Chapter 8SimilaritySection 8.5

Proving Triangles are Similar

USING SIMILARITY THEOREMS

USING SIMILAR TRIANGLES IN REAL LIFE

Postulate

A

C

B

D

F

E

A D and C F

ABC ~ DEF

USING SIMILARITY THEOREMS

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

Proof of Theorem 8.2

GIVEN

PROVE

= = STMN

RSLM

TRNL

RST ~ LMN

SOLUTION

Paragraph Proof

M

NL

R T

S

P Q

Locate P on RS so that PS = LM.

Draw PQ so that PQ RT.

Then RST ~ PSQ, by the AA Similarity Postulate, and .= = ST SQ

RS PS

TR QP

Use the definition of congruent triangles and the AA Similarity Postulate to conclude that RST ~ LMN.

Because PS = LM, you can substitute in the given proportion and find that SQ = MN and QP = NL. By the SSS Congruence Theorem,

it follows that PSQ LMN.

USING SIMILARITY THEOREMS

Compare Side Lengths of LKM and NOP

18

7

62

3

153

5

Ratios Different, triangles not similar

Determine if the triangles are similar

USING SIMILARITY THEOREMS

Compare Side Lengths of LKM and NOP

18 3

30 5

6 3

10 5

15 3

25 5

Ratios Same, triangles are similar

RQS ~ LKM

Determine if the triangles are similar

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

USING SIMILARITY THEOREMS

CED

44°

68°

20

5

2

USING SIMILARITY THEOREMS

Statements Reasons

USING SIMILARITY THEOREMS

Statements Reasons

~

Finding Distance Indirectly

Similar triangles can be used to find distances that are difficult to measure directly.

ROCK CLIMBING You are at an indoor climbing wall. To estimate the height of the wall, you place a mirror on the floor 85 feet from the base of the wall. Then you walk backward until you can see the top of the wall centered in the mirror. You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.

85 ft6.5 ft

5 ft

A

B

C E

DUse similar triangles to estimate the height of the wall.

Not drawn to scale

Finding Distance Indirectly

85 ft6.5 ft

5 ft

A

B

C E

D

Use similar triangles to estimate the height of the wall.

SOLUTION

Using the fact that ABC and EDC are right triangles, you can apply the AA Similarity Postulate to conclude that these two triangles are similar.

Due to the reflective property of mirrors, you can reason that ACB ECD.

85 ft6.5 ft

5 ft

A

B

C E

D

DE65.38

Finding Distance Indirectly

Use similar triangles to estimate the height of the wall.

SOLUTION

= ECAC

DEBA

Ratios of lengths of corresponding sides are equal.

Substitute.

Multiply each side by 5 and simplify.

DE5

= 856.5

So, the height of the wall is about 65 feet.

Finding Distance Indirectly

6 2

24x 2 144x 72x

The Tree is 72 feet tall

Finding Distance Indirectly

6 2

24x 2 144x 72x

The Tree is 72 feet tall4

x

724 2

72 x 4 144x 36x

The mirror would need to be placed 36 feet from the tree

HW Pg :6;9;11;13-17;19-25;27-29;32-

34;39-47