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Warm Up

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Prove Triangles Similar by SSS and SAS

Warm Up

Lesson Presentation

Lesson Quiz

1.ABC: mA =90º, mB =44º;DEF:mD= 90º,mE= 46º.

similar

ANSWER

2.ABC: m A = 132º, m B = 24º; DEF : m D = 90º, m F= 24º.

not similar

ANSWER

Warm-Up

Determine whether the two triangles are similar.

x– 1

6

3.Solve = .

8

12

5

ANSWER

Warm-Up

=

=

=

=

8

12

AB

4

BC

CA

4

16

4

Is either DEF or GHJsimilar to ABC?

FD

9

12

3

3

EF

3

6

DE

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

Compare ABCand DEFby finding ratios of corresponding side lengths.

=

=

Example 1

SOLUTION

Shortest sides

Longest sides

Remaining sides

1

=

=

=

=

16

AB

12

8

6

BC

CA

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

JG

10

GH

16

8

HJ

5

Compare ABCand GHJby finding ratios of corresponding side lengths.

1

=

=

Example 1

Shortest sides

Longest sides

Remaining sides

Find the value of xthat makes ABC ~ DEF.

ALGEBRA

4

x–1

4 18 = 12(x – 1)

12

18

STEP1

Find the value of xthat makes corresponding side lengths proportional.

=

Example 2

SOLUTION

Write proportion.

Cross Products Property

72 = 12x – 12

Simplify.

7 = x

Solve for x.

?

=

=

6

4

AB

AC

4

AB

BC

8

STEP2

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

18

12

24

DF

12

DE

DE

EF

?

=

=

Example 2

BC = x – 1 = 6

DF = 3(x + 1) = 24

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

ANSWER

MLN ~ZYX

Guided Practice

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

15, 16.5

ANSWER

Guided Practice

2. The shortest side of a triangle similar toRSTis 12 units long. Find the other side lengths of the triangle.

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 3

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 3

SOLUTION

Shorter sides

Longer sides

9

3

CA

3

18

BC

5

CD

EC

15

30

5

=

=

=

=

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 4

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.

3. SRT ~ PNQ

ANSWER

R Nand == , therefore the

triangles are similar by the SAS Similarity Theorem.

4

SR

RT

3

PN

NQ

Guided Practice

Explain how to show that the indicated triangles are similar.

4. XZW ~ YZX

XZ

WZ

4

WX

XY

XZ

3

YZ

ANSWER

WZX XZYand

=

=

=

therefore the triangles are similar by either SSS

or SAS Similarity Theorems.

Guided Practice

1. Verify that ABC ~ DEF for the given information.

ABC : AC = 6, AB = 9, BC = 12;

DEF : DF = 2, DE= 3, EF = 4

AC

AB

BC

3

EF

1

DF

DE

ANSWER

. The ratios are equal,

=

=

=

so ABC ~ DEF by the SSS Similarity Theorem.

Lesson Quiz

2.Show that the triangles are similar and write a similarity statement. Explain your reasoning.

=

XY

YZ

3

AB

BC

4

ANSWER

=

=

andYB . So XYZ ~ ABC

by the SAS Similarity Theorem.

Lesson Quiz