Geometry: An Interactive Journey to Mastery

Lesson 13: A Return to Parallelism

10. Consider the following general triple of parallel
    lines theorem.
    A triple of parallel lines cuts all transversal line
    segments in the same ratioͼ௘6HHFigure 13.17௘ͽ
    3URYHWKLVJHQHUDOWKHRUHPE\¿OOLQJLQWKHGHWDLOVRI
    the following steps.
    First, draw a third transversal and label the lengths s
    and t, as shown in Figure 13.18.
    L 8VHVLPLODUWULDQJOHVWRVKRZWKDWyx sty s.
    LL 3HUIRUPDOJHEUDͼ௘FURVVPXOWLSO\௘ͽWRVKRZWKDW
    this means that xtys.
    LLL 8VHVLPLODUWULDQJOHVWRVKRZWKDWst abta.
    LY 3HUIRUPDOJHEUDͼ௘FURVVPXOWLSO\௘ͽWRVKRZWKDWWKLVPHDQVWKDWtasb.
    v. Steps ii and iv then show that xayb.

b

x

y

a

We have: xy ab

Figure 13.17

b

x

y

a

s

t

Figure 13.18