---
book: 1
number: 30
id: "I.30"
kind: "theorem"
uses: ["[[book-1/proposition-29]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.30"
license: "CC-BY-SA-4.0"
---

# I.30

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

![I.30](figures/I-30.svg)

## Proof

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*. [[book-1/proposition-29|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*. [[book-1/proposition-29|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*; [[book-1/common-notions#Common Notion 1|C.N. 1]] and they are alternate.

Therefore *AB* is parallel to *CD*.

Q. E. D.