To construct a rectilinear angle equal to a given rectilinear angle on a given straight line and at a point on it.

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

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

Take the points *D* and *E* at random on the straight lines *CD* and *CE* respectively, and join *DE.* Out of three straight lines which equal the three straight lines *CD, DE,* and *CE* construct the triangle *AFG* in such a way that *CD* equals *AF, CE* equals *AG,* and *DE* equals *FG.*

Since the two sides *DC* and *CE* equal the two sides *FA* and *AG* respectively, and the base *DE* equals the base *FG,* therefore the angle *DCE* equals the angle *FAG.*

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

Q.E.F.

This construction that moves an angle requires a number of steps involving a straightedge and compass. Unless there is some special reason for selecting particular points D and E on the sides of the angle C, they might as well be taken equidistant from C as Apollonius suggested. Then only two distances need to be transferred instead of three.
In order to make |

In all there are ten circles and one line that must be drawn. The lines in the intermediate stages may be suppressed as usual since they’re only needed to verify the construction is correct.

Although it may appear that the triangles are to be in the same plane, that is not necessary. Indeed, the construction in this proposition is used to construct an angle in a different plane in proposition XI.31.