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
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
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
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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