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6.5 – Prove Triangles Similar by SSS and SAS

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6.5 – Prove Triangles Similar by SSS and SAS

Geometry

Ms. Rinaldi

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

=

=

=

=

CA

4

4

8

16

4

AB

12

BC

Is either DEF or GHJsimilar to ABC?

FD

9

DE

3

3

EF

3

12

6

All of the ratios are equal, so ABC~DEF.

ANSWER

Compare ABCand DEFby finding ratios of corresponding side lengths.

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=

EXAMPLE 1

Use the SSS Similarity Theorem

SOLUTION

Remaining sides

Shortest sides

Longest sides

1

=

=

=

=

8

CA

16

BC

12

6

AB

The ratios are not all equal, so ABCand GHJare not similar.

HJ

8

JG

10

GH

5

16

ANSWER

Compare ABCand GHJby finding ratios of corresponding side lengths.

1

=

=

EXAMPLE 1

Use the SSS Similarity Theorem (continued)

Remaining sides

Longest sides

Shortest sides

Which of the three triangles are similar? Write a similarity statement.

Use the SSS Similarity Theorem

EXAMPLE 2

ALGEBRA

Find the value of xthat makes ABC ~ DEF.

4

x–1

4 18 = 12(x – 1)

12

18

STEP1

Find the value of xthat makes corresponding side lengths proportional.

=

EXAMPLE 3

Use the SSS Similarity Theorem

SOLUTION

Write proportion.

Cross Products Property

72 = 12x – 12

Simplify.

7 = x

Solve for x.

?

=

=

ANSWER

AC

6

4

BC

8

4

AB

AB

STEP2

Check that the side lengths are proportional when x = 7.

24

12

12

18

DE

EF

DE

DF

When x = 7, the triangles are similar by the SSS Similarity Theorem.

?

=

=

EXAMPLE 3

Use the SSS Similarity Theorem (continued)

DF = 3(x + 1) = 24

BC = x – 1 = 6

Use the SSS Similarity Theorem

EXAMPLE 4

Find the value of x that makes

Q

Y

20

30

x + 6

21

X

Z

12

P

R

3(x – 2)

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.

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?

EXAMPLE 5

Use the SAS Similarity Theorem

ANSWER

Both m A andm F equal = 53°, so A F. Next, compare the ratios of the lengths of the sides that include A and F.

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

The lengths of the sides that include Aand F are proportional.

15

AB

3

3

9

AC

=

=

FG

2

6

10

2

FH

~

=

=

EXAMPLE 5

Use the SAS Similarity Theorem (continued)

SOLUTION

Shorter sides

Longer sides

18

9

3

CA

3

BC

5

CD

30

5

15

EC

=

=

=

=

The corresponding side lengths are proportional. The included angles ACB and DCEare congruent because they are vertical angles. So, ACB ~DCE by the SAS Similarity Theorem.

EXAMPLE 6

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

Explain how to show that the indicated triangles are similar.

Explain how to show that the indicated triangles are similar.

B. XZW ~ YZX

A. SRT ~ PNQ

Choose a method

EXAMPLE 7