Euclid's Elements Book I

23 definitions · 5 postulates · 5 common notions · 48 propositions

Proposition I.23 · Problem

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

Drag the points circled in red. The figure changes, and every step of the proof still holds.
  1. Let AB be the given straight line, A the point on it, and the angle DCE the given rectilineal angle. It is required to construct on the straight line AB at the point A an angle equal to the angle DCE. (Every point can be dragged.)
  2. On the straight lines CD, CE let the points D, E be taken at random, and let DE be joined.
    Postulate 1
    Postulate 1: To draw a straight line from any point to any point.
  3. Out of three straight lines equal to CD, DE, CE let the triangle AFG be constructed, so that CD is equal to AF, CE to AG, and DE to FG.
    Proposition I.22
    Proposition I.22: Out of three straight lines, which are equal to three given straight lines, to construct a triangle: thus it is necessary that two of the straight lines taken together in any manner should be greater than the remaining one.
  4. Since the two sides DC, CE are equal to the two sides FA, AG, and the base DE is equal to the base FG, the angle DCE is equal to the angle FAG. Therefore on the straight line AB at the point A the angle FAG has been constructed equal to the given rectilineal angle DCE.
    Proposition I.8
    Proposition I.8: If two triangles have the two sides equal to two sides respectively, and have also the base equal to the base, they will also have the angles equal which are contained by the equal straight lines.