---
book: 1
number: 33
id: "I.33"
kind: "theorem"
uses: ["[[book-1/proposition-29]]", "[[book-1/proposition-4]]", "[[book-1/proposition-27]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.33"
license: "CC-BY-SA-4.0"
---

# I.33

The straight lines joining equal and parallel straight lines (at the extremities which are) in the same directions (respectively) are themselves also equal and parallel.

![I.33](figures/I-33.svg)

## Proof

Let *AB*, *CD* be equal and parallel, and let the straight lines *AC*, *BD* join them (at the extremities which are) in the same directions (respectively); I say that *AC*, *BD* are also equal and parallel.

Let *BC* be joined.

Then, since *AB* is parallel to *CD*, and *BC* has fallen upon them, the alternate angles *ABC*, *BCD* are equal to one another. [[book-1/proposition-29|I. 29]]

And, since *AB* is equal to *CD*, and *BC* is common, the two sides *AB*, *BC* are equal to the two sides *DC*, *CB*; and the angle *ABC* is equal to the angle *BCD*; therefore the base *AC* is equal to the base *BD*, and the griangle *ABC* is equal to the triangle *DCB*, and the remaining angles will be equal to the remaining angles respectively, namely those which the equal sides subtend; [[book-1/proposition-4|I. 4]] therefore the angle *ACB* is equal to the angle *CBD*.

And, since the straight line *BC* falling on the two straight lines *AC*, *BD* has made the alternate angles equal to one another, *AC* is parallel to *BD*. [[book-1/proposition-27|I. 27]]

And it was also proved equal to it.

Therefore etc.

Q. E. D.