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U SING S IMILARITY T HEOREMS. U SING S IMILAR T RIANGLES IN R EAL L IFE. Chapter 8 Similarity. Section 8.5 Proving Triangles are Similar. U SING S IMILARITY T HEOREMS. Postulate. E. D. C. F. B. A. A D and C F . ABC ~ DEF. AA Similarity Postulate. W  W

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chapter 8 similarity

USING SIMILARITY THEOREMS

USING SIMILAR TRIANGLESIN REAL LIFE

Chapter 8Similarity

Section 8.5

Proving Triangles are Similar

slide3

USING SIMILARITY THEOREMS

Postulate

E

D

C

F

B

A

A D and C F 

ABC ~ DEF

slide4

AA Similarity Postulate

W  W

WVX  WZY

AA Similarity

slide5

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.

slide6

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 ofABC andDEF

Because all of the ratios are equal, ABC~ DEF

using the sss similarity theorem
Using the SSS Similarity Theorem

Which of the three triangles are similar?

slide8

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.

slide9

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 thatRST ~ PSQ.

SOLUTION

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

slide10

USING SIMILARITY THEOREMS

Parallel lines give congruent angles Use AA ~ Postulate

Only one Angle is Known Use SAS ~ Theorem

slide11

USING SIMILARITY THEOREMS

No, Need to know the included angle.

slide12

USING SIMILARITY THEOREMS

40

No, Need to know the included angle.

Yes, AA ~ Postulate

DRM ~ XST

slide13

USING SIMILARITY THEOREMS

SSS ~ Theorem

AA ~ Theorem

SAS ~ Theorem

checkpoint
Checkpoint

Yes SAS

Yes, SSS

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