Prove Triangles Similar by SSS and SAS
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Prove Triangles Similar by SSS and SAS. Warm Up. Lesson Presentation. Lesson Quiz. 1. ABC : m A = 90º, m B = 44º; DEF : m D = 90º, m E = 46º. similar. ANSWER. 2. ABC : m A = 132º, m B = 24º; DEF : m D = 90º, m F = 24º. not similar. ANSWER. Warm-Up.

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Warm up

Prove Triangles Similar by SSS and SAS

Warm Up

Lesson Presentation

Lesson Quiz


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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

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Determine whether the two triangles are similar.


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x– 1

6

3.Solve = .

8

12

5

ANSWER

Warm-Up


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=

=

=

=

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


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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


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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.


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?

=

=

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.


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ANSWER

MLN ~ZYX

Guided Practice

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


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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.


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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


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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


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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


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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


Warm up

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


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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


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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


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