6.56.5 Prove Triangles Similar by SSS and SASBell Thinger

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

ANSWER 5

3. Solve = .12

6 x – 1 8

6.5

6.5Example 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

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.

Example 1

6.5Example 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.

72 = 12x – 12

7 = x

Cross Products Property

Simplify.

Solve for x.

4 18 = 12(x – 1).

6.5

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.

Example 2

6.5Guided Practice

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

MLN ~ ZYX

ANSWER

6.5

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

Guided Practice

6.5

6.5Example 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

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.

Example 3

6.5Example 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.

6.5Guided 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

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

Guided Practice

6.5CONCEPT SUMMARY

6.5Exit Slip

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

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

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

ANSWER

= =DFAC

DEAB

EFBC

13=

so ABC ~ DEF by the SSS Similarity Theorem.

. The ratios are equal,

6.5

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

ANSWER

= =ABXY

BCYZ

43 and Y B . So XYZ ~ ABC

by the SAS Similarity Theorem.

Exit Slip

6.5

ANSWER 7.5

3. Find the length of BC.

Exit Slip

6.5

4. A tree casts a shadow that is 30 feet long. At the same time a person standing nearby, who is five feet two inches tall, casts a shadow that is 50 inches long. How tall is the tree to the nearest foot?

37 ftANSWER

Exit Slip

6.5

No ANSWERANSWER Yes; XYZ ~ CBA

Exit Slip

6.5

Homework

Pg 409-413#5, 6, 10, 12, 33