1. Determine whether the two triangles are similar.
ABC: m A = 90º, m B = 44º; DEF : m D = 90º, m E = 46º.
ANSWER
similar
ABC: m A = 132º, m B = 24º; DEF : m D = 90º, m F = 24º.
ANSWER
not similar
3. Solve = 12 6
x – 1 8
ANSWER 5

2. Show that triangles are similar. Use SSS and SAS Similarity Theorems.
Target
Show that triangles are similar.
You will…
Use SSS and SAS Similarity Theorems. .

3. Vocabulary
SSS Similarity Theorem 6.2 –
If the corresponding sides of two triangles are proportional, then the triangles are similar.
SAS Similarity Theorem 6.3 –
If an angle of one triangle is congruent to the angle of a second triangle and the sides that include the angle are proportional, then the triangles are similar.

4. 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
Longest sides
Remaining sides AB DE 4 3 8 6 = CA FD 4 3 16 12 = BC EF 4 3 12 9 =
All of the ratios are equal, so ABC ~ DEF.
ANSWER

5. EXAMPLE 1
Use the SSS Similarity Theorem
Compare ABC and GHJ by finding ratios of corresponding side lengths.
Shortest sides
Longest sides
Remaining sides CA JG 16 = 1 AB GH 8 = 1 BC HJ 6 5 12 10 =
The ratios are not all equal, so ABC and GHJ are not similar.
ANSWER

6. Use the SSS Similarity Theorem
EXAMPLE 2
Use the SSS Similarity Theorem
ALGEBRA
Find the value of x that makes ABC ~ DEF.
SOLUTION
STEP 1
Find the value of x that makes corresponding side lengths proportional. 4 12 =
x –1 18
Write proportion.
= 12(x – 1)
Cross Products Property
72 = 12x – 12
Simplify.
7 = x
Solve for x.

7. Use the SSS Similarity Theorem
EXAMPLE 2
Use the SSS Similarity Theorem
Check that the side lengths are proportional when x = 7.
STEP 2 AB DE BC EF = ? 6 18 4 12 =
BC = x – 1 = 6 AB DE AC DF = ? 8 24 4 12 =
DF = 3(x + 1) = 24
When x = 7, the triangles are similar by the SSS Similarity Theorem.
ANSWER

8. GUIDED PRACTICE
for Examples 1 and 2
1. Which of the three triangles are similar? Write a similarity statement.
MLN ~ ZYX.
2 : 3
ANSWER

9. GUIDED PRACTICE
for Examples 1 and 2
2. The shortest side of a triangle similar to RST is 12 units long. Find the other side lengths of the triangle.
ANSWER
15, 16.5

10. EXAMPLE 3
Use the SAS Similarity Theorem
Lean-to Shelter
You are building a lean-to shelter starting from a tree branch, as shown. Can you construct the right end so it is similar to the left end using the angle measure and lengths shown?

11. Use the SAS Similarity Theorem
EXAMPLE 3
Use the SAS Similarity Theorem
SOLUTION
Both m A and m F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F. ~
Shorter sides
Longer sides AB FG 3 2 9 6 = AC FH 3 2 15 10 =
The lengths of the sides that include A and F are proportional.

12. EXAMPLE 3
Use the SAS Similarity Theorem
ANSWER
So, by the SAS Similarity Theorem, ABC ~ FGH. Yes, you can make the right end similar to the left end of the shelter.

13. Tell what method you would use to show that the triangles are similar.
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.
Shorter sides
Longer sides BC EC 3 5 9 15 = CA CD 3 5 18 30 =
The corresponding side lengths are proportional. The included angles ACB and DCE are congruent because they are vertical angles. So, ACB ~ DCE by the SAS Similarity Theorem.

14. GUIDED PRACTICE
for Examples 3 and 4
SRT ~ PNQ
Explain how to show that the indicated triangles are similar.
ANSWER
R  N and = = , therefore the
triangles are similar by the SAS Similarity Theorem. SR PN RT NQ 4 3

15. GUIDED PRACTICE
for Examples 3 and 4
XZW ~ YZX
Explain how to show that the indicated triangles are similar. XZ YZ WZ 4 3 = WX XY
WZX  XZY and
therefore the triangles are similar by either SSS
or SAS Similarity Theorems.
ANSWER