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### Chapter 8Similarity

USING SIMILARITY THEOREMS

USING SIMILAR TRIANGLESIN REAL LIFE

Section 8.5

Proving Triangles are Similar

USING SIMILARITY THEOREMS

THEOREMS

P

A

AB

PQ

BC

QR

CA

RP

Q

R

If = =

B

C

THEOREM 8.2 Side-Side-Side (SSS) Similarity Theorem

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

then ABC ~ PQR.

Using the SSS Similarity Theorem

CA

FD

AB

DE

BC

EF

12

14

A

E

G

J

C

6

4

6

9

6

10

F

D

8

B

H

12

8

= = ,

6

4

9

6

3

2

3

2

3

2

= = ,

= =

Shortest sides

Longest sides

Remaining sides

Which of the following three triangles are similar?

SOLUTION

To decide which of the triangles are similar, consider the ratios of the lengths of corresponding sides.

Ratios of Side Lengths ofABC andDEF

Because all of the ratios are equal, ABC~ DEF

Using the SSS Similarity Theorem

Which of the three triangles are similar?

USING SIMILARITY THEOREMS

THEOREMS

X

M

P

N

Z

Y

XY

MN

ZX

PM

If XM and=

THEOREM 8.3 Side-Angle-Side (SAS) Similarity Theorem

If an 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.

then XYZ ~ MNP.

Using the SAS Similarity Theorem

SR

SP

SP = 4, PR = 12, SQ = 5, QT = 15

GIVEN

RST ~ PSQ

PROVE

S

4

5

P

Q

12

15

ST

SQ

SQ + QT

SQ

5 + 15

5

20

5

= = = = 4

SP + PR

SP

4 + 12

4

16

4

= = = = 4

R

T

Because S is the included angle in both triangles, use the SAS Similarity Theorem to conclude that RST ~ PSQ.

Use the given lengths to prove thatRST ~ PSQ.

SOLUTION

The side lengths SR and ST are proportional to the corresponding side lengths of PSQ.

USING SIMILARITY THEOREMS

Parallel lines give congruent angles Use AA ~ Postulate

Only one Angle is Known Use SAS ~ Theorem

USING SIMILARITY THEOREMS

No, Need to know the included angle.

HWPg 492-494 6-18 Even, 19-28, 30

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