CK-12 Geometry - Second Edition

7.4. Similarity by SSS and SAS http://www.ck12.org

Solution:Compare two triangles at a time. In the proportions, place the shortest sides over each other, the longest
sides over each other, and the middle sides over each other. Then, determine if the proportions are equal.

4 ABCand 4 DEF:^2015 =^2216 =^2418

Reduce each fraction to see if they are equal.^2015 =^43 ,^2216 =^118 , and^2418 =^43. Because^436 =^118 , 4 ABCand 4 DEFare
notsimilar.

4 DEFand 4 GHI:^1530 =^1633 =^1836

15

30 =

1

2 ,

16

33 =

16

33 ,and

18

36 =

1

2. Because

1

26 =

16
33 ,^4 DEFis not similar to^4 GHI.
4 ABCand 4 GHI:^2030 =^2233 =^2436
20
30 =

2

3 ,

22

33 =

2

3 ,and

24

36 =

2

3. Because all three ratios reduce to

2
3 ,^4 ABC∼4GIH.
Example 2:Algebra ConnectionFindxandy, such that 4 ABC∼4DEF.

Solution:According to the similarity statement, the corresponding sides are: ABDE=BCEF=ACDF. Substituting in what
we know, we have:

9

6

=

4 x− 1

10

=

18

y

9

6

=

4 x− 1

10

9

6

=

18

y

9 ( 10 ) = 6 ( 4 x− 1 ) 9 y= 18 ( 6 )

90 = 24 x− 6 9 y= 108

96 = 24 x y= 12

x= 4

SAS for Similar Triangles

SAS is the last way to show two triangles are similar. If we know that two sides are proportional AND the included
angles are congruent, then the two triangles are similar. For the following investigation, you will need to use
Investigation 4-3, Constructing a Triangle with SAS.

Investigation 7-3: SAS Similarity

Tools Needed: paper, pencil, ruler, protractor, compass

1. Using Investigation 4-3, construct a triangle with sides 6 cm and 4 cm and theincludedangle is 45◦.