WalczakMath2 Name_________________________Chapter 6: Similarity

6.1 Ratios and Proportions

6.2 Similar Polygons

6.1-6.2 Review

6.1-6.2 Quiz

6.3 Proving Triangles Similar

6.5 Proportions in Triangles

Chapter 6 Review

Chapter 6 Test

Essential Questions:

1. How do you use proportions to find lengths in similar polygons? 2. How do you show two triangles are similar? 3. How do you identify corresponding parts of similar triangles?

Important review skills:

o Solve equations o Using parallel lines o Finding angles in triangles

2

6.1 Ratios and Proportions

Ratio:

Example 1: A bonsai tree is 18 in wide and stands 2 ft tall. What is the ratio of the width of the bonsai to its height?

Example 2: Write the ratio of the first measurement to the second measurement:

a) diameter of a table tennis ball: 40 mm b) length of a tennis racket: 2 ft 4 in

diameter of a tennis ball: 6.8 cm length of a table tennis paddle: 10 in

Example 3: The measures of two supplementary angles are in the ratio 1 : 4. What are the angle measures?

Example 4: A baseball team played 154 regular season games. The ratio of the number of games they won to the number of games they lost was 5 : 2. How many games did they win? How many games did they lose?

Example 5: The measures of two supplementary angles are in the ratio 5 : 7. What are the angle measures?

Example 6: The lengths of the sides of a triangles are in the extended ratio 4 : 7 : 9. The perimeter is 60 cm. What are the lengths of the sides?

Example 7: The measure of the angles of a triangle are in the extended ratio 4 : 3 : 2. What is the measure of the largest angle?

Useful Conversions: 12 in = 1 ft 3 ft = 1 yd 5280 ft = 1 mi 16 oz = 1 lb 100 cm = 1 m 10 mm = 1 cm

3

Proportion:

Properties of Proportions:

a cb d= is equivalent to:

Example 8:

If 56

xy= , complete each statement.

a. 6x = _______ b. yx=

c. 5x= d. x y

y+

=

Example 9:

Solve each proportion.

a. 125 7x= b. 3

8 4y y+

= c. 13 2x x+

=

4

53°

C

A D

B

127°G

E H

F

1215

18A C

B

2016

24D F

E

6.2 Similar Polygons Two polygons are similar if:

1)

2)

Similarity Ratio:

Example 1: ABCD~EFGH. Complete each statement. a. m E∠ =

b. m B∠ =

c. ?

AB ADEF

=

d. ?

GH FGCD

=

Example 2: Determine whether the triangles are similar. If they are, write a similarity statement and the similarity ratio.

, ,A F B E C D∠ ≅∠ ∠ ≅∠ ∠ ≅∠

5

Example 3:

Sketch ∆XYZ and ∆MNP with , , .X M Y N Z P∠ ≅∠ ∠ ≅∠ ∠ ≅∠ Also, XY = 12, YZ = 14, ZX = 16, MN = 18, NP = 21, and PM = 24. Can you conclude that the two triangles are similar? Explain.

Example 4:

a. ~LMNO QRST . Find the value of x.

b. Find SR.

Example 5:

Pentagon JKLMN ~ Pentagon PQRST. Find each measure.

a. m∠N = b. m∠Q = c. NM = d. SR =

110°

130°80°

50°

14

10

4

N

M L

K

J

80°

170°

50°

2515

10

T

S R

Q

P

130°

2

3.2

5L M

NO x

6Q R

ST

6

6.1-6.2Review

1.DFHN~BMLP.Findthemissingvaluesbelow.

a.m H∠ =

b. x =

2.Thedoorinaroomis8fttall.Anarchitect’smodelofthesamedooris2in.high.Whatistheratiooftheheightofthemodeltotherealheight?

3.Solveeachproportion.

a.48 22

x= b.

126 2x

x= c.

45 3x x+

=

4.If∆ABC~∆LMO,then:

a.m∠C=____________. b.?

AC BCLO

=

5.A3-in.by5-inpictureisenlargedsothatthelongersidemeasures6ft.Whatisthelengthoftheshorterside?

6.Arethepolygonssimilar?Explainwhyorwhynot.

10.125

3 4.5

6.75

100˚3

6.75

2

4.5

80˚

35

30D F

N H40

x

42 MB

74°81°LP

28

7

31

H

G

I

9.63.2

P

C

K

12

8

8

O

N

M

45°

45°

D

C

E

F

B

6.3 Proving Triangles Similar

3 Ways to Prove Triangles Similar

1. Angle-Angle Similarity (AA~) Postulate

Example 1: Are these triangles similar? If so, write a similarity statement.

2. Side-Angle-Side Similarity (SAS ~) Theorem

Example 2: .C G∠ ≅ ∠ Are the triangles below similar? If so, write a similarity statement.

3. Side-Side-Side Similarity (SSS ~) Theorem

Example 3: Are these triangles similar? If so, write a similarity statement.

66

9K L

J

8

Example 4: Are these triangles similar? If so, write a similarity statement.

Example 5: Are these triangles similar? If so, write a similarity statement.

Example 6: Find the value of x. The two triangles are similar.

Example 7: AB||DC . Find the length of WY.

3

6

P

RQ 4

8

Z

YX

12

18

2416

10

USW

V

T

12

x

8

6

45

10

W

D C

BA

YZ

X

9

6.3Practice

1.Arethefollowingtrianglessimilar?Ifso,bywhatpostulateortheorem?Ifthetrianglesaresimilar,writeasimilaritystatementandsolveforx.Ifnecessary,roundtothenearesttenth.

a)Given:!S ≅!K

b)

c)Given:!R ≅!S

d)Given:TS ! RN

x

S

HL

14

10

K

J M

5

C

A

BE

D

12 15

108

17

x

A

N R

x

9

J

E S

5

6

T S

R

G

Nx°

25

x

10

6.5ProportionsinTriangles

Side-SplitterTheorem:Ifalineisparalleltoonesideofatriangleandintersectstheothertwosides,thenitdividesthosesidesproportionally.

Example1: Example2:

Solveforx. Solvefory.

Corollary:Ifthreeparallellinesintersecttwotransversals,thenthesegmentsinterceptedonthetransversalsareproportional.

Example3:

Solveforxandy.

5

10

x

16

A C

E

BD

6

10

12

y

M

L

J

K

N

x30

2615

y16.5

11

Example4:

ThesegmentsjoiningthesidesoftrapezoidRSTUareparalleltoitsbases.Findxandy.

Triangle-Angle-BisectorTheorem:Ifasegmentbisectsanangleofatriangle,thenitdividestheoppositesideintotwosegmentsthatareproportionaltotheothertwosidesofthetriangle.

Example5: Example6:Findthevalueofx. Findthevalueofy.

12.5

y

5

9

x

6

R S

U T

6

x

8

5

5

8

y

3.6

12

8.55

4

12

17

8

A

B

C

FE

D

Chapter6Review

#1–4:Solveforx.

1.10 5

14x= 2.

38 19x=

3.13 2x x+

= 4.2

1 5xx

=+

5.Findvalueofxandy.Ifnecessary,roundtothenearesttenth.

6.Arethetrianglessimilar?Ifso,bywhatpostulateortheorem?Iftheyaresimilar,writeasimilaritystatement.

7.Acrateis1.5ft.highandcastsa2-ftshadow.Atthesametime,anappletreecastsan18-ftshadow.Howtallisthetree?Roundtothenearesttenth.

8.Ascalemodelofaboatis9in.long.Theboat’sactuallengthis60ft.Findtheratioofthelengthofthescalemodeltothelengthoftheboat.

18

yx

30

10

22

13

40

15

16

x

9.Findthevalueofx.Roundtothenearesttenth.

10.Whena6fttallmancastsashadow18ftlong,anearbytreecastsashadow93ftlong.Howtallisthetree?

11-13:Determinewhetherthefollowingtrianglesaresimilar.Ifso,statethepostulateortheoremthatcanbeusedtoprovetheyaresimilar.Then,writethesimilaritystatement.

11.

Similar?___________________________

Thm/Post__________________________

Sim.Statement______________________

12.

Similar?___________________________

Thm/Post__________________________

Sim.Statement______________________

13.

Similar?___________________________

Thm/Post__________________________

Sim.Statement______________________

E

D F130°20°

130°

C

B

A 30°

J

K

ML

N

6

89

12

A

D

K

F

C

14

14.Solveforx. 15.Solveforx.

16.Solveforx.

17.Solveforxandy.

18.A25ftbuildingcastsashadow40ftlong.Atthesametime,agirlcastsashadow8ftlong.Howtallisthegirl?

Challenge:Thetrianglesaresimilar.Theareaofthesmallertriangleis8in2.Findthevalueofxandy.

x

21

7

96

45°6

445°

x+3

x

4

8

y

x 21

6

x

y 4in

16in

x

6 24

830

39