Proposition I-23.
On a given straight line and at a point on it to construct a rectilineal angle
equal to a given rectilineal angle.
Proof.
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Now the two triangles have the same base AG and CE, and also equal sides CD and AF, DE and FG.
Therefore the triangles have the same angles.
(Prop I-8)
Thus angle DCE equals angle FAG, as was to be proved.
QED
Let AB be the given straight line, and the angle DCE the given angle.
To construct on AB an angle equal to angle DCE.
On the straight lines making the given angle, let D and E be selected at random.
Let DE be joined.
Now construct on AB a triangle with the same sides as CD, CE,
and DE,
(Prop I-22)
... in such a way as CD is equal to AF, CE to AG, and DE to FG.
QED
QED