Similar TrianglesSimilar TrianglesChapter 7-3Chapter 7-3

• Identify similar triangles.

• Use similar triangles to solve problems.

Standards 4.0 Students prove basic theorems involving congruence and similarity. (Key)

Standard 5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.

Lesson 3 TH2

Triangle Similarity is:

Writing Proportionality StatementsGiven BTW ~ ETC

• Write the Statement of Proportionality

• Find mTEC

• Find TE and BE

T

WB

CE

34o

79o

20

12

3

BW

EC

TB

TE

TW

TC

mTEC = mTBW = 79o

TB

TE

BW

EC

2012

3 x 5TE

15520 EB

AA Similarity Theorem• If two angles of one triangle are congruent

to two angles of another triangle, then the two triangles are similar.

K

J L

Y

X Z

If K Y and J X,

then JKL ~ XYZ.

Example• Are these two triangles similar? Why?

N M P

Q

R S T

SSS Similarity Theorem• If the corresponding sides of two triangles

are proportional, then the two triangles are similar.

B

A C

Q

P R

PQR ~ ABCthen

if

RP

CA

QR

BC

PQ

AB

Which of the following three triangles are similar?

AC

B

12

69

F

E

D

6

8

4J

H

G

66

14141010

ABC and FDE?

Longest Sides

Shortest Sides

Remaining Sides

2

3

8

12

FE

AC

2

3

4

6

DE

BC

2

3

6

9

FD

AB

ABC~ FDESSS ~ Thm

Scale Factor = 3:2

Which of the following three triangles are similar?

AC

B

12

69

F

E

D

6

8

4J

H

G

6

1410

ABC and GHJ

Longest Sides

Shortest Sides

Remaining Sides

7

6

14

12

GJ

AC

16

6

HJ

BC

10

9

GH

AB

ABC is not similar to DEF

SAS Similarity Theorem• If one angle of one triangle is congruent to an

angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

K

J L

Y

X Z

ΔXYZ ~ ΔJKLthen

and XJ if

XZ

JL

XY

JK

Pantographass

Prove RTS ~ PSQS S (reflexive prop.) S

TR

QP

5

1512

4

16

4

20

5

SPQ SRT

SAS ~ Thm.

ST

SQ

SR

SP

)16(5)20(4 8080

Are the two triangles similar?

3

4

9

12

QT

NQ

N P

Q

R T

10

1512

9Not

Similar

2

3

10

15

RQ

PQ

NQP TQR

How far is it across the river?

42 yds

2 yds

5 yds

42

2

x

5

x yards

2x = 210

x = 105 yds

In the figure, , and ABC and DCB are right angles. Determine which triangles in the figure are similar.

Are Triangles Similar?

Answer: Therefore, by the AA Similarity Theorem, ΔABE ~ ΔCDE.

Vertical angles are congruent,

by the Alternate Interior

Angles Theorem.

Are Triangles Similar?

A. ΔOBW ~ ΔITW

B. ΔOBW ~ ΔWIT

C. ΔBOW ~ ΔTIW

D. ΔBOW ~ ΔITW

In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5. Determine which triangles in the figure are similar.

Parts of Similar Triangles

ALGEBRA Given , RS = 4, RQ = x + 3, QT = 2x + 10, UT = 10, find RQ and QT.

Parts of Similar Triangles

Since

because they are alternate interior angles. By AA

Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar

polygons,

Substitution

Cross products

Parts of Similar Triangles

Answer: RQ = 8; QT = 20

Distributive Property

Subtract 8x and 30 from each side.

Divide each side by 2.

Now find RQ and QT.

Lesson 3 CYP2

A. 2

B. 4

C. 12

D. 14

A. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find AC.

A. 2

B. 4

C. 12

D. 14

B. ALGEBRA Given AB = 38.5, DE = 11, AC = 3x + 8, and CE = x + 2, find CE.

INDIRECT MEASUREMENT Josh wanted to measure the height of the Sears Tower in Chicago. He used a 12-foot light pole and measured its shadow at 1 P.M. The length of the shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow and it was 242 feet at that time.

What is the height of the Sears Tower?

Indirect Measurement

Since the sun’s rays form similar triangles, the following proportion can be written.

Now substitute the known values and let x be the height of the Sears Tower.

Substitution

Cross products

Indirect Measurement

Answer: The Sears Tower is 1452 feet tall.

Simplify.

Divide each side by 2.

Interactive Lab:Cartography and Similarity

Indirect Measurement

Lesson 3 CYP3

A. 196 ft B. 39 ft

C. 441 ft D. 89 ft

INDIRECT MEASUREMENT On her trip along the East coast, Jennie stops to look at the tallest lighthouse in the U.S. located at Cape Hatteras, North Carolina. At that particular time of day, Jennie measures her shadow to be 1 feet 6 inches in length and the length of the shadow of the lighthouse to be 53 feet 6 inches. Jennie knows that her height is 5 feet 6 inches. What is the height of the Cape Hatteras lighthouse to the nearest foot?

Homework Chapter 7-3

• Pg 400

7 – 17, 21, 31 – 38