Geometry: An Interactive Journey to Mastery

The Nature of Parallelism

Lesson 7

Topics
x Parallel lines.
x (XFOLG¶VSDUDOOHOSRVWXODWH
x (UDWRVWKHQHV¶PHDVXUHPHQWRIWKHFLUFXPIHUHQFHRIWKH(DUWK
'H¿QLWLRQV
x alternate interior angles: PQDSDLURIOLQHVDQGDWUDQVYHUVDODSDLURIQRQDGMDFHQWDQJOHVVLWWLQJ
EHWZHHQWKHSDLURIOLQHVDQGSRVLWLRQHGRQRSSRVLWHVLGHVRIDWUDQVYHUVDODUHDOWHUQDWHLQWHULRUDQJOHV
x parallel:7ZROLQHVHDFKLQ¿QLWHLQH[WHQWDUHSDUDOOHOLIWKH\QHYHUPHHW
x same-side interior angles:*LYHQDSDLURIOLQHVDQGDWUDQVYHUVDODSDLURIQRQDGMDFHQWDQJOHVVLWWLQJ
EHWZHHQWKHSDLURIOLQHVDQGSRVLWLRQHGRQWKHVDPHVLGHRIDWUDQVYHUVDODUHVDPHVLGHLQWHULRUDQJOHV
x transversal:$OLQHWKDWFURVVHVDSDLURIOLQHV
Results
,QWKHGLDJUDPVLQFigure 7.1
x ,IWKHWZROLQHVDUHSDUDOOHOWKHQxy ƒ
ͼ௘ZHKDYHVDPHVLGHLQWHULRUDQJOHVVXPPLQJ
WRƒ௘ͽDQGFRQYHUVHO\LIxy ƒ
WKHQWKHWZROLQHVDUHSDUDOOHO
x ,IWKHWZROLQHVDUHSDUDOOHOWKHQa = bͼ௘ZHKDYHFRQJUXHQWDOWHUQDWHLQWHULRUDQJOHV௘ͽDQGFRQYHUVHO\
if a = bWKHQWKHOLQHVDUHSDUDOOHO
Notation
x The notation HJJAB CDG HJJG|| denotes that lines HJJABG and CDHJJGDUHSDUDOOHO
x ,QGLDJUDPVSDUDOOHOOLQHVDUHPDUNHGZLWKDUURZV

y

x a

b

x + y = 180° a = b

Figure 7.1