Is the following property of triangles intuitively obvious?

One property that should be obvious is what is called the triangle inequality. Given 3 side lengths to form a triangle, this will be possible provided that the length of each side is less than the sum of the lengths of the two other sides. For any 3 sides a, b and c, we must have a+b > c.

Now suppose we start with two triangles. Imagine that they are constructed of poles. Let’s take apart the two triangles and form a new triangle such that the sides of the new triangle each has one pole from one of the triangles and one pole from the other. The new lengths will always satisfy the triangle inequality and it will always be possible to construct a new triangle.

This is not at all obvious to me, although the proof is immediate. Designate the lengths of one triangle as a, b and c, and the lengths of the other triangle as x, y and z. Take any pairing of the sides and designate it as (a,x) and (b,y), with the third being (c,z). We want to show that (a+x) + (b+y) > (c+z). Like I said, the proof is immediate. (a+x) + (b+y) = (a+b)+(x+y) > c+z

We can easily generalize. Starting with n triangles, form a new triangle by giving each side of the new triangle exactly one side from each of the n. You can always form the new triangle with the resulting side lengths.