Suppose we have two line segments, AB and CD, which cross at point X.
Now suppose there is an arbitrary point Y somewhere on the segment AX (that is, points A, Y and X are collinear).
What is the correct way to state that the angles AXD and YXD are in fact the same angle in a formal proof?
Context:
I'm asking this question because of an argument I had with my math teacher:
He was proving some geometric stuff on the blackboard, and he simply stated that AXD = YXD and continued with his proof. But it wasn't "sound" enough to me, so I interrupted him and asked how do we know that these angles are in fact the same. He said that it's "obvious" because this is the same angle. I replied that this is exactly what I am concerned about: what particular theorem proves that? He said that it follows from the fact that A, Y, X are collinear. They sure are, but how does it imply that the angles AXD and YXD are in fact the same angle? What particular theorem from Euclidean geometry is used here?
The teacher simply got annoyed and told me to stop distracting him, so I hadn't got any satisfying answer from him. That's why I'm asking it here.
The mere possibility of defining the concept of an angle is based on the following axiom belonging to the set of axioms of congruence:
In the figure below we have two triangles $ABC$ and $A'B'C'$ and one point $D$ on the straight $AB$ and one point $D'$ on the straight $A'B'$:
Axiom: If $AB=A'B'$, $AC=A'C'$, and $CB=C'B'$ and $BD=B'D'$ then $CD=C'D'$.
It follows that the angle at $A$ is not a property of the triangle $ABC$ but that of the two straights $AC$ and $AB$.
The question is still open: Are the straight lines $AB$ and $AD$ the same?
Again, an axiom of the intermediacy group states that
Axiom: If $AB$ is a straight line and $D$ is on this straight line then the two straight lines $AB$ and $AD$ are the same.
So, by asking your question you scratched the very basics of geometry. The teacher's statement did not follow from any theorem of (absolute) geometry -- it was reference to two of the most basic axioms.