---
book: 1
number: 31
id: "I.31"
kind: "construction"
uses: ["[[book-1/proposition-23]]", "[[book-1/proposition-27]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.31"
license: "CC-BY-SA-4.0"
---

# I.31

Through a given point to draw a straight line parallel to a given straight line.

![I.31](figures/I-31.svg)

## Proof

Let *A* be the given point, and *BC* the given straight line; thus it is required to draw through the point *A* a straight line parallel to the straight line *BC*.

Let a point *D* be taken at random on *BC*, and let *AD* be joined; on the straight line *DA*, and at the point *A* on it, let the angle *DAE* be constructed equal to the angle *ADC* [[book-1/proposition-23|I. 23]]; and let the straight line *AF* be produced in a straight line with *EA*.

Then, since the straight line *AD* falling on the two straight lines *BC*, *EF* has made the alternate angles *EAD*, *ADC* equal to one another, therefore *EAF* is parallel to *BC*. [[book-1/proposition-27|I. 27]]

Therefore through the given point *A* the straight line *EAF* has been drawn parallel to the given straight line *BC*.

Q. E. F.