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### Similar Triangles

Chapter 7-3

- 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/VocabLesson 3 TH2

E

C

B

W

Writing Proportionality Statements34o

Given BTW ~ ETC

- Write the Statement of Proportionality
- Find mTEC
- Find TE and BE

3

20

mTEC = mTBW = 79o

79o

12

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.

Q

A

C

P

R

SSS Similarity Theorem- If the corresponding sides of two triangles are proportional, then the two triangles are similar.

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

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

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

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

Are Triangles Similar?

Lesson 3 Ex1by the Alternate Interior Angles Theorem.

Vertical angles are congruent,

Are Triangles Similar?

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

Lesson 3 Ex1In 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 CYP1ALGEBRAGiven , RS = 4, RQ = x + 3, QT= 2x + 10, UT = 10, find RQ and QT.

Parts of Similar Triangles

Lesson 3 Ex2Since 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 Ex2Distributive 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 Ex2A. 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 CYP2B. 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 CYP2INDIRECT 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 Ex3Since 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 Ex3Simplify.

Divide each side by 2.

Answer: The Sears Tower is 1452 feet tall.

Interactive Lab:Cartography and Similarity

Lesson 3 Ex3INDIRECT 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
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