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Similar Triangles. Chapter 7-3. Identify similar triangles. Use similar triangles to solve problems. Standards 4.0 Students prove basic theorems involving congruence and similarity . (Key)

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Similar triangles

Similar Triangles

Chapter 7-3


Lesson 3 mi vocab

  • Use similar triangles to solve problems.

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

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

Lesson 3 MI/Vocab



Writing proportionality statements

T

E

C

B

W

Writing Proportionality Statements

34o

Given BTW ~ ETC

  • Write the Statement of Proportionality

  • Find mTEC

  • Find TE and BE

3

20

mTEC = mTBW = 79o

79o

12


Aa similarity theorem

K

Y

J

L

X

Z

AA  Similarity Theorem

  • If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

If K  Y and J  X,

then  JKL ~  XYZ.


Example

N

M

P

Q

R

S

T

Example

  • Are these two triangles similar? Why?


Sss similarity theorem

B

Q

A

C

P

R

SSS  Similarity Theorem

  • If the corresponding sides of two triangles are proportional, then the two triangles are similar.


Which of the following three triangles are similar

E

6

J

4

H

B

D

8

9

6

6

14

A

10

12

C

F

G

ABC~ FDE

SSS ~ Thm

Scale Factor = 3:2

Which of the following three triangles are similar?

ABC and FDE?

Longest Sides

Shortest Sides

Remaining Sides


Which of the following three triangles are similar1

E

6

J

4

H

B

D

8

9

6

6

14

A

10

12

C

F

G

ABC is not similar to DEF

Which of the following three triangles are similar?

ABC and GHJ

Longest Sides

Shortest Sides

Remaining Sides


Sas similarity theorem

K

Y

J

L

X

Z

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.

ass

Pantograph


Prove rts psq

S

4

5

P

Q

15

12

R

T

Prove RTS ~ PSQ

S  S (reflexive prop.)

SPQ  SRT

SAS  ~ Thm.


Are the two triangles similar

N

P

15

12

Q

10

9

R

T

Are the two triangles similar?

NQP  TQR

Not Similar


How far is it across the river

2 yds

5 yds

How far is it across the river?

x yards

42 yds

2x = 210

x = 105 yds


Lesson 3 ex1

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

Are Triangles Similar?

Lesson 3 Ex1


Lesson 3 ex11

by the Alternate Interior Angles Theorem.

Vertical angles are congruent,

Are Triangles Similar?

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

Lesson 3 Ex1


Lesson 3 cyp1

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

A.ΔOBW ~ ΔITW

B.ΔOBW ~ ΔWIT

C.ΔBOW ~ ΔTIW

D.ΔBOW ~ ΔITW

Lesson 3 CYP1


Lesson 3 ex2

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

Parts of Similar Triangles

Lesson 3 Ex2


Lesson 3 ex21

Since because they are alternate interior angles. By AA Similarity, ΔRSQ ~ ΔTUQ. Using the definition of similar polygons,

Parts of Similar Triangles

Substitution

Cross products

Lesson 3 Ex2


Lesson 3 ex22

Parts of Similar Triangles

Distributive Property

Subtract 8x and 30 from each side.

Divide each side by 2.

Now find RQ and QT.

Answer:RQ = 8; QT = 20

Lesson 3 Ex2


Lesson 3 cyp2

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

Lesson 3 CYP2


Lesson 3 cyp21

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

A. 2

B. 4

C. 12

D. 14

Lesson 3 CYP2


Lesson 3 ex3

Indirect Measurement

INDIRECT MEASUREMENTJosh 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?

Lesson 3 Ex3


Lesson 3 ex31

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

Lesson 3 Ex3


Lesson 3 ex32

Indirect Measurement

Simplify.

Divide each side by 2.

Answer: The Sears Tower is 1452 feet tall.

Interactive Lab:Cartography and Similarity

Lesson 3 Ex3


Lesson 3 cyp3

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?

A. 196 ft B. 39 ft

C. 441 ft D. 89 ft

Lesson 3 CYP3


Homework
Homework

Chapter 7-3

  • Pg 400

    7 – 17, 21, 31 – 38


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