6.5

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Prove Triangles Similar by SSS and SAS

6.5 Warm-Up

ANSWER not similar

Determine whether the two triangles are similar.

1. ABC: m A = 90º, m B = 44º; DEF : m D = 90º, m E = 46º.

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

ANSWER similar

6.5 Warm-Up

ANSWER 5

3. Solve = .12

6 x – 1 8

6.5 Example 1

SOLUTION

Compare ABC and DEF by finding ratios of corresponding side lengths.

Shortest sides

ABDE

43

86 ==

Is either DEF or GHJ similar to ABC?

Longest sides

CAFD

43

1612 ==

Remaining sides

BCEF

43

12 9 ==

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

6.5 Example 1

Shortest sides

ABGH

88 == 1

Compare ABC and GHJ by finding ratios of corresponding side lengths.

Longest sides

CAJG

1616 == 1

Remaining sides

BCHJ

65

1210 ==

The ratios are not all equal, so ABC and GHJ are not similar.

6.5 Example 2

SOLUTION

ALGEBRA Find the value of x that makes ABC ~ DEF.

STEP 1 Find the value of x that makes corresponding side lengths proportional.

412 = x –1

18 Write proportion.

4 18 = 12(x – 1)

72 = 12x – 12

7 = x

Cross Products Property

Simplify.

Solve for x.

6.5 Example 2

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

STEP 2

BC = x – 1 = 6

618

412 =AB

DEBCEF=

?

DF = 3(x + 1) = 24

824

412 =

ABDE

ACDF=

?

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

6.5 Guided Practice

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

MLN ~ ZYX

ANSWER

6.5 Guided Practice

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

6.5 Example 3

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?

6.5 Example 3

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.

~

SOLUTION

Shorter sides Longer sidesABFG

32

96 ==

ACFH

32

1510 ==

The lengths of the sides that include A and F are proportional.

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

6.5 Example 4

Tell what method you would use to show that the triangles are similar.

Find the ratios of the lengths of the corresponding sides.

Shorter sides Longer sides

SOLUTION

CACD

35

1830 ==

BCEC

35

915 ==

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.

6.5 Guided Practice

3. 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.

SRPN

RTNQ

4 3

6.5 Guided Practice

4. XZW ~ YZX

Explain how to show that the indicated triangles are similar.

XZYZ

WZXZ

43= WX

XY= = WZX XZY and

therefore the triangles are similar by either SSSor SAS Similarity Theorems.

ANSWER

6.5 Lesson Quiz

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

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

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

ANSWER

ACDF

ABDE

BCEF

31

== =

so ABC ~ DEF by the SSS Similarity Theorem.

. The ratios are equal,

6.5 Lesson Quiz

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

ANSWER

XYAB

YZBC

34= = and Y B . So XYZ ~ ABC =

by the SAS Similarity Theorem.