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. Side-Side-Side (SSS) Similarity Theorem. If the corresponding side lengths of two triangles are proportional, then the triangles are similar. =. =. =. =. CA. 4. 4. 8. 16. 4. AB. 12. BC.

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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

6.5 – Prove Triangles Similar by SSS and SAS

Geometry

Ms. Rinaldi


Side side side sss similarity theorem

Side-Side-Side (SSS) Similarity Theorem

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


6 5 prove triangles similar by sss and sas

=

=

=

=

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.

=

=

EXAMPLE 1

Use the SSS Similarity Theorem

SOLUTION

Remaining sides

Shortest sides

Longest sides


6 5 prove triangles similar by sss and sas

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


6 5 prove triangles similar by sss and sas

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

Use the SSS Similarity Theorem

EXAMPLE 2


6 5 prove triangles similar by sss and sas

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.


6 5 prove triangles similar by sss and sas

?

=

=

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


6 5 prove triangles similar by sss and sas

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)


Side angle side sas similarity theorem

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 including these angles are proportional, then the triangles are similar.


6 5 prove triangles similar by sss and sas

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


6 5 prove triangles similar by sss and sas

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


6 5 prove triangles similar by sss and sas

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


6 5 prove triangles similar by sss and sas

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


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