Book I · Proposition 27

I.27

If a straight line falling on two straight lines make the alternate angles equal to one another, the straight lines will be parallel to one another.Heath, 1908

Neutral geometry: no appeal to Postulate 5.

Every step, checked

• G lies on AB and H on CDgiven
• the alternate angles AGH and GHD are equalgiven
• were AB and CD to meet, a triangle would arise whose exterior angle equalled an interior and opposite angle, contrary to I.16; so they do not meetby I.16

What it needs, and what needs it

Needs: I.16

Used by: I.28 I.31 I.33

Rests on: C.N.1, C.N.3, C.N.4, C.N.5, Def.10, Def.15, Def.4

Depth: 9 steps of argument above the first principles. Parallel postulate: not needed.

What it takes on trust

Nothing. It draws no intersections and reads nothing off the picture.

The proposition as code

@proposition(
    "I.27",
    THEOREM,
    sample=_transversal,
    note="Neutral geometry: no appeal to Postulate 5.",
)
def prop_I_27(a: Point, b: Point, c: Point, d: Point, g: Point, h: Point) -> Out:
    first, second = line(a, b, "AB"), line(c, d, "CD")
    line(g, h, "the transversal GH")
    hypothesis("G lies on AB and H on CD", on_line(g, first) and on_line(h, second))
    hypothesis("the alternate angles AGH and GHD are equal", eq_angle(a, g, h, g, h, d))

    # The triangle Euclid argues about is the one AB and CD would make if they
    # met. They are parallel, so there is no meeting point to build it from and
    # no figure for I.16 to speak of. Unlike the suppositions of I.14 and I.39,
    # which are drawable, this citation cannot be executed.
    claim("were AB and CD to meet, a triangle would arise whose exterior angle equalled "
          "an interior and opposite angle, contrary to I.16; so they do not meet", "I.16",
          parallel(first, second))
    return Out(first=first, second=second)