Checkpoint Complete the following exercise.
4. Explain how to show JKL ~ LKM.
Show that the corresponding side lengths are proportional, then use the SSS
Similarity Theorem to show JKL ~ LKM.
Homework
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