Euclid's Elements Book I: Proposition 23
On a given straight line and at a point on it to construct a rectilineal angle equal to a given rectilineal angle.
In other words, we are asked to create an angle equivalent to another angle when given a straight line and a point.
Prerequisite Knowledge Needed:
- Proposition 8 states that if two triangles have two equal sides and consist of equal bases, then their angles contained inside the triangle are also equal to their respective angle.
- Proposition 22 persists of creating a triangle given three straight lines
- Rectilineal means in relation to straight lines
1) Let AB be a given straight line, and let angle DCE be a given rectilineal angle labeled as alpha.
2) On straight lines CD and CE, let the points D and E be taken at random respectively to CD and CE.
3) Let points D and E be joined to form segment DE.
4) Using Proposition 22, we then create three straight lines equivalent to the three straight lines CD, CE, and DE. Using these three lines, we create triangle AFG in a way that CD is equal to AF, CE is equal to AG, and DE is equal to FG.
5) Since DC and CE are equal to the two sides FA and AG, and the base DE is equal to the base FG, we can use Proposition 8 and conclude that angle DCE, labeled as alpha, is equivalent to the angle FAG, labeled as beta.
6) Therefore, on the given straight line AB with point A on it, the rectilineal angle FAG has been constructed equal to the given rectilineal angle DCE.
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