1. OBJ: Show that two triangles are similar using the SSS and SAS
7.4
OBJ: Show that two triangles are similar using the SSS and SAS
Similarity Theorems.

2. Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two triangles are proportional, then the triangles are similar.

4. Find the ratios of the corresponding sides. SOLUTION
Example 1
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement and find the scale factor of Triangle B to Triangle A.
Find the ratios of the corresponding sides.
SOLUTION
All three ratios are equal. PR SU 12 6
12 ÷ 6
6 ÷ 6 = 2 1 RQ UT 10 5
10 ÷ 5
5 ÷ 5 QP TS 8 4
8 ÷ 4
4 ÷ 4
By the SSS Similarity Theorem,
PQR ~ STU.
The scale factor of Triangle B to Triangle A is 1/2. 4

5. Is either DEF or GHJ similar to ABC?
Example 2
Use the SSS Similarity Theorem
Is either DEF or GHJ similar to ABC?
SOLUTION
Look at the ratios of corresponding sides in ABC and DEF. 1.
Shortest sides = 6 4 AB DE 3 2
Longest sides 12 8 CA FD
Remaining sides 9 BC EF
ANSWER
Because all of the ratios are equal, ABC ~ DEF. BUT: 5

6. Look at the ratios of corresponding sides in ABC and GHJ. 2.
Example 2
Use the SSS Similarity Theorem
Look at the ratios of corresponding sides in ABC and GHJ. 2.
Shortest sides = 6 AB GH 1
Longest sides 12 14 CA JG 7
Remaining sides BC HJ 9 10
ANSWER
Because the ratios are not equal, ABC and GHJ are not similar. 6

7. Checkpoint
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. 1. 2.

8. Checkpoint
Use the SSS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. 1.
ANSWER
yes; ABC ~ DFE 2.
ANSWER no

9. 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 that include these angles are proportional, then the triangles are similar.

10. The lengths of the sides that include C and F are proportional.
Example 3
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement.
SOLUTION = AC DF 3 5
Shorter sides 6 10 CB FE
Longer sides
The lengths of the sides that include C and F are proportional.
ANSWER
By the SAS Similarity Theorem, ABC ~ DEF. 10

11. V  V by the Reflexive Property of Congruence.
Example 4
Similarity in Overlapping Triangles
Show that VYZ ~ VWX.
SOLUTION
V  V by the Reflexive Property of Congruence.
Shorter sides
4 + 8 4 = VY VW 12 3 1
Longer sides
5 + 10 5 ZV XV 15 11

12. The lengths of the sides that include V are proportional.
Example 4
Continuation….
The lengths of the sides that include V are proportional.
ANSWER
By the SAS Similarity Theorem, VYZ ~ VWX. 12

13. Checkpoint
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning.
ANSWER
No; H  M but 6 8 12 ≠ .

14. so PQR ~ PST by the SAS Similarity Theorem. , 6 3 PS PQ = 2 1 10 5
Checkpoint
Use the SAS Similarity Theorem
Determine whether the triangles are similar. If they are similar, write a similarity statement. Explain your reasoning. 4.
ANSWER
Yes; P  P, and the
so PQR ~ PST by the SAS Similarity Theorem. , 6 3 PS PQ = 2 1 10 5 PT PR ;

15. Review: Determine whether the triangles are similar. If they
are similar, write a similarity statement. 1.
ANSWER
JKL ~ PNM 2.
ANSWER
ABC is not similar to DEF.

16. 3.
Find the value of x.
ANSWER
x = 15

17. Homework
Worksheet 7.4A