Question

Are angles only made of rays? Or are they even made of line segments?

Answer 1

In his Elements (around 300 BCE), Euclid did have angle made of lines. His lines were finite (what we would call line segments) but could always be made longer. (There was no maximum length.)
If two line meet but are not perpendicular, then two angles are formed: one acute and one obtuse. Or, if you count differently, four angles are formed 2 acute (opposite each other) and 2 obtuse (opposite each other and adjacent to the acute angles).
Eventually mathematicians changed the way they talked about such things.
Line segments could be extended because they line on an already infinitely long line. So "line" came to mean "infinitely long line".
And te more easily distinguish the angles formed by intersecting lines, angles were formed from rays. (Infinite in one direction, but sharing a common endpoint.)
We would have to change the way we talk, but it is not clear what mathematicsl would be lost it we changed back to finite lines and angles are formed by (finite) lines sharing a common endpoint.