Geometry: An Interactive Journey to Mastery



Lesson 9: Similarity and Congruence

Results

x 6$6SULQFLSOH ,IWZRWULDQJOHVKDYHDFRPPRQDQJOHDQG

WKHVLGHVDERXWWKDWDQJOHLQHDFKWULDQJOHDUHLQWKHVDPH

UDWLRWKHQWKHWZRWULDQJOHVDUHVLPLODUͼ௘6HHFigure 9.1௘ͽ

This means that

ż DOOWKUHHDQJOHVPDWFK

ż all three sides come in the same ratio.

x $$SULQFLSOH ,IWZRWULDQJOHVKDYHWZRDQJOHVWKDWPDWFK

ͼ௘DQGKHQFHDOOWKUHHDQJOHVPDWFKEHFDXVHWKHUHDUHƒ

LQDWULDQJOH௘ͽWKHQWKH\DUHVLPLODUͼ௘6HHFigure 9.2௘ͽ

This means that all three sides also match in the same ratio.

x LVRVFHOHVWULDQJOHV 7KHWZREDVHDQJOHVRIDQLVRVFHOHVWULDQJOHDUHFRQJUXHQWDQGFRQYHUVHO\LID

WULDQJOHSRVVHVVHVWZRFRQJUXHQWLQWHULRUDQJOHVLWLVLVRVFHOHV

Summary

$WWHPSWVWRFDSWXUHWKHPDWKHPDWLFVRIZKDWLWPHDQVIRUDOLQHWREH³VWUDLJKW ́OHDGVXVWRWZRVLPLODULW\EHOLHIV

DERXWWULDQJOHV,QWKLVOHVVRQZHLQWURGXFHDQGGLVFXVVVLPLODULW\DQGFRQJUXHQFHRISRO\JRQVDQGWKH6$6DQG

$$VLPLODULW\SULQFLSOHVRIWULDQJOHVDQGZHH[SORUHORJLFDOFRQVHTXHQFHVRIWKHVHUHVXOWV

Example 1

The left half of Figure 9.3LVFRQJUXHQWWRWKHULJKWKDOI

)LOOLQWKHEODQNV

D௘ͽ ABCDEF#BBBBBBBBBB

E௘ͽ ‘#C BBBBBBBBBB

F௘ͽ HG#BBBBBBBBBB

G௘ͽ CD BBBBBBBBBB

H௘ͽ AF#BBBBBBBBBB

b

ka

a kb

A

A

Figure 9.1

C

B

A

C

B

A

Figure 9.2

A

J

H I

G

F

E

C D

B

Figure 9.3