---
book: 1
number: 27
id: "I.27"
kind: "theorem"
uses: ["[[book-1/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.27"
license: "CC-BY-SA-4.0"
---

# 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.

![I.27](figures/I-27.svg)

## Proof

For let the straight line *EF* falling on the two straight lines *AB*, *CD* make the alternate angles *AEF*, *EFD* equal to one another;

I say that *AB* is parallel to *CD*.

For, if not, *AB*, *CD* when produced will meet either in the direction of *B*, *D* or towards *A*, *C*.

Let them be produced and meet, in the direction of *B*, *D*, at *G*.

Then, in the triangle *GEF*, the exterior angle *AEF* is equal to the interior and opposite angle *EFG*: which is impossible. [[book-1/proposition-16|I. 16]]

Therefore *AB*, *CD* when produced will not meet in the direction of *B*, *D*.

Similarly it can be proved that neither will they meet towards *A*, *C*.

But straight lines which do not meet in either direction are parallel; [[book-1/definitions#Definition 23|Def. 23]] therefore *AB* is parallel to *CD*.

Therefore etc.

Q. E. D.