Euclid and angle between two lines
Euclid’s definition of angle:
From Euclid’s Elements, Book 1
A plane angle is the inclination to one another of two lines in a plane which meet one another and
do not lie in a straight line.
And when the lines containing the angle are straight, the angle is called rectilinear.
In the diagram we see that angle A can be taken as the inclination, but we can also see that B can be taken as the angle of inclination.
So, which is it?
If the definition is meaningful then the two angles have to be equal in size, regardless of the lack of a measurement system for angles.
My point is that the theorem about vertical angles (Euclid’s Proposition 15) is redundant, and so there is no need to prove it.
This would save students a lot of time and relieve them of the feeling that proof was pointless. This time could be better spent on proving some less obvious things.
Adding angles is a straightforward manipulative activity, but Euclid also uses subtraction of angles, which is not an obvious thing to carry out, and technically requires an additional postulate. See this:
On the formal approach to subtraction