Proving Triangles are Similar8.5

Proving Triangles are Similar8.5

Chapter 8SimilaritySection 8.5

Proving Triangles are Similar

USING SIMILARITY THEOREMS

USING SIMILAR TRIANGLES IN REAL LIFE

Proving Triangles are Similar8.5

Postulate

A

C

B

D

F

E

A D and C F

ABC ~ DEF

USING SIMILARITY THEOREMS

Proving Triangles are Similar8.5

AA Similarity Postulate

W W

WVX WZY

AA Similarity

Proving Triangles are Similar8.5

AA Similarity Postulate

CAB ~ TQR

USING SIMILARITY THEOREMS

AA Similarity Postulate

WSU ~ VTU

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

THEOREM S

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

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

If = =A BPQ

BCQR

CARP

then ABC ~ PQR.

A

B C

P

Q R

Proving Triangles are Similar8.5

E

F D8

6 4A C

B

12

6 9

G J

H

14

6 10

Using the SSS Similarity Theorem

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

= = , 6 4

AB DE

3 2

Shortest sides

= = , 12 8

CA FD

3 2

Longest sides

= = 9 6

BC EF

3 2

Remaining sides

Because all of the ratios are equal, ABC ~ DEF

Proving Triangles are Similar8.5

= = ,1214

CA JG

67

Longest sides

E

F D8

6 4

Using the SSS Similarity Theorem

A C

B

G J

H

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 GHJ

12 14

6 6 109

= = 1, 6 6

AB GH

Shortest sides

= 910

BC HJ

Remaining sides

Because all of the ratios are not equal, ABC and DEF are not similar.

E

F D8

6 4A C

B

12

6 9

G J

H

14

6 10

Since ABC is similar to DEF and ABC is not similar to GHJ, DEF is not similar to GHJ.

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

THEOREM S

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.

ZXPM

XYMN

If X M and =

X

Z Y

M

P N

Proving Triangles are Similar8.5

Using the SAS Similarity Theorem

Use the given lengths to prove that RST ~ PSQ.

SOLUTION PROVE RST ~ PSQ

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

Paragraph Proof Use the SAS Similarity Theorem. Find the ratios of the lengths of the corresponding sides.

= = = = 4 SR SP

16 4

SP + PR SP

4 + 12 4

= = = = 4 ST SQ

20 5

SQ + QT SQ

5 + 15 5

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

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

12

4 5

15

P Q

S

R T

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

Nothing is known about any corresponding congruent angles

SSS ~ Theorem is the only choice9 6

69

46

69

23

SSS ~ Theorem

ABC ~ XYZ

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

Nothing is known about any corresponding congruent angles

SSS ~ Theorem is the only choice9 6

69

46

69

23

SSS ~ Theorem

ABC ~ XYZ

Only one Angle is Known Use SAS ~ Theorem

6 3

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

Only one Angle is Known Use SAS ~ Theorem

Parallel lines give congruent angles Use

AA ~ Postulate

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

No, Need to know the included angle.

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

No, Need to know the included angle.

40

Yes, AA ~ Postulate

DRM ~ XST

Proving Triangles are Similar8.5

USING SIMILARITY THEOREMS

SSS ~ Theorem

AA ~ Theorem

SAS ~ Theorem

Proving Triangles are Similar8.5

HW Pg :6;9;11;13-17;19-25;27-

29;32-34;39-47