(Edited by Sir Thomas L. Heath, 1908)

[Euclid, ed. Heath, 1908, on

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

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.
[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.

Book I: Euclid, Book I (ed. Sir Thomas L. Heath 1st Edition, 1908)

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