---
book: 1
number: 23
id: "I.23"
kind: "construction"
uses: ["[[book-1/proposition-8]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:1.prop.23"
license: "CC-BY-SA-4.0"
---

# I.23

On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.

![I.23](figures/I-23.svg)

## Proof

Let *AB* be the given straight line, *A* the point on it, and the angle *DCE* the given rectilineal angle;

thus it is required to construct on the given straight line *AB*, and at the point *A* on it, a rectilineal angle equal to the given rectilineal angle *DCE*.

On the straight lines *CD*, *CE* respectively let the points *D*, *E* be taken at random; let *DE* be joined, and out of three straight lines which are equal to the three straight lines *CD*, *DE*, *CE* let the triangle *AFG* be constructed in such a way that *CD* is equal to *AF*, *CE* to *AG*, and further *DE* to *FG*.

Then, since the two sides *DC*, *CE* are equal to the two sides *FA*, *AG* respectively, and the base *DE* is equal to the base *FG*, the angle *DCE* is equal to the angle *FAG*. [[book-1/proposition-8|I. 8]]

Therefore on the given straight line *AB*, and at the point *A* on it, the rectilineal angle *FAG* has been constructed equal to the given rectilineal angle *DCE*.

Q. E. F.