parallels are parallel to each other

I.30

Straight lines parallel to the same straight line are also parallel to one another.

—Euclid

../../../../_images/elem.1.prop.30.png

Let each of the straight lines AB, CD be parallel to EF; I say that AB is also parallel to CD.

For let the straight line GK fall upon them;

Then, since the straight line GK has fallen on the parallel straight lines AB, EF,

  • the angle AGK is equal to the angle GHF. [I.29]

Again, since the straight line GK has fallen on the parallel straight lines EF, CD,

  • the angle GHF is equal to the angle GKD. [I.29]

But the angle AGK was also proved equal to the angle GHF;

  • therefore the angle AGK is also equal to the angle GKD; [I.c.n.1]

and they are alternate.

Therefore AB is parallel to CD. [1]

references

[I.29]: /elem.1.29 “Book 1 - Proposition 29” [I.c.n.1]: /elem.1.c.n.1 “Book 1 - Common Notion 1”

footnotes