SSS Similarity
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
Say Thanks to the Authors
Click http://www.ck12.org/saythanks
(No sign in required)
To access a customizable version of this book, as well as other
interactive content, visit www.ck12.org
CK-12 Foundation is a non-profit organization with a mission to
reduce the cost of textbook materials for the K-12 market both
in the U.S. and worldwide. Using an open-content, web-based
collaborative model termed the FlexBook®, CK-12 intends to
pioneer the generation and distribution of high-quality educational
content that will serve both as core text as well as provide an
adaptive environment for learning, powered through the FlexBook
Platform®.
Copyright © 2012 CK-12 Foundation, www.ck12.org
The names “CK-12” and “CK12” and associated logos and the
terms “FlexBook®” and “FlexBook Platform®” (collectively
“CK-12 Marks”) are trademarks and service marks of CK-12
Foundation and are protected by federal, state, and international
laws.
Any form of reproduction of this book in any format or medium,
in whole or in sections must include the referral attribution link
http://www.ck12.org/saythanks (placed in a visible location) in
addition to the following terms.
Except as otherwise noted, all CK-12 Content (including
CK-12 Curriculum Material) is made available to Users
in accordance with the Creative Commons Attribution/NonCommercial/Share Alike 3.0 Unported (CC BY-NC-SA) License
(http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended
and updated by Creative Commons from time to time (the “CC
License”), which is incorporated herein by this reference.
Complete terms can be found at http://www.ck12.org/terms.
Printed: November 26, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
www.ck12.org
C ONCEPT
Concept 1. SSS Similarity
1
SSS Similarity
Here you’ll learn how to determine if triangles are similar using Side-Side-Side (SSS).
What if you were given a pair of triangles and the side lengths for all three of their sides? How could you use this
information to determine if the two triangles are similar? After completing this Concept, you’ll be able to use the
SSS Similarity Theorem to decide if two triangles are congruent.
Watch This
Watch this video beginning at the 2:09 mark.
MEDIA
Click image to the left for more content.
James Sousa:SimilarTriangles
Now watch the first part of this video.
MEDIA
Click image to the left for more content.
James Sousa:SimilarTriangles UsingSSSandSAS
Guidance
By definition, two triangles are similar if all their corresponding angles are congruent and their corresponding sides
are proportional. It is not necessary to check all angles and sides in order to tell if two triangles are similar. In fact, if
you know only that all sides are proportional, that is enough information to know that the triangles are similar. This
is called the SSS Similarity Theorem.
SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two
triangles are similar.
1
www.ck12.org
If
AB
YZ
=
BC
ZX
=
AC
XY ,
then 4ABC ∼ 4Y ZX.
Example A
Determine if the following triangles are similar. If so, explain why and write the similarity statement.
We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with
the longest sides and work down to the shortest sides.
BC
FD
BA
FE
AC
ED
=
=
=
28
20
21
15
14
10
=
=
=
7
5
7
5
7
5
Since all the ratios are the same, 4ABC ∼ 4EFD by the SSS Similarity Theorem.
Example B
Find x and y, such that 4ABC ∼ 4DEF.
According to the similarity statement, the corresponding sides are:
18
we have 96 = 4x−1
10 = y .
9 4x − 1
=
6
10
9(10) = 6(4x − 1)
AB
DE
=
BC
EF
=
AC
DF .
Substituting in what we know,
9 18
=
6
y
9y = 18(6)
90 = 24x − 6
9y = 108
96 = 24x
y = 12
x=4
Example C
Determine if the following triangles are similar. If so, explain why and write the similarity statement.
2
www.ck12.org
Concept 1. SSS Similarity
We will need to find the ratios for the corresponding sides of the triangles and see if they are all the same. Start with
the longest sides and work down to the shortest sides.
AC
ED
=
21
35
=
3
5
BC
FD
AB
EF
=
15
25
10
20
=
3
5
1
2
=
=
Since the ratios are not all the same, the triangles are not similar.
Guided Practice
Determine if any of the triangles below are similar. Compare two triangles at a time.
1. Is 4ABC ∼ 4DEF?
2. Is 4DEF ∼ 4GHI?
3. Is 4ABC ∼ 4GHI?
Answers:
1. 4ABC and 4DEF: Is
20
15
=
22
16
=
24
18 ?
Reduce each fraction to see if they are equal.
4
3
6=
11
8 , 4ABC
= 21 , 16
33 =
16
33 ,
and
18
36
2
= 32 , 22
33 = 3 , and
15
30
= 12 .
=
1
2
16
33
=
11
8,
and
24
18
= 43 .
18
36 ?
16
33 , 4DEF
22
24
33 = 36 ?
6=
20
30 =
24
2
36 = 3 . All
3. 4ABC and 4GHI: Is
20
30
= 43 , 22
16 =
and 4DEF are not similar.
2. 4DEF and 4GHI: Is
15
30
20
15
is not similar to 4GHI.
three ratios reduce to 23 , 4ABC ∼ 4GIH.
Practice
Fill in the blanks.
1. If all three sides in one triangle are __________________ to the three sides in another, then the two triangles
are similar.
3
www.ck12.org
2. Two triangles are similar if the corresponding sides are _____________.
Use the following diagram for questions 3-5. The diagram is to scale.
3. Are the two triangles similar? Explain your answer.
4. Are the two triangles congruent? Explain your answer.
5. What is the scale factor for the two triangles?
Fill in the blanks in the statements below. Use the diagram to the left.
6. 4ABC ∼ 4_____
BC
AC
7. AB
? = ? = ?
8. If 4ABC had an altitude, AG = 10, what would be the length of altitude DH?
9. Find the perimeter of 4ABC and 4DEF. Find the ratio of the perimeters.
Use the diagram to the right for questions 10-15.
10.
11.
12.
13.
14.
15.
4
4ABC ∼ 4_____
Why are the two triangles similar?
Find ED.
BD
?
DE
? = BC = ?
AD
Is DB
= CE
EB true?
AD
AC
Is DB = DE
true?
www.ck12.org
Concept 1. SSS Similarity
Find the value of the missing variable(s) that makes the two triangles similar.
16.
5