In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence.
In the euclidean plane $\mathbb{E}^2$, for example, we define a point as an element of $\mathbb{R}^2$, a line as a set $\{(x, y) \in \mathbb{R}^2 \;|\; ax+by=c\}$ for some $a, b, c \in\mathbb{R}$ with $(a, b) \neq(0, 0)$ and so on, which is all fine.
My question is: what about angles?
Even the most rigorous textbooks I've found treat angles as a natural thing, but to me, it seems something hard to explain formally. How can we actually define an angle? I'm not talking about sines or cosines (in $\mathbb{E}^2$ for example, we could define the cosine of the angle between vectors $u$ and $v$ as $cos(\theta):=\frac{<u,v>}{||u||.||v||}$), but really about THE angle between two lines, the real number $\theta$ itself. I haven't found any convincing definition so far.
Since angles are so important in geometry, I don't understand why they are not presented in Hilbert's axioms, or at least clearly defined somewhere along the way.
Thanks!
Best Answer
First I'll talk about Hilbert axioms.
Angles themselves are not an undefined term. Angle is defined with the notion of line and betweenness. Namely an angle is an unordered pair $\{A,B\}$ where $A$ and $B$ are halflines having the same origin (we also exclude the situation when $A$ and $B$ are contained in the same line) and halfline is a set $\{b:B(oba)\vee a=b\vee B(oab)\}$ for some $a\neq o$ ($o$ is called the origin of the angle).
What is an undefined term is the congruence relation between angles, which tells us which angles are in a sense 'equal', but nothing is said about real numbers assigned to the angles.
However there is a notion of measure which correlates with every angle a positive real number in such way that congruent angles have equal measures and roughly speaking the measure is additive (strict definition requires some other definitons).
The following theorem is true:
Let $\{A,B\}$ be the angle and $x_0>0$. There exists exactly one measure of angles $\psi$ such that $\psi(\{A,B\})=x_0$.
We usually use this theorem with $\{A,B\}$ being the right angle and $x_0=\frac{\pi}{2}$ or $x_0=90$.
The proof is somewhat unconstructive because it doesn't give the formula for the measure (it uses the continuity of the real line). You can find the proof, all lemmas and definitions and generally much more detail in the book K. Borsuk, W. Szmielew, Foundations of geometry. Unfortunately in english edition they changed the system of axioms greatly in comparison with original Hilbert's axioms and for example they removed the congruence relation between angles from undefined terms. However the part about measure of angles stays the same and doesn't depend on axioms that much.
Now let's go to our euclidian model $\mathbb{R}^2$. Once we define the notions of line and betweenness we also have the notion of angle. Next we define the notion of congruence between angles in such way:
The angle $\{A,B\}$ with origin $o$ is congruent to the angle $\{A_1,B_1\}$ with origin $o_1$ if and only if there exist $a\in A, b\in B, a_1\in A_1, b_1\in B_1$ such that $|oa|=|o_1a_1|, |ob|=|o_1b_1|, |ab|=|a_1b_1|$.
Once we prove all axioms we know that all theorems which can be derived from the axioms are true in the model, in particular the theorem about measure of angles.