Plane geometry.

Question

I was reading proposition 14 of Euclid's elements and there is only one thing which I find weird:

Why do we need postulate 4 to conclude that “the sum of the angles $\angle CBA$ and $\angle ABE$ equals the sum of the angles $\angle CBA$ and $\angle ABD$.”

Why can't we just use common notion 1? It seems useless to me to use the postulate…

Thank you!

Answer 1

This is a massive rewrite of my answer, see history for details.

The text you are referring to seems to be this one here:

[…] therefore the sum of the angles $ABC$ and $ABE$ equals two right angles. But the sum of the angles $ABC$ and $ABD$ also equals two right angles, therefore the sum of the angles $CBA$ and $ABE$ equals the sum of the angles $CBA$ and $ABD$.

The idea is that you need

$$ \angle ABC = \angle ABE = \angle ABC = \angle ABD $$

to conclude

$$ \angle ABC+\angle ABE = \angle ABC+\angle ABD $$

This fact, that all four right angles are indeed equal to one another, is where postulate 4 comes into play.

The way I read it, you'd use common notion 1 to justify the sum equality: if the left hand side is equal to two right angles, and the right hand side is equal to two right angles, then they have to be equal to one another. But for that you need “two right angles” to be equal to one another no matter which two right angles you are talking about, which in turn boils down to the statement that all four single right angles are equal to one another.